AIOU Solved Assignments code 805  Autumn & Spring 2024 Assignment 1 & 2  Course: Advanced Microeconomics (805)  Spring 2024. AIOU past papers

ASSIGNMENT No:  1& 2
Advanced Microeconomics (805)    Semester
Autumn & Spring 2024

AIOU Solved Assignment 1& 2 Code 805 Autumn & Spring 2024

Q.No.1Illustrate graphically the general profit maximizing rules for a monopolist ?

Ans:- A perfectly competitive firm acts as a price taker, so we calculate total revenue taking the given market price and multiplying it by the quantity of output that the firm chooses. The demand curve as it is perceived by a perfectly competitive firm appears in (Figure) (a). The flat perceived demand curve means that, from the viewpoint of the perfectly competitive firm, it could sell either a relatively low quantity like Ql or a relatively high quantity like Qh at the market price P.

The Perceived Demand Curve for a Perfect Competitor and a Monopolist

(a) A perfectly competitive firm perceives the demand curve that it faces to be flat. The flat shape means that the firm can sell either a low quantity (Ql) or a high quantity (Qh) at exactly the same price (P). (b) A monopolist perceives the demand curve that it faces to be the same as the market demand curve, which for most goods is downward-sloping. Thus, if the monopolist chooses a high level of output (Qh), it can charge only a relatively low price (PI). Conversely, if the monopolist chooses a low level of output (Ql), it can then charge a higher price (Ph). The challenge for the monopolist is to choose the combination of price and quantity that maximizes profits.

What defines the market?

A monopoly is a firm that sells all or nearly all of the goods and services in a given market. However, what defines the “market”?

In a famous 1947 case, the federal government accused the DuPont company of having a monopoly in the cellophane market, pointing out that DuPont produced 75% of the cellophane in the United States. DuPont countered that even though it had a 75% market share in cellophane, it had less than a 20% share of the “flexible packaging materials,” which includes all other moisture-proof papers, films, and foils. In 1956, after years of legal appeals, the U.S. Supreme Court held that the broader market definition was more appropriate, and it dismissed the case against DuPont.

Questions over how to define the market continue today. True, Microsoft in the 1990s had a dominant share of the software for computer operating systems, but in the total market for all computer software and services, including everything from games to scientific programs, the Microsoft share was only about 14% in 2014. The Greyhound bus company may have a near-monopoly on the market for intercity bus transportation, but it is only a small share of the market for intercity transportation if that market includes private cars, airplanes, and railroad service. DeBeers has a monopoly in diamonds, but it is a much smaller share of the total market for precious gemstones and an even smaller share of the total market for jewelry. A small town in the country may have only one gas station: is this gas station a “monopoly,” or does it compete with gas stations that might be five, 10, or 50 miles away?

In general, if a firm produces a product without close substitutes, then we can consider the firm a monopoly producer in a single market. However, if buyers have a range of similar—even if not identical—options available from other firms, then the firm is not a monopoly. Still, arguments over whether substitutes are close or not close can be controversial.

While a monopolist can charge any price for its product, nonetheless the demand for the firm’s product constrains the price. No monopolist, even one that is thoroughly protected by high barriers to entry, can require consumers to purchase its product. Because the monopolist is the only firm in the market, its demand curve is the same as the market demand curve, which is, unlike that for a perfectly competitive firm, downward-sloping.

(Figure) illustrates this situation. The monopolist can either choose a point like R with a low price (Pl) and high quantity (Qh), or a point like S with a high price (Ph) and a low quantity (Ql), or some intermediate point. Setting the price too high will result in a low quantity sold, and will not bring in much revenue. Conversely, setting the price too low may result in a high quantity sold, but because of the low price, it will not bring in much revenue either. The challenge for the monopolist is to strike a profit-maximizing balance between the price it charges and the quantity that it sells. However, why isn’t the perfectly competitive firm’s demand curve also the market demand curve? See the following Clear It Up feature for the answer to this question.

What is the difference between perceived demand and market demand?

The demand curve as perceived by a perfectly competitive firm is not the overall market demand curve for that product. However, the firm’s demand curve as perceived by a monopoly is the same as the market demand curve. The reason for the difference is that each perfectly competitive firm perceives the demand for its products in a market that includes many other firms. In effect, the demand curve perceived by a perfectly competitive firm is a tiny slice of the entire market demand curve. In contrast, a monopoly perceives demand for its product in a market where the monopoly is the only producer.

Total Cost and Total Revenue for a Monopolist

We can illustrate profits for a monopolist with a graph of total revenues and total costs, with the example of the hypothetical HealthPill firm in (Figure). The total cost curve has its typical shape that we learned about in Production, Costs and Industry Structure, and that we used in Perfect Competition; that is, total costs rise and the curve grows steeper as output increases, as the final column of (Figure) shows.

Total Revenue and Total Cost for the HealthPill Monopoly

Total revenue for the monopoly firm called HealthPill first rises, then falls. Low levels of output bring in relatively little total revenue, because the quantity is low. High levels of output bring in relatively less revenue, because the high quantity pushes down the market price. The total cost curve is upward-sloping. Profits will be highest at the quantity of output where total revenue is most above total cost. The profit-maximizing level of output is not the same as the revenue-maximizing level of output, which should make sense, because profits take costs into account and revenues do not.

Total Costs and Total Revenues of HealthPill
Quantity Q	Price P	Total Revenue TR	Total Cost TC
1	1,200	1,200	500
2	1,100	2,200	750
3	1,000	3,000	1,000
4	900	3,600	1,250
5	800	4,000	1,650
6	700	4,200	2,500
7	600	4,200	4,000
8	500	4,000	6,400

Total revenue, though, is different. Since a monopolist faces a downward sloping demand curve, the only way it can sell more output is by reducing its price. Selling more output raises revenue, but lowering price reduces it. Thus, the shape of total revenue isn’t clear. Let’s explore this using the data in (Figure), which shows quantities along the demand curve and the price at each quantity demanded, and then calculates total revenue by multiplying price times quantity at each level of output. (In this example, we give the output as 1, 2, 3, 4, and so on, for the sake of simplicity. If you prefer a dash of greater realism, you can imagine that the pharmaceutical company measures the pharmaceutical company measures these output levels and the corresponding prices per 1,000 or 10,000 pills.) As the figure illustrates, total revenue for a monopolist has the shape of a hill, first rising, next flattening out, and then falling. In this example, total revenue is highest at a quantity of 6 or 7.

However, the monopolist is not seeking to maximize revenue, but instead to earn the highest possible profit. In the HealthPill example in (Figure), the highest profit will occur at the quantity where total revenue is the farthest above total cost. This looks to be somewhere in the middle of the graph, but where exactly? It is easier to see the profit maximizing level of output by using the marginal approach, to which we turn next.

Marginal Revenue and Marginal Cost for a Monopolist

In the real world, a monopolist often does not have enough information to analyze its entire total revenues or total costs curves. After all, the firm does not know exactly what would happen if it were to alter production dramatically. However, a monopolist often has fairly reliable information about how changing output by small or moderate amounts will affect its marginal revenues and marginal costs, because it has had experience with such changes over time and because modest changes are easier to extrapolate from current experience. A monopolist can use information on marginal revenue and marginal cost to seek out the profit-maximizing combination of quantity and price.

(Figure) expands (Figure) using the figures on total costs and total revenues from the HealthPill example to calculate marginal revenue and marginal cost. This monopoly faces typical upward-sloping marginal cost and downward sloping marginal revenue curves, as (Figure) shows.

Notice that marginal revenue is zero at a quantity of 7, and turns negative at quantities higher than 7. It may seem counterintuitive that marginal revenue could ever be zero or negative: after all, doesn’t an increase in quantity sold not always mean more revenue? For a perfect competitor, each additional unit sold brought a positive marginal revenue, because marginal revenue was equal to the given market price. However, a monopolist can sell a larger quantity and see a decline in total revenue. When a monopolist increases sales by one unit, it gains some marginal revenue from selling that extra unit, but also loses some marginal revenue because it must now sell every other unit at a lower price. As the quantity sold becomes higher, at some point the drop in price is proportionally more than the increase in greater quantity of sales, causing a situation where more sales bring in less revenue. In other words, marginal revenue is negative.

Marginal Revenue and Marginal Cost for the HealthPill Monopoly

For a monopoly like HealthPill, marginal revenue decreases as it sells additional units of output. The marginal cost curve is upward-sloping. The profit-maximizing choice for the monopoly will be to produce at the quantity where marginal revenue is equal to marginal cost: that is, MR = MC. If the monopoly produces a lower quantity, then MR > MC at those levels of output, and the firm can make higher profits by expanding output. If the firm produces at a greater quantity, then MC > MR, and the firm can make higher profits by reducing its quantity of output.

Costs and Revenues of HealthPill
Quantity

QTotal Revenue
TRMarginal Revenue
MRTotal Cost
TCMarginal Cost
MC11,2001,20050050023,2001,00077527533,0008001,00023543,6006001,25025054,0004001,65040064,2002002,50085074,20004,0001,50084,000–2006,4002,400

A monopolist can determine its profit-maximizing price and quantity by analyzing the marginal revenue and marginal costs of producing an extra unit. If the marginal revenue exceeds the marginal cost, then the firm should produce the extra unit.

For example, at an output of 4 in (Figure), marginal revenue is 600 and marginal cost is 250, so producing this unit will clearly add to overall profits. At an output of 5, marginal revenue is 400 and marginal cost is 400, so producing this unit still means overall profits are unchanged. However, expanding output from 5 to 6 would involve a marginal revenue of 200 and a marginal cost of 850, so that sixth unit would actually reduce profits. Thus, the monopoly can tell from the marginal revenue and marginal cost that of the choices in the table, the profit-maximizing level of output is 5.

