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Unit 8 Supply and demand: Markets with many buyers and sellers
8.4 Firms in competitive equilibrium
In the second-hand textbook example, both buyers and sellers are individual actors. Now we analyse markets in which firms sell identical goods.
Unit 7 explains how firms choose their price and quantity when producing differentiated goods. The more competition there is from other firms producing similar products, the more restricted is the choice of price: the firm’s demand curve is then quite flat (elastic) because raising the price would cause consumers to switch to other similar brands.
If products are identical and consumers can easily switch from one firm to another, the choice of price is extremely restricted: firms will be price-takers in equilibrium.
Imagine a city where many small bakeries produce bread and sell it directly to consumers. Figure 8.7 shows the market demand curve for one product, a large baguette: the total amount demanded at each price by all consumers in the city. It is downward-sloping as usual because at higher prices, fewer consumers will be willing to buy.
Suppose that you are the owner of one small bakery that specializes in baguette production. You have to decide what price to charge, and how many to produce each morning. Suppose that neighbouring bakeries are selling loaves identical to yours at €2.35. This is the prevailing market price, and you will not be able to sell loaves at a higher price than other bakeries, because no one would buy—you are a price-taker. But at €2.35, there are many consumers willing to buy, so you can sell as many loaves as you wish.
What you should do depends on your costs of production—and, in particular, on your marginal costs. You have some fixed costs—the costs associated with your premises and equipment—but you have to pay these irrespective of the number of loaves you produce. It is the additional costs of actually making each loaf of bread—the cost of the ingredients, and what you have to pay your employees for the time it takes to bake a loaf—that determine whether you should produce 30, 50, or 100 loaves per day. Once you have installed mixers, ovens, and other equipment, the marginal cost of each extra loaf may be relatively low, as long as you don’t exceed the capacity of your equipment.
Figure 8.8 illustrates this situation. You have the capacity to produce up to 120 loaves per day at a constant marginal cost of €1.50. If you want to produce more than 120 loaves with your current equipment, you will have to operate overnight, paying overtime wage rates and higher energy costs. You could then produce up to 60 more loaves, at a marginal cost of €2.60.
The horizontal line at P = €2.35 represents the demand for bread from your bakery—because you are a price-taker, each loaf you produce can be sold for €2.35.
In this example, it is easy to find the profit-maximizing price and quantity without drawing isoprofit curves. Work through Figure 8.8 to understand how to do it.
To produce more than 120 loaves, you have to operate beyond normal working hours, at a marginal cost of €2.60, up to a maximum capacity of 180 loaves. Your marginal cost function steps up at 120.
The market price is P = €2.35. If you choose a higher price, customers will go to other bakeries. Your feasible set of prices and quantities is the shaded area below the horizontal line at P, where the price is less than or equal to €2.35, and the quantity is less than or equal to 180.
On every loaf you produce up to 120, you can make a surplus of €2.35 − €1.50 = €0.85. Each additional loaf increases your profit. Above 120, you will make a loss of €2.60 − €2.35 = €0.25 on each additional loaf. So your profit-maximizing quantity is Q* = 120.
Your surplus is the shaded area between the line P = €2.35 and the marginal cost. Surplus = (2.35 − 1.50) × 120 = €102.
Figure 8.8
The profit-maximizing price and quantity.
Your best choice is P = €2.35 and Q = 120; you maximize profit by making as many loaves as possible at a marginal cost below the market price. Your profit will be the total surplus on 120 loaves minus your fixed costs.
Importantly, when you are a price-taker, it is not the market demand curve in Figure 8.7 that determines how many loaves you can sell at different prices; it is the price charged by your competitors. So we can think of the horizontal line at P = €2.35 in Figure 8.8 as your firm’s demand curve: as in Unit 7 it is the feasible frontier for your price and quantity. If you charge more than €2.35, your demand will be zero—but at €2.35 or less, you can sell as many loaves as you like.
The firm’s supply curve
A firm in a competitive market equilibrium does not choose a price. It accepts the market price, and chooses a quantity that depends on its marginal cost. Figure 8.8 shows how many loaves you will produce when the market price is €2.35. What would you do if the price changed? Your profit-maximizing quantity would depend on how the price compared with your marginal cost:
If the price falls below €1.50, you should immediately stop making bread. You would make a loss on every loaf produced.
As long as the price remains between €1.50 and €2.60, your profit-maximizing quantity remains the same. You should produce 120 loaves.
