Unit 8 Supply and demand: Markets with many buyers and sellers

8.5 Gains from trade in competitive equilibrium: Allocation and distribution

gains from trade, gains from exchange
The benefits that each party gains from a transaction compared to how they would have fared without the transaction.

Buyers and sellers of bread voluntarily engage in trade because both benefit. The gains from trade can be measured in the same way as in Unit 7, by the surpluses obtained by buyers and sellers. There is a potential surplus whenever a buyer will pay more for a loaf than the marginal cost of producing it.

Figure 8.12 shows the market supply and demand curves for bread. Remember that bakeries differ in their marginal costs of producing bread, and the supply curve tells us the marginal cost of each loaf produced. The demand curve shows the consumers’ willingness to pay. For every loaf up to the 5,000th one produced, there is a consumer who is willing to pay more than the marginal cost of producing it.

In the competitive equilibrium at point A, 5,000 loaves are sold at €2 each. All consumers up to the 5,000th receive a monetary surplus equal to their WTP minus the price; and producers of all loaves up to the 5,000th receive a surplus equal to the price minus the marginal cost of producing the loaf.

In this diagram, the horizontal axis shows quantity of loaves, denoted Q, and ranges from 0 to 10,000. The vertical axis shows sellers’ reservation price in dollars, denoted P (price), ranging from 0 to 5. Coordinates are (quantity of loaves, price). An upward-sloping line originates from point (0, 1) and passes through point A (5,000, 2). A downward-sloping, convex line connects the points (0, 4.75), (5,000, 2) and (10,000, 0.5) and is labelled demand curve. The demand curve intersects the supply curve at point (5,000, 2). The area between the demand curve, the vertical axis, and the horizontal line passing through price €2 is the consumer surplus. The area between the supply curve, the vertical axis, and the horizontal line passing through point €2 is the producer surplus.
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Figure 8.12 Equilibrium in the bread market: gains from trade.

Consumer surplus: In this diagram, the horizontal axis shows quantity of loaves, denoted Q, and ranges from 0 to 10,000. The vertical axis shows sellers’ reservation price in dollars, denoted P (price), ranging from 0 to 5. Coordinates are (quantity of loaves, price). An upward-sloping line originates from point (0, 1) and passes through point A (5,000, 2). A downward-sloping, convex line connects the points (0, 4.75), (5,000, 2) and (10,000, 0.5) and is labelled demand curve. The demand curve intersects the supply curve at point (5,000, 2).
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Consumer surplus

At the equilibrium price of €2 in the bread market, a consumer who is willing to pay €3.25 obtains a surplus of €1.25.

Total consumer surplus: In this diagram, the horizontal axis shows quantity of loaves, denoted Q, and ranges from 0 to 10,000. The vertical axis shows sellers’ reservation price in dollars, denoted P (price), ranging from 0 to  255. Coordinates are (quantity of loaves, price). An upward-sloping line originates from point (0, 1) and passes through point A (5,000, 2). A downward-sloping, convex line connects the points (0, 4.75), (5,000, 2) and (10,000, 0.5) and is labelled demand curve. The demand curve intersects the supply curve at point (5,000, 2). The area between the demand curve, the vertical axis, and the horizontal line passing through price €2 is the consumer surplus.
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Total consumer surplus

The shaded area above €2 shows total consumer surplus—the sum of all the buyers’ gains from trade.

Producer surplus: In this diagram, the horizontal axis shows quantity of loaves, denoted Q, and ranges from 0 to 10,000. The vertical axis shows sellers’ reservation price in dollars, denoted P (price), ranging from 0 to 5 25. Coordinates are (quantity of loaves, price). An upward-sloping line originates from point (0, 1) and passes through point A (5,000, 2). A downward-sloping, convex line connects the points (0, 4.75), (5,000, 2) and (10,000, 0.5) and is labelled demand curve. The demand curve intersects the supply curve at point (5,000, 2).
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Producer surplus

The producer’s surplus on a unit of output is the difference between the price at which it is sold, and the marginal cost of producing it. The marginal cost of the 1,000th loaf is €1.11; since it is sold for €2, the producer obtains a surplus of €0.89.