The monopoly could seek out the profit-maximizing level of output by increasing quantity by a small amount, calculating marginal revenue and marginal cost, and then either increasing output as long as marginal revenue exceeds marginal cost or reducing output if marginal cost exceeds marginal revenue. This process works without any need to calculate total revenue and total cost. Thus, a profit-maximizing monopoly should follow the rule of producing up to the quantity where marginal revenue is equal to marginal cost—that is, MR = MC. This quantity is easy to identify graphically, where MR and MC intersect.

Maximizing Profits

If you find it counterintuitive that producing where marginal revenue equals marginal cost will maximize profits, working through the numbers will help.

Step 1. Remember, we define marginal cost as the change in total cost from producing a small amount of additional output.

MC=change in total costchange in quantity producedMC=change in total costchange in quantity produced

Step 2. Note that in (Figure), as output increases from 1 to 2 units, total cost increases from ?1500 to ?1800. As a result, the marginal cost of the second unit will be:

MC=?775–?5001=?275MC=?775–?5001=?275

Step 3. Remember that, similarly, marginal revenue is the change in total revenue from selling a small amount of additional output.

MR=change in total revenuechange in quantity soldMR=change in total revenuechange in quantity sold

Step 4. Note that in (Figure), as output increases from 1 to 2 units, total revenue increases from ?1200 to ?2300. As a result, the marginal revenue of the second unit will be:

MR=?2300–?12001=?1000MR=?2300–?12001=?1000

Marginal Revenue, Marginal Cost, Marginal and Total Profit

P11,20050070070023,0002757251,42538002355752,00046002503502,350540040002,3506200850−6501,700701,500−1,5002008−2002,400−2,600−2,400

(Figure) repeats the marginal cost and marginal revenue data from (Figure), and adds two more columns: Marginal profit is the profitability of each additional unit sold. We define it as marginal revenue minus marginal cost. Finally, total profit is the sum of marginal profits. As long as marginal profit is positive, producing more output will increase total profits. When marginal profit turns negative, producing more output will decrease total profits. Total profit is maximized where marginal revenue equals marginal cost. In this example, maximum profit occurs at 5 units of output.

A perfectly competitive firm will also find its profit-maximizing level of output where MR = MC. The key difference with a perfectly competitive firm is that in the case of perfect competition, marginal revenue is equal to price (MR = P), while for a monopolist, marginal revenue is not equal to the price, because changes in quantity of output affect the price.

Illustrating Monopoly Profits

It is straightforward to calculate profits of given numbers for total revenue and total cost. However, the size of monopoly profits can also be illustrated graphically with (Figure), which takes the marginal cost and marginal revenue curves from the previous exhibit and adds an average cost curve and the monopolist’s perceived demand curve. (Figure) shows the data for these curves.

Quantity

QDemand
PMarginal Revenue
MRMarginal Cost
MCAverage Cost
AC11,2001,20050050023,1001,00027538831,000800235333490060025031358004004003306700200850417760001,5005718500–2002,400800

Illustrating Profits at the HealthPill Monopoly

This figure begins with the same marginal revenue and marginal cost curves from the HealthPill monopoly from (Figure). It then adds an average cost curve and the demand curve that the monopolist faces. The HealthPill firm first chooses the quantity where MR = MC. In this example, the quantity is 5. The monopolist then decides what price to charge by looking at the demand curve it faces. The large box, with quantity on the horizontal axis and demand (which shows the price) on the vertical axis, shows total revenue for the firm. The lighter-shaded box, which is quantity on the horizontal axis and average cost of production on the vertical axis shows the firm’s total costs. The large total revenue box minus the smaller total cost box leaves the darkly shaded box that shows total profits. Since the price charged is above average cost, the firm is earning positive profits.

(Figure) illustrates the three-step process where a monopolist: selects the profit-maximizing quantity to produce; decides what price to charge; determines total revenue, total cost, and profit.

Step 1: The Monopolist Determines Its Profit-Maximizing Level of Output

The firm can use the points on the demand curve D to calculate total revenue, and then, based on total revenue, calculate its marginal revenue curve. The profit-maximizing quantity will occur where MR = MC—or at the last possible point before marginal costs start exceeding marginal revenue. On (Figure), MR = MC occurs at an output of 5.

Step 2: The Monopolist Decides What Price to Charge

The monopolist will charge what the market is willing to pay. A dotted line drawn straight up from the profit-maximizing quantity to the demand curve shows the profit-maximizing price which, in (Figure), is ?800. This price is above the average cost curve, which shows that the firm is earning profits.

Step 3: Calculate Total Revenue, Total Cost, and Profit

Total revenue is the overall shaded box, where the width of the box is the quantity sold and the height is the price. In (Figure), this is 5 x ?800 = ?4000. In (Figure), the bottom part of the shaded box, which is shaded more lightly, shows total costs; that is, quantity on the horizontal axis multiplied by average cost on the vertical axis or 5 x ?330 = ?1650. The larger box of total revenues minus the smaller box of total costs will equal profits, which the darkly shaded box shows. Using the numbers gives ?4000 – ?1650 = ?2550.
In a perfectly competitive market, the forces of entry would erode this profit in the long run. However, a monopolist is protected by barriers to entry. In fact, one obvious sign of a possible monopoly is when a firm earns profits year after year, while doing more or less the same thing, without ever seeing increased competition eroding those profits.

How a Profit-Maximizing Monopoly Decides Price

In Step 1, the monopoly chooses the profit-maximizing level of output Q₁, by choosing the quantity where MR = MC. In Step 2, the monopoly decides how much to charge for output level Q₁ by drawing a line straight up from Q₁ to point R on its perceived demand curve. Thus,
the monopoly will charge a price (P₁). In Step 3, the monopoly identifies its profit. Total revenue will be Q₁ multiplied by P₁. Total cost will be
Q₁ multiplied by the average cost of producing Q₁, which point S shows on the average cost curve to be P₂. Profits will be the total revenue rectangle minus the total cost rectangle, which the shaded zone in the figure shows.

Why is a monopolist’s marginal revenue always less than the price?

The marginal revenue curve for a monopolist always lies beneath the market demand curve. To understand why, think about increasing the quantity along the demand curve by one unit, so that you take one step down the demand curve to a slightly higher quantity but a slightly lower price. A demand curve is not sequential: It is not that first we sell Q₁ at a higher price, and then we sell Q₂ at a lower price. Rather, a demand curve is conditional: If we charge the higher price, we would sell Q₁. If, instead, we charge a lower price (on all the units that we sell), we would sell Q₂.

When we think about increasing the quantity sold by one unit, marginal revenue is affected in two ways. First, we sell one additional unit at the new market price. Second, all the previous units, which we sold at the higher price, now sell for less. Because of the lower price on all units sold, the marginal revenue of selling a unit is less than the price of that unit—and the marginal revenue curve is below the demand curve. Tip: For a straight-line demand curve, MR and demand have the same vertical intercept. As output increases, marginal revenue decreases twice as fast as demand, so that the horizontal intercept of MR is halfway to the horizontal intercept of demand. You can see this in the (Figure).

The Monopolist’s Marginal Revenue Curve versus Demand Curve

Because the market demand curve is conditional, the marginal revenue curve for a monopolist lies beneath the demand curve.

The Inefficiency of Monopoly

Most people criticize monopolies because they charge too high a price, but what economists object to is that monopolies do not supply enough output to be allocatively efficient. To understand why a monopoly is inefficient, it is useful to compare it with the benchmark model of perfect competition.

Allocative efficiency is an economic concept regarding efficiency at the social or societal level. It refers to producing the optimal quantity of some output, the quantity where the marginal benefit to society of one more unit just equals the marginal cost. The rule of profit maximization in a world of perfect competition was for each firm to produce the quantity of output where P = MC, where the price (P) is a measure of how much buyers value the good and the marginal cost (MC) is a measure of what marginal units cost society to produce. Following this rule assures allocative efficiency. If P > MC, then the marginal benefit to society (as measured by P) is greater than the marginal cost to society of producing additional units, and a greater quantity should be produced. However, in the case of monopoly, price is always greater than marginal cost at the profit-maximizing level of output, as you can see by looking back at (Figure). Thus, consumers will suffer from a monopoly because it will sell a lower quantity in the market, at a higher price, than would have been the case in a perfectly competitive market.

The problem of inefficiency for monopolies often runs even deeper than these issues, and also involves incentives for efficiency over longer periods of time. There are counterbalancing incentives here. On one side, firms may strive for new inventions and new intellectual property because they want to become monopolies and earn high profits—at least for a few years until the competition catches up. In this way, monopolies may come to exist because of competitive pressures on firms. However, once a barrier to entry is in place, a monopoly that does not need to fear competition can just produce the same old products in the same old way—while still ringing up a healthy rate of profit. John Hicks, who won the Nobel Prize for economics in 1972, wrote in 1935: “The best of all monopoly profits is a quiet life.” He did not mean the comment in a complimentary way. He meant that monopolies may bank their profits and slack off on trying to please their customers.

When AT&T provided all of the local and long-distance phone service in the United States, along with manufacturing most of the phone equipment, the payment plans and types of phones did not change much. The old joke was that you could have any color phone you wanted, as long as it was black. However, in 1982, government litigation split up AT&T into a number of local phone companies, a long-distance phone company, and a phone equipment manufacturer. An explosion of innovation followed. Services like call waiting, caller ID, three-way calling, voice mail through the phone company, mobile phones, and wireless connections to the internet all became available. Companies offered a wide range of payment plans, as well. It was no longer true that all phones were black. Instead, phones came in a wide variety of shapes and colors. The end of the telephone monopoly brought lower prices, a greater quantity of services, and also a wave of innovation aimed at attracting and pleasing customers.

The Rest is History

In the opening case, we presented the East India Company and the Confederate States as a monopoly or near monopoly provider of a good. Nearly every American schoolchild knows the result of the “unwelcome visit” the “Mohawks” bestowed upon Boston Harbor’s tea-bearing ships—the Boston Tea Party. Regarding the cotton industry, we also know Great Britain remained neutral during the Civil War, taking neither side during the conflict.