If the price rises above €2.60, it is higher than your marginal cost of producing additional loaves overnight. You now maximize profit by expanding output to a total of 180 loaves.
It may seem surprising that we say that ‘price equals marginal cost’ in Figure 8.8. The profit-maximizing quantity is 120, and marginal cost jumps from €1.50 to €2.60 at this point. In the case of a jump in marginal costs, the point where the marginal cost function crosses the price line lies between the two marginal costs.
supply curve
A supply curve shows the number of units of output that would be supplied to the market at any given price. The firm’s supply curve shows the units supplied by an individual firm, and the market (or industry) supply curve shows the total number of units supplied by all sellers in the market (or firms in the industry). Also known as: supply function.
So the marginal cost function in Figure 8.8 is your firm’s supply curvesupply curve A supply curve shows the number of units of output that would be supplied to the market at any given price. The firm’s supply curve shows the units supplied by an individual firm, and the market (or industry) supply curve shows the total number of units supplied by all sellers in the market (or firms in the industry). Also known as: supply function.: it tells you the number of loaves to produce for each level of the market price. Your profit-maximizing quantity can be found at the point where your marginal cost function crosses a horizontal line at the level of the market price. This result—that price equals marginal cost—is true for any firm in a competitive equilibrium when marginal cost is constant or increasing with quantity. We discuss another example in the extension to this section.
But remember that you also have fixed costs. It is possible that even at the profit-maximizing quantity, your surplus is too small to cover the fixed costs and leave you with a profit overall. This is most likely to happen if the market price is only just above €1.50. 120 loaves is still the best choice, because the surplus covers some of your fixed costs. If you expect the price to rise soon, it might be worth continuing to produce bread despite the short-term losses, so that the revenue helps you to cover the costs of maintaining your premises and retaining staff. If not, you may need to consider closing the business.
The market supply curve
The market for bread in the city has many consumers and many bakeries. Let’s suppose initially there are 15 bakeries, differing in marginal costs and production capacity for large baguettes depending on what other products they make and sell. Those who specialize in producing baguettes have more suitable premises, equipment, and staff skills, so they can produce at lower marginal costs than others.
Each bakery has a supply curve shaped similarly to the one in Figure 8.8, with a maximum capacity at constant marginal cost under normal operations. It may have higher steps corresponding to amounts it could produce at higher marginal cost by introducing extra shifts or switching production from other types of bread. It will produce the maximum number of loaves it can make at a marginal cost below the prevailing market price.
To find the market supply curve, we just add up the total amount that all the bakeries will supply at each price. Figure 8.9 shows how to do this, starting with the quantity produced at lowest marginal cost, and successively adding further quantities in order of increasing marginal production cost.
Higher steps show the amounts that can be produced at higher marginal cost by bakeries switching production from other types of bread, or introducing overtime shifts. If all produce at maximum capacity, they can produce 4,000 loaves.
If the price is P, only the bakeries with marginal cost less than or equal to P will produce bread. If the price was €3, the graph shows that total market supply would be 3,000 loaves.
Figure 8.9
The market supply curve: 15 firms.
If there were 50 bakeries in the city, more bread would be produced, with many more ‘steps’ on the supply curve. Rather than drawing them all, we approximate market supply with a smooth curve. Figure 8.10 shows an approximate market supply curve with 50 firms.
The supply curve tells us two different things. If we choose any price, it tells us the total number of loaves the bakeries would produce. But to construct it, we plotted the marginal cost of each loaf of bread in increasing order of marginal costs. So, if we choose a particular quantity (7,000, say) and use the curve to find the corresponding value on the vertical axis (€2.74), this tells us that the marginal cost of the 7,000th loaf is €2.74. In other words, the market supply curve is the marginal cost curve for all the bread produced in the city.
Competitive equilibrium in the bread market
Now we know both the demand curve Figure 8.7 and the market supply curve (Figure 8.10). Figure 8.11 shows that the market-clearing price is exactly €2.00: consumers demand and firms supply 5,000 loaves per day. The bread market is in competitive equilibrium.
Since the equilibrium is the point where the demand curve crosses the marginal cost curve, we know that—in equilibrium—both the willingness to pay of the 5,000th consumer, and the marginal cost of the 5,000th loaf, are equal to the market price.
Extension 8.4 Supply, demand, and competitive equilibrium
In the main part of this section, we derived the supply function for a firm in a competitive market under the assumption that the firm’s marginal cost is constant until it reaches its production capacity. We used this to find market supply, and the equilibrium price and quantity.