Total producer surplus: In this diagram, the horizontal axis shows quantity of loaves, denoted Q, and ranges from 0 to 10,000. The vertical axis shows sellers’ reservation price in dollars, denoted P (price), ranging from 0 to 5 25. Coordinates are (quantity of loaves, price). An upward-sloping line originates from point (0, 1) and passes through point A (5,000, 2). A downward-sloping, convex line connects the points (0, 4.75), (5,000, 2) and (10,000, 0.5) and is labelled demand curve. The demand curve intersects the supply curve at point (5,000, 2). The area between the supply curve, the vertical axis, and the horizontal line passing through point €2 is the producer surplus.
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Total producer surplus

The shaded area below €2 is the sum of the bakeries’ surpluses on every loaf produced.

The total surplus in the bread market: In this diagram, the horizontal axis shows quantity of loaves, denoted Q, and ranges from 0 to 10,000. The vertical axis shows sellers’ reservation price in dollars, denoted P (price), ranging from 0 to 5. Coordinates are (quantity of loaves, price). An upward-sloping line originates from point (0, 1) and passes through point A (5,000, 2). A downward-sloping, convex line connects the points (0, 4.75), (5,000, 2) and (10,000, 0.5) and is labelled demand curve. The demand curve intersects the supply curve at point (5,000, 2). The area between the demand curve, the vertical axis, and the horizontal line passing through price €2 is the consumer surplus. The area between the supply curve, the vertical axis, and the horizontal line passing through point €2 is the producer surplus.
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The total surplus in the bread market

Adding together the consumer and producer surplus, we get the total surplus on all 5,000 loaves sold.

The whole shaded area in Figure 8.12—the sum of consumer and producer surplus—is the total surplus from trade in the bread market. All potential gains from trade are exhausted at the competitive equilibrium—in other words, the total surplus is maximized.

Joel Waldfogel, an economist, gave his chosen discipline a bad name by suggesting that gift-giving at Christmas may result in a deadweight loss.1 If you receive a gift that is worth less to you than it cost the giver, you could argue that the surplus from the transaction is negative. Do you agree?2

  • If fewer loaves were produced, there would be unexploited gains from trade: some consumers without bread would be willing to pay more than the cost of producing another loaf.
  • If any more than 5,000 loaves were produced, the total surplus would decrease, because the surplus on the extra loaves would be negative: they would cost more to make than consumers would pay.
  • At the equilibrium, the marginal cost of the last loaf produced—the 5,000th—is equal to the willingness to pay of the last consumer who buys, so all potential gains from trade are exploited.

This property holds in general: a competitive equilibrium allocation maximizes the total surplus available in the market. It contrasts with the allocation of a differentiated good (Figure 7.20), where there is a deadweight loss because the producer sets a price above the marginal cost of the last item produced.

Pareto efficiency

Since all potential gains from trade are exploited at the competitive equilibrium, we know that it is not possible to change the allocation (the amount of bread produced, and who buys and sells it) to make a consumer or firm better off without making any of them worse off.

Pareto efficient, Pareto efficiency
An allocation is Pareto efficient if there is no feasible alternative allocation in which at least one person would be better off, and nobody worse off.
external effect, externality
An external effect occurs when a person’s action confers a benefit or imposes a cost on others and this cost or benefit is not taken into account by the individual taking the action. External effects are also called externalities.
complete contract
A contract is complete if it a) covers all of the aspects of the exchange in which any party to the exchange has an interest, and b) is enforceable (by the courts) at close to zero cost to the parties.

Does this mean the equilibrium allocation is Pareto efficient? The answer is yes, provided that no one other than the consumers and bakeries is affected by trade in the bread market. But if, for example, producing bread involved a lot of noise or pollution which affected the people living nearby, then to find a Pareto efficient allocation, we would have to allow for these additional costs of bread production (known as external effects).

The Pareto efficiency of a competitive equilibrium allocation is often interpreted as a powerful argument in favour of markets as a means of allocating resources. But we should be careful not to exaggerate the value of this theoretical result. It holds only under the following stringent conditions:

  • In a market with many buyers and sellers of identical goods
  • At the equilibrium, when all participants are price-takers
  • When trade in the market has no external effects
  • Where there is a complete contract between each buyer and seller.

The fourth condition can be assumed to hold in our example: the exchange of a loaf of bread for money is governed by a complete contract. If you find there is no loaf of bread in the bag marked ‘bread’ when you get home, you can get your money back. But in other cases, it may be impossible to ensure that all the important aspects of the exchange are covered by a legally enforceable contract.

In practice:

  • Most firms sell goods that are somehow differentiated from those of other firms—even bakeries selling identical loaves differ in location, and the service and range of goods they offer.
  • Evidence of price-taking is hard to find (Section 8.10).
  • Many goods have external effects—such as carbon emissions that contribute to climate change.
  • Contracts are often incomplete: for example, the buyer of a second-hand car may not know whether it is reliable and roadworthy.