Did the monopoly nature of these business have unintended and historical consequences? Might the American Revolution have been deterred, if the East India Company had sailed the tea-bearing ships back to England? Might the southern states have made different decisions had they not been so confident “King Cotton” would force diplomatic recognition of the Confederate States of America? Of course, it is not possible to definitively answer these questions. We cannot roll back the clock and try a different scenario. We can, however, consider the monopoly nature of these businesses and the roles they played and hypothesize about what might have occurred under different circumstances.

Perhaps if there had been legal free tea trade, the colonists would have seen things differently. There was smuggled Dutch tea in the colonial market. If the colonists had been able to freely purchase Dutch tea, they would have paid lower prices and avoided the tax.

What about the cotton monopoly? With one in five jobs in Great Britain depending on Southern cotton and the Confederate States as nearly the sole provider of that cotton, why did Great Britain remain neutral during the Civil War? At the beginning of the war, Britain simply drew down massive stores of cotton. These stockpiles lasted until near the end of 1862. Why did Britain not recognize the Confederacy at that point? Two reasons: The Emancipation Proclamation and new sources of cotton. Having outlawed slavery throughout the United Kingdom in 1833, it was politically impossible for Great Britain, empty cotton warehouses or not, to recognize, diplomatically, the Confederate States. In addition, during the two years it took to draw down the stockpiles, Britain expanded cotton imports from India, Egypt, and Brazil.

Monopoly sellers often see no threats to their superior marketplace position. In these examples did the power of the monopoly blind the decision makers to other possibilities? Perhaps. As a result of their actions, this is how history unfolded.

Key Concepts and Summary

A monopolist is not a price taker, because when it decides what quantity to produce, it also determines the market price. For a monopolist, total revenue is relatively low at low quantities of output, because it is not selling much. Total revenue is also relatively low at very high quantities of output, because a very high quantity will sell only at a low price. Thus, total revenue for a monopolist will start low, rise, and then decline. The marginal revenue for a monopolist from selling additional units will decline. Each additional unit a monopolist sells will push down the overall market price, and as it sells more units, this lower price applies to increasingly more units.

The monopolist will select the profit-maximizing level of output where MR = MC, and then charge the price for that quantity of output as determined by the market demand curve. If that price is above average cost, the monopolist earns positive profits.

Monopolists are not productively efficient, because they do not produce at the minimum of the average cost curve. Monopolists are not allocatively efficient, because they do not produce at the quantity where P = MC. As a result, monopolists produce less, at a higher average cost, and charge a higher price than would a combination of firms in a perfectly competitive industry. Monopolists also may lack incentives for innovation, because they need not fear entry.

Self-Check Questions

Suppose demand for a monopoly’s product falls so that its profit-maximizing price is below average variable cost. How much output should the firm supply? Hint: Draw the graph.

If price falls below AVC, the firm will not be able to earn enough revenues even to cover its variable costs. In such a case, it will suffer a smaller loss if it shuts down and produces no output. By contrast, if it stayed in operation and produced the level of output where MR = MC, it would lose all of its fixed costs plus some variable costs. If it shuts down, it only loses its fixed costs.

Imagine a monopolist could charge a different price to every customer based on how much he or she were willing to pay. How would this affect monopoly profits?

This scenario is called “perfect price discrimination.” The result would be that the monopolist would produce more output, the same amount in fact as would be produced by a perfectly competitive industry. However, there would be no consumer surplus since each buyer is paying exactly what they think the product is worth. Therefore, the monopolist would be earning the maximum possible profits.

Review Questions

How is the demand curve perceived by a perfectly competitive firm different from the demand curve perceived by a monopolist?

How does the demand curve perceived by a monopolist compare with the market demand curve?

Is a monopolist a price taker? Explain briefly.

What is the usual shape of a total revenue curve for a monopolist? Why?

What is the usual shape of a marginal revenue curve for a monopolist? Why?

How can a monopolist identify the profit-maximizing level of output if it knows its total revenue and total cost curves?

How can a monopolist identify the profit-maximizing level of output if it knows its marginal revenue and marginal costs?

When a monopolist identifies its profit-maximizing quantity of output, how does it decide what price to charge?

Is a monopolist allocatively efficient? Why or why not?

How does the quantity produced and price charged by a monopolist compare to that of a perfectly competitive firm?

Critical Thinking Questions

Imagine that you are managing a small firm and thinking about entering the market of a monopolist. The monopolist is currently charging a high price, and you have calculated that you can make a nice profit charging 10% less than the monopolist. Before you go ahead and challenge the monopolist, what possibility should you consider for how the monopolist might react?

If a monopoly firm is earning profits, how much would you expect these profits to be diminished by entry in the long run?

Problems

Draw the demand curve, marginal revenue, and marginal cost curves from (Figure), and identify the quantity of output the monopoly wishes to supply and the price it will charge. Suppose demand for the monopoly’s product increases dramatically. Draw the new demand curve. What happens to the marginal revenue as a result of the increase in demand? What happens to the marginal cost curve? Identify the new profit-maximizing quantity and price. Does the answer make sense to you?

Draw a monopolist’s demand curve, marginal revenue, and marginal cost curves. Identify the monopolist’s profit-maximizing output level. Now, think about a slightly higher level of output (say Q₀ + 1). According to the graph, is there any consumer willing to pay more than the marginal cost of that new level of output? If so, what does this mean?

AIOU Solved Assignment 1& 2 Code 805 Autumn & Spring 2024

Q.No.2     What is meant by contract curve for exchange and how it is derived ?

Ans:-  In microeconomics, the contract curve is the set of points representing final allocations of two goods between two people that could occur as a result of mutually beneficial trading between those people given their initial allocations of the goods. All the points on this locus are Pareto efficient allocations, meaning that from any one of these points there is no reallocation that could make one of the people more satisfied with his or her allocation without making the other person less satisfied. The contract curve is the subset of the Pareto efficient points that could be reached by trading from the people’s initial holdings of the two goods. It is drawn in the Edgeworth box diagram shown here, in which each person’s allocation is measured vertically for one good and horizontally for the other good from that person’s origin (point of zero allocation of both goods); one person’s origin is the lower left corner of the Edgeworth box, and the other person’s origin is the upper right corner of the box. The people’s initial endowments (starting allocations of the two goods) are represented by a point in the diagram; the two people will trade goods with each other until no further mutually beneficial trades are possible. The set of points that it is conceptually possible for them to stop at are the points on the contract curve. However, some authors^[1] identify the contract curve as the entire Pareto efficient locus from one origin to the other.

Any Walrasian equilibrium lies on the contract curve. As with all points that are Pareto efficient, each point on the contract curve is a point of tangency between an indifference curve of one person and an indifference curve of the other person. Thus, on the contract curve the marginal rate of substitution is the same for both people.

Contents

• 1Example
• 2Mathematical explanation
• 3See also
• 4References

Example[edit]

Assume the existence of an economy with two agents, Octavio and Abby, who consume two goods X and Y of which there are fixed supplies, as illustrated in the above Edgeworth box diagram. Further, assume an initial distribution (endowment) of the goods between Octavio and Abby and let each have normally structured (convex) preferences represented by indifference curves that are convex toward the people’s respective origins. If the initial allocation is not at a point of tangency between an indifference curve of Octavio and one of Abby, then that initial allocation must be at a point where an indifference curve of Octavio crosses one of Abby. These two indifference curves form a lens shape, with the initial allocation at one of the two corners of the lens. Octavio and Abby will choose to make mutually beneficial trades — that is, they will trade to a point that is on a better (farther from the origin) indifference curve for both. Such a point will be in the interior of the lens, and the rate at which one good will be traded for the other will be between the marginal rate of substitution of Octavio and that of Abby. Since the trades will always provide each person with more of one good and less of the other, trading results in movement upward and to the left, or downward and to the right, in the diagram.

The two people will continue to trade so long as each one’s marginal rate of substitution (the absolute value of the slope of the person’s indifference curve at that point) differs from that of the other person at the current allocation (in which case there will be a mutually acceptable trading ratio of one good for the other, between the different marginal rates of substitution). At a point where Octavio’s marginal rate of substitution equals Abby’s marginal rate of substitution, no more mutually beneficial exchange is possible. This point is called a Pareto efficient equilibrium. In the Edgeworth box, it is a point at which Octavio’s indifference curve is tangent to Abby’s indifference curve, and it is inside the lens formed by their initial allocations.

Thus the contract curve, the set of points Octavio and Abby could end up at, is the section of the Pareto efficient locus that is in the interior of the lens formed by the initial allocations. The analysis cannot say which particular point along the contract curve they will end up at — this depends on the two people’s bargaining skills.

In the case of two goods and two individuals, the contract curve can be found as follows. Here {\displaystyle x_{2}^{1}} refers to the final amount of good 2 allocated to person 1, etc., {\displaystyle u^{1}} and {\displaystyle u^{2}} refer to the final levels of utility experienced by person 1 and person 2 respectively, {\displaystyle u_{0}^{2}} refers to the level of utility that person 2 would receive from the initial allocation without trading at all, and {\displaystyle \omega _{1}^{tot}} and {\displaystyle \omega _{2}^{tot}} refer to the fixed total quantities available of goods 1 and 2 respectively.

{\displaystyle \max _{x_{1}^{1},x_{2}^{1},x_{1}^{2},x_{2}^{2}}u^{1}(x_{1}^{1},x_{2}^{1})}

subject to:

{\displaystyle x_{1}^{1}+x_{1}^{2}\leq \omega _{1}^{tot}}

{\displaystyle x_{2}^{1}+x_{2}^{2}\leq \omega _{2}^{tot}}

{\displaystyle u^{2}(x_{1}^{2},x_{2}^{2})\geq u_{0}^{2}}

This optimization problem states that the goods are to be allocated between the two people in such a way that no more than the available amount of each good is allocated to the two people combined, and the first person’s utility is to be as high as possible while making the second person’s utility no lower than at the initial allocation (so the second person would not refuse to trade from the initial allocation to the point found); this formulation of the problem finds a Pareto efficient point on the lens, as far as possible from person 1’s origin. This is the point that would be achieved if person 1 had all the bargaining power. (In fact, in order to create at least a slight incentive for person 2 to agree to trade to the identified point, the point would have to be slightly inside the lens.)