In this extension, we build on the analysis in Extensions 7.4, 7.5, and 7.6, and use calculus (differentiation) to find the supply function for a firm with a smoothly increasing marginal cost function. Then, we find the equilibrium price and quantity algebraically.
In our model of competitive equilibrium in a city with many small bakeries and many consumers, the marginal cost (MC) for each bakery is a step function: marginal cost is constant until normal capacity is reached. Beyond that, marginal cost is higher: the function steps up to a higher level. Bakeries differ in their marginal costs, and each one produces the maximum number of loaves it can make at a marginal cost below the prevailing market price, P. Adding up the total amount at each price gives us the market supply curve.
Now consider a bakery with cost function, C(Q); we assume quantity Q is a continuous variable and obtain the marginal cost by differentiating:
MC=C′(Q)
Suppose that the cost function is also convex: C″(Q)>0. Then, marginal cost increases with quantity: MC is an upward-sloping line. We showed in Extension 7.6 that in this case, the firm’s average cost (AC) and isoprofit curves are U-shaped, and the MC curve passes through the minimum point of each one.
The cost function in Figure 8.1 is C(Q)=F+a1Q+a2Q2+a3Q3, where F=35,a1=1,a2=0.00203, and a3=0.00002. If you wish, you can find the MC and AC curves algebraically and check their properties.
Figure E8.1 shows the MC and isoprofit curves for a hypothetical bakery. They are similar to the ones in Extension 7.6, but we have not assumed that MC is a straight line.
Figure E8.1
Marginal cost and isoprofit curves for a bakery with increasing marginal cost.
The profit-maximizing quantity
In a competitive equilibrium, the bakery is a price-taker. If the market price is P, how many loaves of bread should the bakery supply? Its profit is a function of quantity, Q:
Π(Q)=PQ−C(Q)
We can find the value of Q that maximizes profit by differentiating, and setting the derivative equal to zero to obtain the first-order condition:
dΠdQ=P−C′(Q)=0⇒P=C′(Q)
This is an important result: when the firm is a price-taker, it chooses its quantity, Q, so that the marginal cost is equal to the market price.
You can check, by finding the second derivative of profit, that the point where P=C′(Q) is a maximum point.
Figure E8.2 demonstrates the same result using the graphical approach to profit maximization that we followed in Unit 7. In a competitive market, the bakery cannot choose a price above the market price because it would not attract any customers. So if the market price is P∗, its feasible frontier is the line P=P∗ and its feasible set is the shaded area below it. To achieve maximum profit, it should find the point in the feasible set that reaches the highest isoprofit curve—that is, a tangency point with the line P=P∗.
The slope of the isoprofit curve at the tangency point is zero. And since we know the marginal cost curve passes through the minimum points of all the the isoprofit curves, the profit-maximizing point must lie on the marginal cost curve: it satisfies the condition P∗=C′(Q∗).
Figure E8.2
The bakery maximizes profit where the feasible frontier is tangent to an isoprofit curve, which lies on the marginal cost curve.
The firm’s supply function
Since for each value of P, the bakery chooses the quantity of loaves that satisfies the condition, P=C′(Q), we can say the equation, P=C′(Q), is the bakery’s inverse supply function. Graphically, the marginal cost function tells us how many loaves the firm will supply at any given price. If the equilibrium price changes, the firm will move to a different point on its marginal cost curve.
If we rearrange the equation, P=C′(Q), to write Q in terms of P, we obtain the direct supply function: the value, Q, that the firm will choose for a given value of P. We will write the firm’s supply function as:
Q=S(P)
where Q=S(P) is the inverse of the marginal cost function, P=C′(Q).
The market supply function
Suppose the total number of bakeries in the market is m. We will now write Qi=Si(P) for the supply function of the ith bakery, for i=1,…,m, Q, and Q=S(P) for the market supply function (the total quantity of bread supplied to the market by all the bakeries together).
When the market price is P, the total amount of bread supplied to the market is the sum of the amounts Qi supplied by each bakery. So the market supply is:
Q=S(P) where S(P)=m∑i=1Si(P)
In Figure E8.3, we have drawn the supply functions for an example in which there are 50 bakeries (m=50) with identical upward-sloping supply functions, so that S(P)=50×Si(P). The supply function, Si(P), for an individual bakery is shown on the left, and the total market supply, S(P), is on the right. The curves appear to have exactly the same shape. But they are drawn using different scales; the quantity of loaves supplied at each price, on the horizontal axis, is 50 times as high in the right-hand panel. If we drew them on the same scale, the market diagram would be 50 times as wide so the market supply curve would be much flatter (more elastic). The increase in price for each extra loaf produced is much lower on the market supply curve.