The implications for Pareto efficiency of external effects and incomplete contracts are analysed in Unit 10.

The distribution of the gains from trade

There are two criteria for assessing an allocation: efficiency and fairness (Unit 5). Even if we think that a market allocation is Pareto efficient, we should not conclude that it is necessarily a desirable one. As we explain in Unit 7, consumer and producer surplus do not tell us about fairness, because the sum of monetary gains is a poor measure of the overall benefits. But we can assess how the monetary gains are distributed between the producers and the consumers, by comparing the consumer and producer surplus. In Figure 8.12, consumer surplus is slightly higher than producer surplus. This happens because the demand curve is relatively steep (inelastic).

Just as we measure the responsiveness of consumers to price changes using the elasticity of demand, we can use the elasticity of supply to measure the responsiveness of producers. In Figure 8.12, demand is less elastic than supply. In general, the distribution of the total surplus between consumers and producers depends on the relative elasticities of demand and supply.

Exercise 8.3 Maximizing the surplus

Consider a market for the tickets to a football match. Six supporters of the Blue team would like to buy tickets; their valuations of a ticket (willingness to pay) are 8, 7, 6, 5, 4, and 3. The diagram below shows the demand ‘curve’. Six supporters of the Red team already have tickets, for which their reservation prices (willingness to accept) are 2, 3, 4, 5, 6, and 7.

In this diagram, the horizontal axis shows the number of supporters, ranging from 0 to 6, and the vertical axis shows willingness to pay, ranging from 0 to 10. A step function, connecting the points (0, 10), (0, 8), (1, 8), (1, 7), (2, 7), (2, 6), (3, 6), (3, 5), (4, 5), (4, 4), (5, 4), (5, 3), (6, 3) and (6, 0) represents the demand curve.
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Suppose all trades are to take place at a single price as in a competitive market where buyers and sellers are price-takers.

  1. Draw the supply and demand ‘curves’ on a single diagram. (Hint: the supply curve is also a step function, like the demand curve.)
  2. Show that four trades take place in equilibrium.
  3. What is the equilibrium price?
  4. Calculate the following by adding up the surpluses of each individual trade: consumer (buyer) surplus, producer (seller) surplus, total surplus in equilibrium.

Now suppose that the market operates through bargaining between individual buyers and sellers.

  1. Find a way of matching the buyers and sellers so that more than four trades occur. (Hint: suppose the highest WTP buyer buys from the highest WTA seller.)
  2. Using the scenario you found in step 5, work out the surplus from each trade and compare it to the equilibrium surplus from step 4.
  3. Starting from the allocation of tickets you obtained through bargaining, in which at least five tickets are owned by Blue supporters, is there a way through further trade to make one of the supporters better off without making anyone worse off?

Exercise 8.4 Gains from trade, deadweight loss, and elasticity of supply

  1. Consider the bread market in Figure 8.12, where the equilibrium price is €2.00 and 5,000 loaves are sold. Suppose that the bakeries get together to form a cartel (a group of firms that collude in order to increase their joint profits). They agree to raise the price to €2.70, and jointly cut production to supply the number of loaves that consumers demand at that price. Sketch a diagram to illustrate the market outcome under the cartel. Shade the areas on your diagram to show the consumer surplus, producer surplus, and deadweight loss caused by the cartel.
  2. Describe some features of goods that are likely to have highly elastic supply curves, and give some examples.
  3. Draw two diagrams, one where the supply curve is highly elastic, and one where the supply curve is highly inelastic. Use your diagrams to explain how the share of the gains from trade obtained by producers depends on the elasticity of the supply curve.

Question 8.6 Choose the correct answer(s)

In Figure 8.12, the market equilibrium output and price of the bread market is shown to be at (Q*, P*) = (5,000, €2). Suppose that the mayor decrees that bakeries must sell their bread at a price of €1.50. Based on this information, read the following statements and choose the correct option(s).

  • The consumer and producer surpluses both increase.
  • The producer surplus increases, but the consumer surplus decreases.
  • The consumer surplus increases, but the producer surplus decreases.
  • The total surplus is lower than at the market equilibrium.
  • Producer surplus is lower, because producers who would have sold bread at prices between €1.50 and €2 no longer sell their bread.
  • Producer surplus is lower, because producers who would have sold bread at prices between €1.50 and €2 no longer sell their bread.
  • Consumer surplus is lower, because fewer trades occur (consumers are no longer able to buy bread from the producers who would have sold at prices between €1.50 and €2).
  • There is a deadweight loss, equal to the area of the triangle between the supply and demand curves to the left of equilibrium.