In order to trace out the entire contract curve, the above optimization problem can be modified as follows. Maximize a weighted average of the utilities of persons 1 and 2, with weights b and 1 – b, subject to the constraints that the allocations of each good not exceed its supply and subject to the constraints that both people’s utilities be at least as great as their utilities at the initial endowments:

{\displaystyle \max _{x_{1}^{1},x_{2}^{1},x_{1}^{2},x_{2}^{2}}b\cdot u^{1}(x_{1}^{1},x_{2}^{1})+(1-b)\cdot u^{2}(x_{1}^{2},x_{2}^{2})}

subject to:

{\displaystyle x_{1}^{1}+x_{1}^{2}\leq \omega _{1}^{tot}}

{\displaystyle x_{2}^{1}+x_{2}^{2}\leq \omega _{2}^{tot}}

{\displaystyle u^{1}(x_{1}^{1},x_{2}^{1})\geq u_{0}^{1}}

{\displaystyle u^{2}(x_{1}^{2},x_{2}^{2})\geq u_{0}^{2}}

where {\displaystyle u^{1}} is the utility that person 1 would experience in the absence of trading away from the initial endowment. By varying the weighting parameter b, one can trace out the entire contract curve: If b = 1 the problem is the same as the previous problem, and it identifies an efficient point at one edge of the lens formed by the indifference curves of the initial endowment; if b = 0 all the weight is on person 2’s utility instead of person 1’s, and so the optimization identifies the efficient point on the other edge of the lens. As b varies smoothly between these two extremes, all the in-between points on the contract curve are traced out.

Note that the above optimizations are not ones that the two people would actually engage in, either explicitly or implicitly. Instead, these optimizations are simply a way for the economist to identify points on the contract curve.

AIOU Solved Assignment 1& 2 Code 805 Autumn & Spring 2024

Q.No.3     What is the relationship between partial and general equilibrium analysis ?

Ans:- Partial Equilibrium Analysis • General Equilibrium Analysis • Comparative Statics • Welfare Analysis Advanced Microeconomic Theory 2 Partial Equilibrium Analysis • In a competitive equilibrium (CE), all agents must select an optimal allocation given their resources: – Firms choose profit-maximizing production plans given their technology; – Consumers choose utility-maximizing bundles given their budget constraint. • A competitive equilibrium allocation will emerge at a price that makes consumers’ purchasing plans to coincide with the firms’ production decision. Advanced Microeconomic Theory 3 Partial Equilibrium Analysis • Firm: – Given the price vector firm ’s equilibrium output level % ∗ must solve max )*+, which yields the necessary and sufficient condition ∗ ≤ % 3 (% ∗ ), with equality if % ∗ > 0 – That is, every firm  produces until the point in which its marginal cost, % 3 (% ∗ ), coincides with the current market price. Advanced Microeconomic Theory 4 Partial Equilibrium Analysis • Consumers: – Consider a quasilinear utility function 7 7, 7 = 7 + 7(7) where 7 denotes the numeraire, and �7 3 7 > 0, 7 33 7 < 0 for all 7 > 0. – Normalizing, 7 0 = 0. Recall that with quasilinear utility functions, the wealth effects for all non-numeraire commodities are zero. Advanced Microeconomic Theory 5 Partial Equilibrium Analysis – Consumer ’s UMP is max @A∈ℝD, EA∈ℝD 7 + 7(7) s.t. 7 + ∗7 IJKLM NOPNQR. ≤ @A + ∑ 7%(∗% ∗ − %(% ∗ ) VWJXYKZ ) [ %\] IJKLM WNZJ^W_NZ (NQRJ`aNQKbPWJXYKZ) – The budget constraint must hold with equality (by Walras’ law). Hence, 7 = −∗7 + @A + ∑ 7% ∗% ∗ − %(% ∗ ) [ %\] Advanced Microeconomic Theory 6 Partial Equilibrium Analysis – Substituting the budget constraint into the objective function, max EA∈ℝD 7 7 − ∗7 + @A + ∑ 7% ∗% ∗ − %(% ∗ ) [ %\] – FOCs wrt 7 yields 7 3 7 ∗ ≤ ∗, with equality if 7 ∗ > 0 – That is, consumer increases the amount he buys of good  until the point in which the marginal utility he obtains exactly coincides with the market price he has to pay for it. Advanced Microeconomic Theory 7 Partial Equilibrium Analysis – Hence, an allocation (] ∗ , c ∗ , … , e ∗ , ] ∗ , c ∗ , … , [ ∗) and a price vector ∗ ∈ ℝf constitute a CE if: ∗ ≤ % 3 (% ∗ ), with equality if % ∗ > 0 7 3 7 ∗ ≤ ∗ , with equality if 7 ∗ > 0 ∑ 7 e ∗ 7\] = ∑ % [ ∗ %\] – Note that the these conditions do not depend upon the consumer’s initial endowment. Advanced Microeconomic Theory 8 Partial Equilibrium Analysis • The individual demand curve, where 7 3 7 ∗ ≤ ∗ Advanced Microeconomic Theory 9 Partial Equilibrium Analysis • Horizontally summing individual demand curves yields the aggregate demand curve. Advanced Microeconomic Theory 10 Partial Equilibrium Analysis • The individual supply curve, where ∗ ≤ % 3 (% ∗ ) Advanced Microeconomic Theory 11 Partial Equilibrium Analysis Advanced Microeconomic Theory 12 • Horizontally summing individual supply curves yields the aggregate supply curve. Partial Equilibrium Analysis • Superimposing aggregate demand and aggregate supply curves, we obtain the CE allocation of good  • To guarantee that a CE exists, the equilibrium price ∗ must satisfy max 7 7 3 0 ≥ ∗ ≥ min % % 3 0 Advanced Microeconomic Theory 13 Partial Equilibrium Analysis • Also, since 7 3 7 is downward sloping in 7, and % 3 (7) is upward sloping in 7, then aggregate demand and supply cross at a unique point. – Hence, the CE allocation is unique. Advanced Microeconomic Theory 14 Partial Equilibrium Analysis • If we have max 7 7 3 0 < min % % 3 0 , then there is no positive production or consumption of good . Advanced Microeconomic Theory 15 Partial Equilibrium Analysis • Example 6.1: – Assume a perfectly competitive industry consisting of two types of firms: 100 firms of type A and 30 firms of type B. – Short-run supply curve of type A firm is k  = 2 – Short-run supply curve of type B firm is m  = 10 – The Walrasian market demand curve is  = 5000 − 500 Advanced Microeconomic Theory 16 Partial Equilibrium Analysis • Example 6.1 (continued): – Summing the individual supply curves of the 100 type-A firms and the 30 type-B firms,   = 100 q 2+ 30 q 10 = 500 – The short-run equilibrium occurs at the price at which quantity demanded equals quantity supplied, 5000 − 500 = 500, or  = 5 – Each type-A firm supplies: k  = 2 q 5 = 10 – Each type-B firm supplies: m  = 10 q 5 = 50

AIOU Solved Assignment 1& 2 Code 805 Autumn & Spring 2024

Q.No.4     Contrast the general equilibrium in exchanges with general equilibrium in production.