Figure E8.3
Supply functions for the firm and the market (identical bakeries).
Just as in the case in the main part of this section, where the individual bakeries’ marginal costs are step functions, the market supply curve here can be interpreted as the marginal cost curve for the market as a whole. This is true whether or not the bakeries have the same marginal cost function. At any given price, each bakery chooses to produce a quantity where the marginal cost is equal to the price. So at the total market quantity, the cost of one more loaf would be the same, whichever bakery produced it.
Equilibrium in the bread market
If we know the supply and demand functions for any market, we can find the competitive equilibrium price and quantity graphically at the point where they cross, as in Figure 8.11 in the main part of this section.
We can find the equilibrium mathematically by solving a pair of simultaneous equations—the market supply and demand curves—for the equilibrium values of P and Q.
If the demand and supply curves are expressed in terms of the direct demand and supply functions, Q=S(P) and Q=D(P), we can start by finding an equilibrium price—that is, a price that clears the market, equalizing the quantities demanded and supplied. The equilibrium price satisfies the equation:
S(P)=D(P)
If instead you start from the indirect supply function, in which price, P, equals marginal cost, which is a function of Q, you can find the equilibrium quantity first, by substituting for P in the demand function.
The demand curve is downward-sloping; that is to say, D(P) is a strictly decreasing function. Then, if S(P) is a strictly increasing function (like the one shown in Figure E8.3, and also in Figure 8.10 in the main part of this section), there is at most one equilibrium price (one crossing point). Having found the equilibrium price by solving this equation, the equilibrium quantity may be found by substituting the equilibrium price back into the supply or demand equation.
Remember that the slope of the supply function comes from the cost function: market supply is upward-sloping if marginal costs increase with quantity. What happens when all firms have the same, constant, marginal cost? As discussed in Section 8.7, this is what we might expect to happen in the long run when firms are able to increase capacity, and enter or leave the market. All firms in the market will have the same marginal cost, and no fixed costs. In this case, the market supply curve is flat at a price, P0, equal to the marginal cost.
If the supply curve is flat, you can’t interpret it as saying: ‘If the price is P then the quantity supplied is Q(P)′. Instead, we can say that for any price less than P0 the quantity supplied is zero and for any price greater than P0 the quantity supplied is undefined. (Technically it is infinite, but we would assume as a practical matter that the flat supply curve would eventually hit some kind of quantity constraints.)
But a flat supply curve presents no problems for finding the market equilibrium. The supply and demand curve are P=P0 and Q=S(P). Solving them simultaneously is straightforward: the equilibrium price and quantity are P=P0 and Q=D(P0). Market supply determines the price, and the demand determines the quantity.
You may be wondering what happens for the case in which costs decrease with quantity. The answer is that decreasing average costs (whether arising from fixed costs, or decreasing marginal costs) limit competition, as discussed in Section 7.11. Markets for goods with decreasing average costs cannot be in competitive equilibrium—they do not satisfy the conditions we used to derive the competitive market supply curve, S(P). Different models are required to analyse these cases.
Exercise E8.1 Equilibrium in markets with identical firms
Suppose there are 80 identical bakeries, each with the cost function, C(Q)=240+6q+2q2, where q refers to the number of loaves supplied by an individual bakery. The indirect market demand function is P=170–2Q, where Q refers to the total number of loaves in the market.
Find an expression for market supply. (Hint: Derive the supply function of an individual firm as a function of the market price.)
Find the equilibrium price, market quantity, and quantity supplied by each bakery.
Repeat Questions 1 and 2, but use the variable, m, to represent the number of bakeries in the market. Explain how the equilibrium price, market quantity, and quantity supplied by each bakery changes with m.
Exercise E8.2 Market equilibrium with linear functions
Suppose that the market demand and supply functions are both linear:
D(P)=a−bP,S(P)=c+dP
where a,b,c,d are constants. Assume that b>0,d>0,a>0, and a>c.
Explain the reasoning behind these assumptions.
Find expressions for equilibrium price and quantity, as a function of these constants. What condition is required for a market equilibrium with Q>0?
Read more (on inverse functions): Sections 7.4 of Malcolm Pemberton and Nicholas Rau. Mathematics for Economists: An Introductory Textbook (4th ed., 2015 or 5th ed., 2023). Manchester: Manchester University Press.