Question 8.7 Choose the correct answer(s)

Read the following statements about a competitive equilibrium allocation and choose the correct option(s).

  • It is the best possible allocation for everyone in the market.
  • No buyer’s or seller’s surplus can be increased without reducing someone else’s surplus.
  • The allocation must always be Pareto efficient.
  • The total surplus from trade is maximized.
  • The allocation maximizes the total surplus, but that does not mean it is best for everyone in the market—for example, we may think it is unfair.
  • This must be true, since the allocation maximizes the total surplus.
  • The equilibrium allocation may not be Pareto efficient if it affects someone other than the buyers or sellers.
  • This is a general property of competitive equilibrium.

Extension 8.5 Gains from trade

In the main part of this section, we described the gains from trade in the bread market diagrammatically, and argued that total surplus would be maximized at the competitive equilibrium. Now, we explain how to calculate consumer and producer surplus using calculus (integration) and prove the maximization result algebraically.

Figure 8.12a, reproduced below as Figure E8.4, shows the gains from trade at the equilibrium in the market for bread in one city. The surplus obtained by consumers is represented by the area below the demand curve and above the horizontal line at the level of the market price. Producer surplus is the area above the supply curve and below the horizontal price line. The sum of these two areas is the total gain from trading in this market, relative to the outside option of no bread being produced.

In this diagram, the horizontal axis shows quantity of loaves, denoted Q, and ranges from 0 to 10,000. The vertical axis shows sellers’ reservation price in dollars, denoted P (price), ranging from 0 to 5. Coordinates are (quantity of loaves, price). An upward-sloping line originates from point (0, 1) and passes through point A (5,000, 2). A downward-sloping, convex line connects the points (0, 4.75), (5,000, 2) and (10,000, 0.5) and is labelled demand curve. The demand curve intersects the supply curve at point (5,000, 2). The area between the demand curve, the vertical axis, and the horizontal line passing through price €2 is the consumer surplus. The area between the supply curve, the vertical axis, and the horizontal line passing through point €2 is the producer surplus.
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Figure E8.4 Equilibrium in the bread market: gains from trade.

The demand and supply functions

To determine the gains from trade mathematically, we need to specify the equations of the demand and supply curves. As in other extensions, we treat \(Q\) as a continuous variable so that we can use calculus for this analysis, rather than thinking of \(Q\) as representing discrete numbers of loaves.

Suppose the demand for bread is described by the inverse demand function, \(P=f(Q)\), where \(P\) is the price and \(Q\) is the quantity of bread. Under the usual assumption that demand curves slope downward (the Law of Demand), \(f\) is a decreasing function: \(f'(Q)<0\).

We know from Extension 8.4 that the inverse supply curve is the marginal cost curve for bread production in this market. If we let \(C(Q)\) be the total cost (for all the bakeries together) of producing a total quantity, \(Q\), then the marginal cost is \(C'(Q)\), and \(P=C'(Q)\) is the equation of the inverse market supply function. We assume \(C'(Q)\) is positive and increases with \(Q\), so that the supply curve slopes upward, which means that \(C(Q)\) is an increasing, convex function.

Marginal cost is the derivative of total costs. So, we can find the total cost at quantity, \(Q\), by integrating the marginal cost function. Integrating between quantities 0 and \(Q\), we obtain:

\[C(Q) = C(0)+ \int_0^Q C'(q) \, dq\]

where \(C(0)\) is the total cost at quantity zero—that is, the fixed costs. This equation tells us that the total cost, \(C(Q)\), is the fixed costs of all firms plus the area under the marginal cost curve for quantities less than or equal to \(Q\), which is the total variable costs.

Calculating consumer and producer surplus

Remember that the demand function tells us the willingness to pay (WTP) for bread. If consumers are lined up in order of willingness to pay for a loaf, then the \(q{^t}{^h}\) consumer is willing to pay \(P = f(q)\). Any buyer whose willingness to pay for a good is higher than the price they pay receives a surplus. Suppose that the \(q{^t}{^h}\) consumer pays \(P_0\) for a loaf of bread. Then the surplus of this consumer will be \(f(q) - P_0\). In Figure E8.4, where all consumers pay a price of €2, this is the vertical distance at the quantity, \(q\), between the demand curve and the horizontal line, \(P=2\).

consumer surplus
Each consumer who buys a good receives a surplus equal to their willingness to pay minus the price. The term ‘consumer surplus’ normally refers to the sum of these surpluses across all consumers.