Ans:- “From the time of Adam Smith’s Wealth of Nations in 1776, one recurrent theme of economic analysis has been the remarkable degree of coherence among the vast numbers of individual and seemingly separate decisions about the buying and selling of commodities. In everyday, normal experience, there is something of a balance between the amounts of goods and services that some individuals want to supply and the amounts that other, differerent individuals want to sell [sic]. Would-be buyers ordinarily count correctly on being able to carry out their intentions, and would-be sellers do not ordinarily find themselves producing great amounts of goods that they cannot sell. This experience of balance is indeed so widespread that it raises no intellectual disquiet among laymen; they take it so much for granted that they are not disposed to understand the mechanism by which it occurs.” Kenneth Arrow (1973) 1 Introduction General equilibrium analysis addresses precisely how these “vast numbers of individual and seemingly separate decisions” referred to by Arrow aggregate in a way that coordinates productive effort, balances supply and demand, and leads to an efficient allocation of goods and services in the economy. The answer economists have provided, beginning with Adam Smith and continuing through to Jevons and ∗Various sections of these notes draw heavily on lecture notes written by Felix Kubler; some of the other sections draw on Mas-Colell, Whinston and Green. 1 Walras is that it is the price system plays the crucial coordinating and equilibrating role: the fact the everyone in the economy faces the same prices is what generates the common information needed to coordinate disparate individual decisions. You doubtless are familiar with the standard treatment of equilibrium in a single market. Price plays the role of equilibrating demand and supply so that all buyers who want to buy at the going price can, and do, and similarly all sellers who want to sell at the going price also can and do, with no excess or shortages on either side. The extension from this partial equilibrium in a single market to general equilibrium reflects the idea that it may not be legitimate to speak of equilibrium with respect to a single commodity when supply and demand in that market depend on the prices of other goods. On this view, a coherent theory of the price system and the coordination of economic activity has to consider the simultaneous general equilibrium of all markets in the economy. This of course raises the questions of (i) whether such a general equilibrium exists; and (ii) what are its properties. A recurring theme in general equilibrium analysis, and economic theory more generally, has been the idea that the competitive price mechanism leads to outcomes that are efficient in a way that outcomes under other systems such as planned economies are not. The relevant notion of efficiency was formalized and tied to competitive equilibrium by Vilfredo Pareto (1909) and Abram Bergson (1938). This line of inquiry culminates in the Welfare Theorems of Arrow (1951) and Debreu (1951). These theorems state that there is in essence an equivalence between Pareto efficient outcomes and competitive price equilibria. Our goal in the next few lectures is to do some small justice to the main ideas of general equilibrium. We’ll start with the basic concepts and definitions, the welfare theorems, and the efficiency properties of equilibrium. We’ll then provide a proof that a general equilibrium exists under certain conditions. From there, we’ll investigate a few important ideas about general equilibrium: whether equilibrium is unique, how prices might adjust to their equilibrium levels and whether these levels are stable, and the extent to which equilibria can be characterized and changes in exogenous preferences or endowments will have predictable consequences. Finally we’ll discuss how one can incorporate production into the model and then time 2 and uncertainty, leading to a brief discussion of financial markets. 2 The Walrasian Model We’re going to focus initially on a pure exchange economy. An exchange economy is an economy without production. There are a finite number of agents and a finite number of commodities. Each agent is endowed with a bundle of commodities. Shortly the world will end and everyone will consume their commodities, but before this happens there will be an opportunity for trade at some set prices. We want to know whether there exist prices such that when everyone tries to trade their desired amounts at these prices, demand will just equal supply, and also what the resulting outcome will look like – whether it will be efficient in a well-defined sense and how it will depend on preferences and endowments. 2.1 The Model Consider an economy with I agents i ∈ I = {1, …, I} and L commodities l ∈ L = {1, …, L}. A bundle of commodities is a vector x ∈ RL +. Each agent i has an endowment ei ∈ RL + and a utility function ui : RL + → R. These endowments and utilities are the primitives of the exchange economy, so we write E = ((ui , ei )i∈I). Agents are assumed to take as given the market prices for the goods. We won’t have much to say about where these prices come from, although we’ll say a bit later on. The vector of market prices is p ∈ RL +; all prices are nonnegative. Each agent chooses consumption to maximize her utility given her budget constraint. Therefore, agent i solves: max x∈RL + ui (x) s.t. p · x ≤ p · ei . The budget constraint is slightly different than in standard price theory. Recall that the familiar budget constraint is p · x ≤ w, where w is the consumer’s initial wealth. Here the consumer’s “wealth” is p · ei , the amount she could get if she sold 3 her entire endowment. We can write the budget set as Bi (p) = {x : p · x ≤ p · ei }. We’ll occasionally use this notation below. 2.2 Walrasian Equilibrium We now define a Walrasian equilibrium for the exchange economy. A Walrasian equilibrium is a vector of prices, and a consumption bundle for each agent, such that (i) every agent’s consumption maximizes her utility given prices, and (ii) markets clear: the total demand for each commodity just equals the aggregate endowment. Definition 1 A Walrasian equilibrium for the economy E is a vector (p,(xi )i∈I) such that: 1. Agents are maximizing their utilities: for all i ∈ I, xi ∈ arg max x∈Bi(p) ui (x) 2. Markets clear: for all l ∈ L, X i∈I xi l = X i∈I ei l. 2.3 Pareto Optimality The second important idea is the notion of Pareto optimality, due to the Italian economist Vilfredo Pareto. This notion doesn’t have anything to do with equilibrium per se (although we’ll see the close connection soon). Rather it considers the set of feasible allocations and identifies those allocations at which no consumer could be made better off without another being made worse off. Definition 2 An allocation (xi )i∈I ∈ RI·L + is feasible if for all l ∈ L: P i∈I xi l ≤ P i∈I ei l. 4 Definition 3 Given an economy E, a feasible allocation x is Pareto optimal (or Pareto efficient) if there is no other feasible allocation xˆ such that ui (ˆxi ) ≥ ui (xi ) for all i ∈ I with strict inequality for some i ∈ I. You should note that Pareto efficiency, while it has significant content, says essentially nothing about distributional justice or equity. For instance, it can be Pareto efficient for one guy to have everything and everyone else have nothing. Pareto efficiency just says that there aren’t any “win-win” changes around; it’s quiet on how social trade-offs should be resolved. 2.4 Assumptions As we go along, we’re going to repeatedly invoke a bunch of assumptions about consumers’ preferences and endowments. We summarize the main ones here. (A1) For all agents i ∈ I, ui is continuous. (A2) For all agents i ∈ I, ui is increasing, i.e. ui (x0 ) > ui (x) whenever x0 À x. (A3) For all agents i ∈ I, ui is concave. (A4) For all agents i ∈ I, ei À 0. The first three assumptions – continuity, monotonicity and concavity of the utility function – should be familiar from consumer theory. Some of these are a bit stronger than necessary (e.g. monotonicity can be weakened to local nonsatiation, concavity to quasi-concavity), but we’re not aiming for maximum generality. The last assumption, about endowments, is new and is a big one. It says that everyone has a little bit of everything. This turns out to be important and you’ll see where it comes into play later on. 3 A Graphical Example General equilibrium theory can quickly get into the higher realms of mathematical economics. Nevertheless a lot of the big ideas can be expressed in a simple 5 two-person two-good exchange economy. A useful graphical way to study such economies is the Edgeworth box, after F. Edgeworth, a famous Cambridge (U.K.) economist of the 19th century.1 Figure 1(a) presents an Edgeworth box. The bottom left corner is the origin for agent 1. The bottom line is the x-axis for Agent 1 and the left side is the y-axis. In the picture, agent 1’s endowment is e1 = (e1 1, e1 2). For agent 2, the origin is the top right corner and everything is flipped upside down and backward. Every point in the box represents a (non-wasteful) allocation of the two goods. x1 1 x2 1 Agent 1 Agent 2 e e1 1 e2 2 e1 2 e2 1 0 0 Agent 1 Agent 2 e1 1 B1(p) B2(p) x1(p,p•e1) x2 2 x1 2 e Figures 1(a) and 1(b): The Edgeworth Box Figure 1(b) adds prices into the picture. Given prices p1, p2 for the two goods, the budget line for agent 1 is the line with slope p1/p2 through the endowment point e. This is also the budget line for agent 2. So this line divides the Edgeworth box into the two budget sets B1(p) and B2(p). Each agent will then choose consumption to maximize utility given prices. In Figure 1(b), agent 1’s Marshallian demand x1(p, p · e) is represented by the familiar tangency condition. 1Apparently the name is something of a misnomer, as it seems that Edgeworth boxes were first drawn by Pareto – or so I read on the internet. 6 As we change prices, the Marshallian demands of the two agents will also change. Note that what matters, of course, is the relative prices of the two goods, as these determine the slope of the budget line. Figure 2 traces out the Marshallian demand of agent 1 as we vary the relative prices. The dotted line is called agent 1’s offer curve. Agent 1 Agent 2 OC1 e Figure 2: Offer Curve for Agent 1 Walrasian equilibrium requires that both agents consume their Marshallian demands given prices and also that these demands are compatible. So what we want to do is set relative prices, find the Marshallian demands of the two agents, and see whether or not demand equals supply in the two markets. Figure 3(a) represents a situation where prices do not simultaneously clear the two markets. In this picture, at the given prices, agent 2 is willing to supply some amount of good 2, but less than agent 1 wants to consume. So good 2 is in excess demand. In contrast, agent 1 is willing to supply more of good 1 than agent 2 demands. So good 2 is in excess supply. In Figure 3(b), prices do clear the market and we have a Walrasian equilibrium at the point x. In equilibrium, starting from the endowment point e, agent 1 7 sells good 1 to buy good 2; agent 2 does the reverse. The crucial point is that both markets clear. Note that the Walrasian equilibrium allocation is the intersection of the two offer curves. That the point x lies on the offer curve of agent i means that x it represents the Marshallian demand of that agent given prices p and endowment e. That the point x is the intersection of the two offer curves means that at the given prices, demands are compatible and markets clear. These are conditions (1) and (2) in the definition of Walrasian equilibrium. Agent 1 Agent 2 OC1 e Agent 1 Agent 2 OC1 e x1(p,p•e1) x2(p,p•e2) OC2 Figures 3(a) and 3(b): Dis-equilibrium and Equilibrium in the Edgeworth Box Two natural questions to ask about Walrasian equilibrium are (i) is it unique? and (ii) does it always exist? Both questions have negative answers. Figure 4(a) presents an example with multiple Walrasian equilibria (we’re revisit this example later). In the figure, given the endowment e, the offers curves of the two agents intersect three times. So there are three Walrasian equilibria. 8 Agent 1 Agent 2 e Agent 1 Agent 2 OC1 OC2 e Direction of increasing preference for agent 2 Direction of increasing preference for agent 1 Figures 4(a) and (b): Non-uniqueness and Non-existence of Equilibrium Figure 4(b) presents a different example where Walrasian equilibrium does not exist. In this example, Agent 2 starts with all of good 1 and this is the only good she cares about. Agent 1 starts with all of good 2 and none of good 1. He cares about both goods, but the slope of his indifference curve when he has none of good 1 is infinite. That is, he has infinite marginal utility for his very first unit of good 1. In this example, for any prices p, agent 2 will insist on consuming her endowment – that is, all of good 1. Moreover, there are no prices p at which agent 1 would not insist on buying at least a little bit of good 1. Therefore for any prices p good 1 will always be in excess demand and there cannot be a Walrasian equilibrium. Note that this example violates assumption (A4), which requires that the endowment be an interior point in the Edgeworth box. It is also possible to use the Edgeworth box to depict the idea of Pareto optimality. This is done in Figure 5. The Pareto set in this picture is the set of all allocations such that to make one agent better off would require making the other agent worse off. Figure 5 also shows the contract curve. This is the part of the Pareto set that both agents prefer to the endowment e. It seems natural to expect that if the agents were to start at their endowments and strike a mutually agreeable bargain, they would reach a point on the contract curve assuming that 9 bargaining does not leave mutual gains from trade on the table. Agent 1 Agent 2 e Pareto Set Contract Curve Figure 5: The Contract Curve Figure 5 also provides some intuition for a key result in general equilibrium theory: any Walrasian equilibrium is Pareto optimal (or lies on the Pareto set). The reason is as follows. At a Walrasian equilibrium, the budget line will separate the two “as good as” sets of the agents (as we saw in Figure 3(b)). Thus, there will be no alternative to the Walrasian outcome that would make both agents better off. Therefore any Walrasian equilibrium is Pareto optimal. The Pareto set, of course, is the set of all Pareto optimal allocations, so an alternative statement is that any Walrasian equilibrium allocation lies on the Pareto set. This result is known as the first theorem of welfare economics. 4 The Welfare Theorems We now turn to a more formal statement of the theorem suggested above – that every Walrasian equilibrium allocation is a Pareto optimal allocation. We then prove a converse result that if an initial allocation is Pareto optimal, there is a Walrasian equilibrium at which no trade occurs.

