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

Extension 8.5 Gains from trade

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.

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.

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:

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

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.

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.