We defined consumer surplus as the sum of the surpluses (in monetary terms) for all the consumers who purchase bread; when quantity \(Q\) is a continuous variable, it is the integral of the surpluses over all these consumers.

Suppose the price is \(P_0\) and the total quantity sold is \(Q_0\). Figure E8.5 illustrates this situation for the case, \(P_0=2.5, Q_0=3,000\). Then, consumer surplus corresponds to the shaded area between the demand curve and \(P=P_0\). We are assuming that all loaves are sold at the same price, but not that the market is in equilibrium.

In this diagram, the horizontal axis shows quantity of loaves, denoted Q, and ranges from 0 to 10,000. The vertical axis shows price in euros, denoted P, ranging from 0 to 5. Coordinates are (quantity of loaves, price). An upward-sloping line that originates from point (0, 1) and passes through point A (5,000, 2) is labelled ‘Supply, P = C’(Q)’. A downward-sloping, convex line connects the points (0, 4.75), (5,000, 2) and (10,000, 0.5) and is labelled ‘Demand, P = f(Q)’. The demand curve intersects the supply curve at point (5,000, 2). At a price of 2.5, denoted P_0, consumer surplus is the area enclosed by the points (0, 4.75), (3000, 2.98), (3000, 2.5), and (0, 2.5). Producer surplus is the area enclosed by the points (0, 1), (3000, 1.46), and (3000, 2.5), and (0, 2.5).
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Figure E8.5 Consumer and producer surplus when \(Q=Q_0\) and \(P=P_0\).

To find the shaded area that represents consumer surplus, we need to integrate the surpluses, \((f(q)-P_0)\), over all values of \(q\) between \(q=0\) and \(q=Q_0\):

\[\mbox{consumer surplus} = \int_0^{Q_0} (f(q) - P_0) \, dq = \int_0^{Q_0} f(q) \, dq - P_0Q_0 = F(Q_0) - P_0Q_0\]

In this expression, we have introduced the notation, \(F(Q)\), to denote the integral of the function, \(f\). That is, the area under the demand curve for quantities between 0 and \(Q\). By the fundamental theorem of calculus:

\[F'(Q) = f(Q)\]

We have specified that \(f(Q)\) is a decreasing function, so it follows that function \(F\) is concave.

Likewise, since the market supply curve corresponds to the marginal costs of loaves of bread in order of increasing marginal cost, the \(q{^t}{^h}\) loaf costs \(C'(q)\) to produce. If the firm is paid \(P_0\) for this loaf, it receives a surplus equal to \(P_0 - C'(q)\). In the diagram, this is the vertical distance at the quantity, \(q\), between the supply curve and the horizontal line, \(P=P_0\).

producer surplus
The producer of a good receives a surplus on each unit, equal to the price minus the marginal cost of producing it. The term ‘producer surplus’ normally refers to the sum of these surpluses across all units sold.

When the price is \(P_0\) and the quantity sold is \(Q_0\), we find total producer surplus (the shaded area between the supply curve and \(P=P_0\) in Figure E8.5) by integrating the surpluses, \(P_0 - C'(q)\), that are received by the producers of each loaf sold:

\[\mbox{producer surplus} = \int_0^{Q_0} (P_0 - C'(q)) \, dq = P_0Q - \int_0^{Q_0} C'(q) \, dq = P_0Q_0 - C(Q_0)+C(0)\]

As explained in Section 7.7, producer surplus is the firm’s economic rent relative to the outside option of not producing any output, but still incurring the fixed costs. Profit is the firm’s rent relative to the outside option of leaving the market altogether.

This expression shows that the producer surplus is equal to the firms’ profits plus their fixed costs. Equivalently, profit equals producer surplus minus fixed costs.

Maximizing consumer and producer surplus

The expressions we have obtained for consumer surplus, \(F(Q_0)-P_0Q_0\), and producer surplus, \(P_0Q_0 - C(Q_0)+C(0)\), give the value of consumer surplus for any price, \(P_0\), and any quantity, \(Q_0\); they apply whether or not the price is at the market-clearing level.