AIOU Solved Assignment 1& 2 Code 805 Autumn & Spring 2024

Q.No.5     Explain I detail the concept of Pareto optimality.

Ans:- This paper aims at reconstructing, in non-formal terms, the development of Vilfredo Pareto’s writings on what was to become known as “Pareto optimality”. After some references to the context in which Pareto begun to undertake investigations into welfare economics (1), I shall reveal the early versions of the definition of maximum ophelimity for the society (2). After recounting the first significant academic reactions, by Walras and Wicksell, to Pareto’s innovative concept (3), I then elaborate on the development of Pareto’s final specification of his economic optimum (4-5). I shall close the paper with a few comments on the sociological adaptation that Pareto made to his economic optimum (6).

• 1I recall that from Cours d’Économie politique onwards, Pareto distinguishes between the concepts of (…)

2Pareto’s interest to welfare economics stems from his personal political views, which favoured free trade and liberty in economic, political and social matters more generally. In short, he was motivated to apply the tools of pure economics to demonstrate that a liberal economy is, according to new and non Ricardian criteria, better than an economy in which state intervention plays an important role. However, one of the important, and possibly unexpected, results of his research was to demonstrate that the application of pure economics alone is incapable of establishing the superiority of the liberal economy relative to a socialist economy; a conclusion that Pareto reached more than a decade before Enrico Barone came to the same conclusion in his famous 1908 paper. Among the innovations associated with Pareto’s work on welfare issues, there are a number of conceptual and terminological clarifications. Notable in that regard are: the definition of maximum ophelimity for the society in economics1; and, in sociology, the definition of maximum utility for the society (which refers to the utility of the single members of the society) as distinct from maximum utility of the society (which refers to the utility of the society as a whole).

1. Background and assumptions

• 2For a study on the formal aspects of this topic refer to Montesano (1991).

3Before I start to reconstruct the evolution of what would become known as Pareto optimality2, it is interesting to make reference to the background of such Paretian investigation.

4The first study in which Pareto reflected on the economic properties of freedom (whose proxy is free competition) was an article of 1891, Socialismo e Libertà (1974), where he stated that contemporary society was neither founded on free competition nor on private property. Rather, it was founded on the action that the State carried out in favour of the affluent classes, thus creating a bourgeois socialism (Pareto, 1974 [1891], p.384). Having specified this, he still favours the liberal system on the basis of empirical evidence that presents “freedom as lesser evil” (ibid., p.404).

5Around the same time, Pantaleoni and his student Angelo Bertolini accomplished an original attempt of liberal theoretical foundation, in the essay Cenni sul concetto di massimi edonistici individuali e collettivi (1892).

6The two authors, who explicitly refer to Edgeworth’s Mathematical Psychics (1881), stated that through individual or collective optimal action (that is to say, action performed according to the law of the minimum means), both individual and collective hedonistic maximum could be achieved (Bertolini, Pantaleoni, 1892, p.293-294, note 1). Individual hedonistic maximum means “the fullest state of well-being” (ibid., p.301), collective hedonistic maximum means the equitable distribution of goods, if all individuals have the same attitude “towards felicitation” (ibid., p.302). On the contrary, if individuals do not have the same attitude cited above, collective maximum means the unequal distribution of goods that gives more to those who possess this attitude to a greater extent (ibid., p.303). Among the four possible combinations between type of action (individual or collective) and type of resulting maximum (individual or collective), Pantaleoni and Bertolini dwelt mainly on free competition, defined as an “outstanding form of individual work” that enables the achievement of both individual and collective hedonistic maximum (ibid., p.297-298).

7Pantaleoni gave a draft copy of this essay to Pareto. Initially Pareto regarded the hedonistic principle as a stimulus to further epistemological study, such as in the Considerazioni fondamentali sui principi dell’economia politica pura (1892-1893; see Pareto, 1984, p.101-102 – letter to Pantaleoni of December 6^th, 1891). With regard to Cenni sul concetto…, Pareto merely pointed out that those who knew the collective hedonistic maximum did not necessarily want to accomplish it (ibid., p.99-100).

8But Pareto soon became aware of the Walrasian approach to this issue, that is to say, the competitive general equilibrium which implies, but does not demonstrate, the situation that Pareto shall define as the maximum ophelimity for the society (1894a, p.149). Pareto immediately noticed that such an approach involved a degree of circularity (1964 [1896-1897], §65). A theorem, of course, cannot imply its premises: so the Walrasian theorem cannot imply that the private propriety gives rise to the maximum ophelimity for the society as the private property is one of the premises of this theorem. Many years later, Pareto (1968 [1916], §2329¹) would extend his criticism by agreeing with his student Pierre Boven, who had argued that Walras reasoned in circle when suggesting that maximising behaviour by individuals leads to collective maximisation because he had been inspired by a sentiment favourable to free competition and contrary to monopoly. As demonstrated later in this paper, Pareto escaped this Walrasian circularity by developing an entirely new analysis of collective welfare.

• 3Similar to the representative agent of microeconomics of our times.

9It is finally interesting to note that Pareto highlighted that if on one hand it is not possible to compare interpersonally, and therefore not possible to add the ophelimities of different individuals (1966 [1906], chapter IV, §32), instead it is possible to compare the ophelimities of individuals which do not deviate much from an average type. In a nutshell, Pareto considered the practical possibility of comparing ophelimities that the average man3 feels in different conditions, empirically represented by actual individuals who differ very little from such an average man (1964 [1896-1897], volume II, §646). As I shall see infra, Pareto subsequently followed up on the notion of comparability among individuals, although by means of a theoretical stratagem in economics and by resorting, in sociology, to a comparison made by the government.

10On balance, I believe that Pareto’s focus on the issue of maximum ophelimity for society has its origin in two distinct sources: first, from his passion for liberalism; and second, from the attempts by Pantaleoni, Bertolini and Walras to investigate collective welfare, which Pareto found interesting but logically and methodologically unsatisfactory. As a consequence, he set out to develop a new theoretical demonstration of the optimizing property of free competition.

2. The first three versions of Paretian optimality

11Pareto supplied his first version of Paretian optimality in the article Il massimo di utilità dato dalla libera concorrenza (1894b), which began by recognizing that Walras had shown that free competition maximizes collective ophelimity in the hypothesis of constancy of both prices and production coefficients. While Pareto considered the case of variable consumer prices to be largely irrelevant from a theoretical perspective, he remained of the original view that it is important to study the effect of variations in the production coefficients on economic welfare (ibid., p.48). He undertook his study by starting from Walras’s assumption that the production coefficients minimize the cost of production under conditions of free competition (ibid. – quotes Walras, 1889, p.323). Pareto accomplices the minimization of production cost by: envisaging that the prices of the n factors of production p_t, p_s… are constant; obtaining “from the technical conditions of production” the implicit functions that link the n production coefficients of every m goods produced F_a(a_t, a_s,…)=0…; obtaining from such production functions, that every production coefficient for the production of a good depends on the other n-1 (ibid., p.50). All of the above stated, Pareto set out the minimization first order conditions for every equation of the cost of production differentiating it with respect to the n-1 production coefficients: therefore the m(n-1) differential equations p_t∂a_t/∂a_s+p_s=0… are obtained. Along with the m production functions, they give rise to a system of mn equations whose solution creates just as many production coefficients to minimize production costs.

• 4If an individual consumes 1 kg of wheat and 1 kg of meat, one can say that he benefits directly of (…)

12The first step of this second phase of his demonstration involves the application of the method previously adopted by Walras in his The geometrical theory of the determination of price (1892; ibid., p.50-51). Given the quantities of goods consumed by an individual and the associated prices of those goods, this method facilitates the calculation of the ophelimity that corresponds to the quantity of a numéraire good4. By summing all the ophelimities, so determined, of every individual, one obtains the ophelimity which the society made by those individuals benefits from. Similarly, ophelimities that individuals sacrifice to produce goods can be expressed as quantities of a numéraire good.

• 5Pareto supplies the following example: the free competition coefficients grant the maximum ophelimi (…)

13Pareto does not deal with the exact allocation among individuals of overall ophelimity enjoyed as a result of consumption and sacrificed as a result of production. But he asserts that the ophelimity enjoyed (sacrificed) by every individual is all the greater (lower) the greater (lower) is the overall ophelimity enjoyed (sacrificed). Pareto shows also that if one wants to deliberately alter the distribution of goods produced, it would be more “convenient” (more efficient in a welfare sense) to re-allocate directly the ophelimity from individual to individual than altering the production coefficients of free competition5.

• 6The prices p_b, p_c,… express the quantities of good a necessary to purchase one unit of goods b, c… (…)

14That being stated, since the ophelimity by each individual depends on the quantities of goods consumed r_a, r_b, r_c,… , an increase in production coefficient a_s by da_s (ibid., p.52)leads to an increase in ophelimity enjoyed by the individual. This increase, when expressed as quantities of the numéraire good a, is given by (∂r_a/∂a_s+p_b∂r_b/∂a_s+p_c∂r_c/∂a_s+…)da_s6. The ophelimity enjoyed by the society as a whole depends on the quantities globally consumed of m goods R_a, R_b, R_c,… Consequently, an increase in production coefficient a_s by da_s, increases by (∂R_a/∂a_s+p_b∂R_b/∂a_s+p_c∂R_c/∂a_s+…)da_s the ophelimity for the society expressed in quantities of good a. However, an increase in production coefficient a_s also means that ophelimity sacrificed by society increases by (p_t∂R_t/∂a_s+p_s∂R_s/∂a_s+…)da_s. For society as a whole, welfare is maximised when the positive difference between total ophelimity enjoyed and total ophelimity sacrificed is greatest, which is achieved when the value of a_s is conditioned by the maximising requirement that ∂R_a/∂a_s+p_b∂R_b/∂a_s+p_c∂R_c/∂a_s+…-p_t∂R_t/∂a_s-p_s∂R_s/∂a_s -…=0. Considering also the m production functions one therefore has a system of m(n-1)+m=mn equations whose solution determines just as many production coefficients, which maximize the net ophelimity enjoyed by society (ibid., p.54).