Since they measure the gains from trade, it is useful to know what conditions make them as large as possible. Continuing to assume that the price is fixed at \(P=P_0\), we will consider how consumer surplus (CS) varies with quantity, \(Q\):

\[\text{CS}(Q) = F(Q) - P_0Q\]

The value of \(Q\) that maximizes consumer surplus can be found by setting the derivative of CS to zero:

\[F'(Q) = P_0\]

Note that since \(F(Q)\) is concave, the second derivative of CS is negative, which confirms that this condition gives us a maximum point.

This equation tells us that if the price is \(P_0\), then CS is maximized when the quantity sold is on the demand curve at \(P_0\)—that is, when all consumers whose willingness to pay is greater than or equal to \(P_0\) participate in the market. If fewer consumers participate (as at \(Q_0\) in Figure E8.5), there are unexploited gains. And if any other consumers bought bread, they would receive a negative surplus, thereby decreasing the aggregate consumer surplus.

In exactly the same way, you can show that producer surplus,

\[\text{PS}(Q) = P_0Q-C(Q)\]

is maximized when

\[P_0 =C'(Q)\]

So whatever the price, producers maximize their surplus if the marginal cost of bread is equal to the price.

Maximizing total surplus

The sum of the producer and consumer surplus is the total surplus. When the price is \(P_0\) and the quantity sold is \(Q\), total surplus is \(N(Q)\):

\[\begin{align} N(Q) &= F(Q) - P_0 Q + P_0 Q - C(Q)\\ \text{total surplus} &= \text{consumer surplus} +\text{producer surplus} \end{align}\]

which can be simplified to:

\[N(Q) = F(Q) - C(Q)\]

Note that the total surplus depends only on the quantity sold. Whatever the price, the amount paid for bread is a loss for consumers and an equal gain for firms, so the two cancel out when we add their surpluses together to evaluate the total surplus from the market.

To find the quantity, \(Q^*\), that maximizes the total surplus, we set the derivative of N(Q) to zero. Then \(Q^*\) is the quantity that satisfies the equation:

\[F'(Q^*) =C'(Q^*)\]

To be sure that \(Q^*\) maximizes \(N\), we need to consider the second derivative. Remember that \(F\) is concave, and \(C\) is convex. So the second derivative of \(F\) is negative, and the second derivative of \(C\) is positive. We can deduce that the second derivative of \(N\) is negative, and hence that \(Q^*\) corresponds to a maximum point.

Since \(F'(Q^*) =f(Q^*)\), this equation tells us that \(Q^*\) is at the point where the inverse demand curve, \(P=f(Q)\), meets the inverse supply curve, \(P=C'(Q)\). \(Q^*\) is the level of output at which demand and supply curves cross. This is the level of output achieved when the market is in competitive equilibrium. Therefore, we have proved that in the competitive equilibrium allocation, in which the market clears at the equilibrium price \(P^* = f(Q^*) =C'(Q^*)\), the quantity sold maximizes the total gains from trade.

Exercise E8.3 Calculating surplus

Suppose the (indirect) market supply function is \(P = 2 + 4Q\) and the (indirect) market demand function is \(P = 80 - 2Q\).

  1. Find the equilibrium price and quantity, and calculate the corresponding producer and consumer surplus. Draw a diagram to illustrate this outcome.
  2. Now suppose the market price is fixed at \(P = 44\). Calculate producer and consumer surplus when (i) market quantity is determined by supply, and (ii) market quantity is determined by demand. For both scenarios, use a diagram to indicate these areas of surplus.

Now suppose the (indirect) market supply function is \(P = 0.4Q^2 + Q + 14.7\) and the (indirect) market demand function is \(P = 0.1Q^2 – 8Q + 120\).

  1. Find the equilibrium price and quantity, and calculate the corresponding producer and consumer surplus. (Hint: Use integration.) Draw a diagram to illustrate this outcome.
  2. Now suppose the market price is fixed at \(P = 44.1\). Calculate producer and consumer surplus when (i) market quantity is determined by supply, and (ii) market quantity is determined by demand.

Read more: Sections 8.4 (on convexity and concavity) and 19.1 (on integration) of Malcolm Pemberton and Nicholas Rau. Mathematics for Economists: An Introductory Textbook (4th ed., 2015 or 5th ed., 2023). Manchester: Manchester University Press.

  1. Joel Waldfogel. 1993. ‘The Deadweight Loss of Christmas’. The American Economic Review 83 (5) (December): pp. 1328–1336. 

  2. The Economist. 2009. ‘Is Santa a Deadweight Loss?’. Updated 14 December 2009.