15The third and last phase of the demonstration consists in checking that the production coefficients determined under conditions of free competition coincide with those maximizing the net ophelimity for society. The demonstration involves differentiating the equations that specify the quantities of every production factor (required to produce the quantity of goods in the collective) with respect to a coefficient of production (ibid., p.55-56).

16For instance, when equations R_t=a_tR_a+b_tR_b+…; R_s=a_sR_a+b_sR_b+…;… are differentiated with respect to da_s , one derives equations ∂R_t/∂a_s=R_a∂a_t/∂a_s+a_t∂R_a/∂a_s+b_t∂R_b/∂a_s+…; ∂R_s/∂a_s=R_a+a_s∂R_a/∂a_s+b_s∂R_b/∂a_s+… When these are multiplied respectively by p_t, p_s,… and recognising that the price of individual goods produced equals the cost of producing them (e.g. p_a=a_tp_t+a_sp_s+…), we obtain the equation ∂R_a/∂a_s+p_b∂R_b/∂a_s+…-p_t∂R_t/∂a_s-p_s∂R_s/∂a_s-…= R_a(p_t∂a_t/∂a_s+p_s).

• 7To maximize the net ophelimity for the society.

17As the expression of the second term between parentheses above is equal to zero7, this solution is consistent with cost minimising production coefficients and, thereby, maximizes the net ophelimity for the society. However, Pareto also emphasized that coefficients of free competition were not those in force in contemporary society (1964 [1896-1897], §723). But even more importantly, he stressed that the theorem “can be accepted equally by liberals and socialists”, as it relates to production and not to the distribution of goods (1894b, p.61).

18Pantaleoni and Barone informally objected to Pareto’s demonstration because he summed the ophelimities of individuals, which were heterogeneous and therefore not capable of summation. Pareto then prepared a revised demonstration of the result (ibid., p.58-60).

19Pareto observes that when production coefficient a_s varies by da_s, the ophelimity of any consumer varies by (φ_a∂r_a/∂a_s+φ_b∂r_b/∂a_s+φ_c∂r_c/∂a_s+…)da_s which, taking into account the conditions of maximum ophelimity for the individual φ_b/φ_a=p_b ; φ_c/φ_a=p_c … can be reformulated as (∂r_a/∂a_s+p_b∂r_b/∂a_s+p_c∂r_c/∂a_s+…)φ_a da_s. Such equation must be set to equal zero to meet the first order condition of individual maximum ophelimity. As goods have positive ophelimity, φ_a has a positive value, so maximisation is predicated on expression between parentheses summing to zero.

• 8Which gives the aggregate net value of welfare gains, in terms of good a, that result from a change (…)

20However, the number of individuals is θ, so there is a system of θ equations, but only one unknown quantity to determine (the production coefficient a_s). Consequently, the system would be over determined if the bracketed term was equal to zero for all individuals. So, in the face of da_s, some individuals may experience a loss in ophelimity as represented by the numéraire good a (a negative value for the term in parenthesis) or a gain in ophelimity (a positive value for the term in parenthesis). Pareto indicated non-zero values for the term in parenthesis with λ. If all λ are positive (negative), all λφ_a are positive (negative) which implies that the ophelimity for the collective certainly increases (decreases) and therefore it is advisable to let a_s increase (decrease) further. If, however, some λs are positive and other are negative, some individuals would benefit from this while others would be disadvantaged. Of course, for different individuals, welfare, λφ_a, is not comparable because the unit of φ_a, is inconsistent between people. However, as φ_a is positive for goods and λ is accounted for objectively (as all prices in equation ∂r_a/∂a_s+p_b∂r_b/∂a_s+p_c∂r_c/∂a_s+…=λ are expressed in terms of quantity of good a), Pareto points out that it is still possible for those who gain to economically compensate those who lose as long as the sum of the λs8 is positive. Compensation is no longer possible when the sum of the λs is zero or negative – and variations in a_sshould end when the sum of all λ is zero as at that point the ophelimity for the society is maximised.

23In his subsequent Cours d’Économie politique (1964 [1896-1897], §385², §723²), Pareto proposed a new, and clearer, version of the demonstration. He starts from the notion that an individual, thanks to a small increase dλ of the quantity of gold at his disposal, can increase quantities of goods consumed and reduce quantities of productive services sacrificed. The subsequent variations of ophelimity are dU=φ_adr_a+φ_bdr_b+…-φ_sdr_s-…, which, when account is taken of the condition for maximising individual ophelimity p_b/p_a=φ_b/φ_a… can be reformulated as dU=(p_adr_a+p_bdr_b+…-p_sdr_s-…)φ_a/p_a or assuming (p_adr_a+p_bdr_b+…-p_sdr_s‑…)=dλ and φ_a/p_a= μ, as dU=μdλ.

23As φ_a is positive, the sign of dU depends on the sign of dλ. If dλ of all individuals is positive (negative), then dU of all individuals is positive (negative) and the ophelimity for the society increases (decreases). So, if one intends to maximize the ophelimity for the society (which for instance would be the objective of a futuristic socialist state) it is necessary to continue and let the quantities of goods and productive factors vary along the direction (opposite to that) followed so far. As already demonstrated in 1894 paper, the maximum ophelimity of society is reached when the sum of dλ is zero. In fact, until the sum is positive (negative), by varying the amounts of goods and services along the direction (opposite to that) followed so far, one obtains an additional amount of goods with which one could further increase the ophelimity for society. Finally, Pareto showed that one reaches the zero sum of dλ also adopting production coefficients that minimize production costs that is to say, adopting free competition coefficients (ibid., §725).

25As shown in the next paragraph, Pareto’s mathematical skill appealed to Walras. It is interesting to stress the cumbersome but rather clear strategy followed by Pareto in his demonstration. In the first trial, he successively detects the coefficients of free competition, the coefficients maximizing the ophelimity for the society and, at the end, he demonstrates that both correspond. In the second and in the third trial, Pareto, in an innovative way, begins by the variation of the ophelimity of individuals; subsequently he demonstrates that the variations are indirectly homogeneous (due to a clever Walrasian stratagem) and summable. In the end, he finds the maximum ophelimity for the society by using the customary first order condition for maximising a function.

3. Walras and Wicksell’s comments

25The Paretian attempts to identify the conditions for an efficient economic organization of society gave rise to some contemporary comments by Walras and Wicksell.

25Walras qualifies Il massimo di utilità dato dalla libera concorrenza as “illuminating”. The minimization of production costs, the maximization, with regard to ophelimity, of the functions of demand and supply of services and the indication of “coherence” of both optimization procedures “was undoubtedly the way forward” to attain the demonstration of the optimality of free competition (Walras, 1965, p.605 – letter to Pareto of July 9^th 1894). Nonetheless, Walras adds that he had already given such a demonstration. In that regard, the prices of general equilibrium under free competition, determined through the tâtonnement process, maximize the ophelimities of the owners of production services, whose sum gives the maximum total ophelimity for the society. Therefore, the Paretian demonstration is simply a confirmation of the Walrasian one and of the “excellence of the theory as a whole” (ibid., p.607). But Pareto replies that the Walrasian demonstration is applicable only to the case of fixed production coefficients, whereas his own demonstration constitutes a more general treatment because production coefficients are variable. Moreover he aims at indicating, contrary to what was requested by socialists, that there is no greater collective ophelimity than the one “obtained by the game of the entrepreneurs’ competition” (Pareto, 1975, p.250 – letter to Walras of July 20^th, 1894). With specific regard to Pareto’s analysis based on λ, Walras agrees that it represents “a pure creation of political economy” that will enable Pareto to construct a comprehensive applied political economy (Walras, 1965, p.611 – letter to Pareto of July 25^th, 1894).

26In contrast, Wicksell stressed that trading at prices different from those given by free competition equilibrium could yield advantages that are greater than those achieved at competitive equilibrium prices. Actually, φ is the function of total ophelimity of any trader and depends on the available amounts x and y of goods A and B (the first received and the second transferred); the budget constraint is y=px, where p is the price of A (in terms of B) given under free competition (Wicksell, 1958 [1897], p.143, note 1). If the price increases by Δp, so that the new price system no longer correspond to that given under free competition, the change in ophelimity may be represented as Δφ=(∂φ/∂x)Δx-(∂φ/∂y)Δy=(∂φ/∂x)Δx-(∂φ/∂y)(pΔx+xΔp) which, after accounting for the condition that the maximum ophelimity for individual traders ∂φ/∂x=p∂φ/∂y, becomes Δφ=-(∂φ/∂y)xΔp. Summing such Δφ for all traders one obtains ∑-(∂φ/∂y)xΔp, which, in general, is not zero. Therefore, Wicksell found that the change in ophelimity might be positive when prices are not those given by free competition. He found that it is only under special circumstances that prices given under free competition will maximize ophelimity, such in the case where all the traders own the same quantities of goods. In fact, according to Wicksell, the economic equality implies that ∂φ/∂y is equal for all the individuals, which transforms ∑-(∂φ/∂y)xΔp into -(∂φ/∂y)Δp∑x that in free competition, where ∑x=0, is cancelled (ibid.).

27The agreement between Pareto and Walras on the Paretian demonstration of maximum ophelimity for the society is one of the rare positive episodes in their intellectual relations. However, Pareto does not appear to be aware of the above mentioned Wicksellian criticisms which, as we have seen, does not deal with the intricate question of comparability among individual ophelimities and, most of all, aims at destroying the original Paretian support for free competition.

4. The Pareto-Scorza controversy

• 9On the controversy see the important reconstruction by McLure (2000).

28In 1902, the mathematician Gaetano Scorza (1902a, p.300) stated that “we are still very far from being able to give a mathematical proof of the laissez faire, laissez passer principle”9. Pareto responded vehemently, stating that the ophelimity maximization condition of individuals represents freedom of exchange, but this alone does not demonstrate maximisation under free competition (Pareto, 1902, p.404). He later reminded that if Φ₁, Φ₂… are total utilities of the individual traders, the change in individuals ophelimity will be positive for some traders and negative for other traders when the original state is defined by the following conditions: dΦ₁/φ_1a+dΦ₂/φ_2a+…=0 and φ_1a , φ_2a… are positive. It is therefore on this occasion that for the first time Pareto explicitly and formally defined his definitive criterion of maximum of ophelimity for the society (ibid., p.410-413). Pareto then showed with a long algebraic example (ibid., p.415-420), that this result is achieved when the following conditions are valid: maximum ophelimity of individuals, positivity of degrees of ophelimity of each commodity, and zeroed sum of quantities traded. At this point he underlined that these conditions subsisting in case of free competition, this latter “leads to” maximum ophelimity for society (ibid., p.420). This maximum is described geometrically by the point of tangency between two indifference curves (one for each trader). There is anyhow a locus of ophelimity maxima upon every point one reaches, either through free competition (defined as the organization where prices are constant) or through “other different regulations, provided that they are duly selected”, for instance “a tyrant who compelled those who barter to get the same results as free competition” (ibid., p.423-423, p.431). As I shall see in the following paragraphs, Pareto largely restricted himself to repeat, in his future works, the results obtained in this paper, although with some additions and clarifications.

5. Paretian optimality in the Manuel d’Économie politique

• 10Pareto shows that instead from a point of intersection among indifference curves (a situation which (…)

29In his Manuel d’Économie politique (1966 [1906]) Pareto envisages the maximum of ophelimity for the society as a property of the general economic equilibrium and he defines it as the position from which any small variation increases the ophelimity of some and reduces the ophelimity of others (ibid., chapter VI, §32-33). Being Φ₁, Φ₂,… Φ_n the total ophelimities of the n individuals; δ all “the variations […] that are compatible with the constraints of the system” (ibid., appendix, §89); φ_1a, φ_2a,… φ_na the marginal ophelimities of good a for each individual; δΦ/φ_a=δq_a the variation of the quantity of good a owned by anyone among n individuals, with such quantities that, it is explicitly said, are therefore homogeneous among them and summable (ibid., §127-129). He therefore maintains the formal definition of maximum ophelimity already found in the controversy with Scorza, specifying that it implies that δΦ are nil, for some individuals, positive, for other individuals, and negative for others (ibid., §89). Similarly Pareto, for the case in which traders accept market prices, restates that if equilibrium takes place in a point of tangency between indifference curves, the society benefits of the maximum ophelimity: in fact if one shifts from such point, the ophelimity of a trader increases, while the other one decreases (ibid., chapter VI, §34-35, §43)10.

30Further studying the results of 1902, Pareto highlights that if the sales revenues of a commodity are greater than its total production cost (as in the case of monopoly), this positive difference could be spread among members of society, which implies that the δΦ of beneficiaries are positive (those of the remaining members of society being zero) and that, therefore, δΦ₁/φ_1a +δΦ₂/φ_2a +… +δΦ_n/φ_na>0.

31The maximum of ophelimity for society is obtained when (i) total revenues equal total costs and (ii) prices reflect marginal costs (ibid., appendix §90, §92). This double condition is not met by free competition in the important case where prices are constant and there are positive fixed costs (ibid., §92). Actually it would need to have consumers pay a fixed amount for a commodity, selling them later the units of commodities at their marginal costs (ibid., chapter VI, §45-48, §58): the private firms (and therefore free competition) cannot do it whilst collectivism can do it.

32So, Pareto’s conclusion remains in contradiction with the liberal passion, which had been at the origin of his interest for the maximum ophelimity for the society. In order to achieve the maximum ophelimity for the society, pure economics does not enable us to say whether it is preferable for the economy to be based on private property and free competition or on a collectivist one (ibid., §60-61).

6. Maximum utility in sociology

33In view of the impossibility of comparing the ophelimities of individuals, in economics there can only be a maximum ophelimity for the society (that, as I have discussed amply above, takes into account the only variations of the individuals’ ophelimities) and not the maximum ophelimity of the society (which should precisely consider the ophelimity of society taken as a whole).

34But in sociology, however, Pareto relaxes this restriction and allows for interpersonal subjective comparisons, albeit with reference to sentiments. Therefore, in sociology there are the maximum utility for the society and maximum utility of the society (Pareto, 1968 [1916], §2333-233). By way of example, a situation of prosperity with strong national income inequality may be an example of a maximum utility of the society while a situation involving a weaker national income and less inequality in the distribution of income might be an example of maximum utility for the society, based on subjective interpersonal comparisons (ibid., §2335).

• 11The first expresses the evaluations that 1 gives of the utility variations of every individual, the (…)
• 12β¹ (α¹₁δΦ₁+α¹₂δΦ₂+…)=0 represents the evaluation that the government gives to the evaluations tha (…)
• 13Where M₁ represents the sum of the evaluations that the government provides of the evaluations that (…)

35More formally, in Pareto’s sociological treatment of collective welfare, one should look for positive quantities α₁, α₂,… (each one of them assigned to every member of the society) that enable to get to the equation α₁δΦ₁+α₂δΦ₂+…=0 (Pareto, 1913, p.339). The amounts α₁, α₂,… are subjectively assigned welfare weights for individuals 1, 2 and so on: for instance a humanitarian may assign high α to himself (e.g. individual 1) and to the criminals he supports (e.g. individual 2) but a very low α to the victims of crime (ibid., p.340). This process is then repeated for each member of the society, so there is a system of the equations α₁δΦ₁+α₂δΦ₂+…=0, one for each member of the society. However, the equations for each individual are still heterogeneous. To reduce this system of equations to a single homogeneous social welfare relationship, the assigned weighting αof the various equations are multiplied by β¹, β², which are politically allocated values “determined in view of an objective end […] for instance, the prosperity of the community”(ibid.). This work of homogenization might be accomplished by the government. If, for instance, they considered that “it is necessary, for the prosperity of the community, to punish criminals”, the government shall attribute low β to the equations of humanitarians and criminals and high β to those of their victims (ibid.). Thus the individual equations of individual assessment α¹₁δΦ₁+α¹₂δΦ₂+…=0, α²₁δΦ₁+α²₂δΦ₂+…=0…11 are transformed into the following homogeneous ones, β¹(α¹₁δΦ₁+α¹₂δΦ₂+…)=0, β²(α²₁δΦ₁+α²₂δΦ₂+…)=0…12. These latter equations, once summed, lead to the equation (β¹α¹₁+β²α²₁+…)δΦ₁+(β¹α¹₂+β²α²₂+…)δΦ₂+…=0, that can be reformulated as M₁δΦ₁+M₂δΦ₂+…=013 similar to the known δΦ₁/φ_1a+δΦ₂/φ_2a+…=0. The government that aims at the maximum utility of the society as a whole, shall not exceed the situation corresponding to the condition M₁δΦ₁+M₂δΦ₂+…=0 as if he did it, one would contradict himself “sacrificing those who were not to be sacrificed”. In fact in a sub-optimal situation, the individuals protected by the government would be in a worse situation than in an optimal situation (ibid., p.341).

36There is no space in the present paper to investigate the relationship between economics and sociology in Pareto’s thought. However, a few observations on the issue can be made. Based on a study of Pareto’s entire scientific output, one can venture the idea that Pareto was neither an economist nor a sociologist; rather, he remained always a social scientist. His interests focussed, over and over again, on economics or on sociology due to his temporary moods and, most of all, due to his teaching load: the Cours d’Économie politique (1964 [1896-1897]) and the Systèmes socialistes (1965 [1901-1902]) are the outgrowth of long series of lectures. Surely, over time, the underlying schema of equilibrium, if often interwoven and refined, unified Pareto’s treatment of economic and sociological issues. Finally Pareto’s economics is the study of economic equilibrium whilst Pareto’s sociology is the study of social equilibrium. The concepts of economic and sociological maxima support the above thesis to Pareto’s complementary approach to economics and sociology.

37Pareto began to deal with the maximum ophelimity for the society in the 1890 when he was driven by an enduring passion for economic liberalism and a dissatisfaction with the way Pantaleoni, Bertolini and Walras had dealt with the topic. This paper has attempted to conceptually reconstruct Pareto’s various efforts to define the conditions for optimality. Initially he successively equated optimality, as a maximum of ophelimity for the society, with production coefficients obtained under free competition (those which minimize production cost). But following the critics by Pantaleoni and Barone, Pareto alters his demonstration strategy by gradually reaching the definition of maximum as a situation in which movement away from a point would advantage someone, but disadvantage someone else. To reach this result, taking into account that in economics, individual ophelimities are not comparable, Pareto in various manners makes use of the stratagem borrowed from Walras, to transform the individual ophelimities variations into comparable quantity variations of the same commodity. So, Pareto defines the maximum by using the customary first order conditions for maximising a function. In a well-known article of 1902 and in the Manuel d’Économie politique, Pareto develops the formal presentation of what was to become universally known as a Pareto optimum. Moreover he provides a geometrical representation of his results. But, it is very important to stress too that in the Manuel d’Économie politique Pareto finds that pure economics is unable to establish whether free competition or collectivism is the best form of economic organization from the point of view of reaching the maximum ophelimity for the society.

38Finally, while Pareto’s economic analysis of optimality avoids interpersonal comparisons of welfare, he retreats from that position in sociology. Pareto’s sociology recognises that people do make interpersonal comparisons. In view of this, he considers how social welfare might be logically maximised if we account for the fact that individuals, and governments, frequently use subjective sentiments to assess their own welfare and the welfare of other members of the society. As a result, Pareto elaborates new notions, different from the maximum ophelimity for the society but complementary to it, of maximum utility for and of the society.