Partial Equilibrium (Economics) PDF
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This document discusses partial equilibrium in economics, covering market demand and supply curves. It includes examples using Cobb-Douglas functions and explains how to derive market demand curves from individual consumer demands, and short-run and long-run market supply.
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12 Partial Equilibrium Throughout the class, we have studied the consumer’s problem and the firm’s problem separately. Now it is time to start to connect consumers and firms. This connection will lead us to a key concept in economics: equilibrium. In these notes, we wil...
12 Partial Equilibrium Throughout the class, we have studied the consumer’s problem and the firm’s problem separately. Now it is time to start to connect consumers and firms. This connection will lead us to a key concept in economics: equilibrium. In these notes, we will focus on the idea of partial equilibrium, which will look at one market in isolation while holding all other markets fixed. In future classes we will look at general equilibrium, which explores interactions between markets. Concepts Covered The Market Demand Curve The Short Run Market Supply Curve Short Run Equilibrium Long Run Equilibrium The Long Run Supply Curve 12.1 The Market Demand Curve Definition In the first part of the class, we discussed how to derive a consumer’s Marshallian Demand. As a reminder, the Marshallian demand for a good x is a function that tells us how much a consumer will purchase to maximize their utility, as a function of prices and income. x∗m = x(px , py , I) So far, we have always phrased the problem as if there were only one consumer making this purchasing decision. In reality, there will always be many consumers, potentially with different utility functions, buying each product. To find the total demand for the entire market, we need to add all of these demands. Let’s denote the total demand for good x as XD. Then n X XD = xi (px , py , Ii ) i=1 where n is the number of consumers in the economy. Note that while consumers can have different demand functions and different incomes, we assume they all face the same set of prices. If we wanted to plot the market demand curve for good x, all we need to do is plot the market demand XD for every possible px. This process gives us a quantity demanded for each price. 1 Example Assume there are 3 consumers in the economy that all have the same Cobb-Douglas utility function U (x, y) = x1/2 y 1/2 They all face the same prices for x and y but have different incomes. In particular, assume I1 = 50, I2 = 100, and I3 = 150. Let’s find the market demand curve for good x. First, we can find Marshallian demand for consumer i as 1 Ii xim = 2 px Plugging in incomes and summing across consumers gives us a market demand of 1 I1 + I2 + I3 150 XD = = 2 px px One thing to point out here is that Cobb-Douglas (and any homothetic preference) allows for easy aggregation of consumers. In particular, notice that the distribution of income does not affect market demand at all. All that matters is the total amount of income in the economy. For example, if we plugged in I1 = 100, I2 = 100, and I3 = 100, we would get exactly the same market demand. 12.2 The Short Run Market Supply Curve Definition Now let’s turn to the supply side. Recall from the last notes that the short run supply curve is the portion of the marginal cost curve that lies above the average variable cost curve. The individual supply of a producer of good x depends on the price of good x and the price of labor and capital. Denote the market supply by Xs. Then the market supply is described by n X XS = Qix (px , w, v) i=1 where here n is the number of firms in this market. Since we will stick to our assumption of perfect competition, the firm takes prices as given (including wage and rental rate) so the only potential difference among firms would be their production functions. Example Two firms produce in a market (even though it’s only 2 we will assume they are price takers). They both have the same production function Q = F (K, L) = K 1/2 L1/2 2 Assume capital is fixed in the short run and each firm has different amounts of capital. Firm 1 has K1 = 40 and Firm 2 has K2 = 20. Assume the wage is w = 20 We want to find the market supply curve for good x. First, we have that the contingent demands for labor are Q2 L∗ = K Plugging this into cost gives our cost function Q2 C(Q) = vK + w K Marginal cost is ∂C(Q) Q = 2w ∂Q K Note that Q AV C = w K is always less than marginal cost. Therefore the supply curve is given by setting P=MC Qx K px = 2w =⇒ Qx = px K 2w Summing across our two firms and plugging in the wage, we would get the market supply curve px XS = (K1 + K2 ) = (3/2)px 40 12.3 Short Run Equilibrium An equilibrium occurs when the market supply is equal to the market demand. In other words, an equilibrium in the market for good x will occur when X D = XS Using the example above, that implies 150 3 = px px 2 This equation is solved when px = 10. Therefore, p∗x = 10 is the short run equilibrium price for x. We can then plug this price into either supply or demand to get the equilibrium quantity X ∗ = 15. If we wanted to, we could then go back and find the individual supplies and demands for each firm and consumer. From above, an individual’s consumer demand was 1 Ii xim = 2 px Plugging in the incomes from above and the equilibrium price, we get that x1 = 5/2, x2 = 5, x3 = 15/2 (summing these up again gives 15, confirming we calculated the market demand correctly). On the firm side, we have individual supply (with the wage plugged in) K Qx = px 40 Plugging in the capital for each firm and the price gives us Q1x = 10, Q2x = 5 3 12.4 Long Run Equilibrium In the last notes, we talked about the firm’s entry and exit decision. In particular, we talked about how a firm’s decision to stay in the market depended on whether they were making positive profit or not. We calculated the cutoff and said that the firm would stay in the market as long as P = M C > AC However, as firms enter and exit the market, the equilibrium price changes. For example, imagine firms are currently making positive profit. In the long run, entrepreneurs will realize there are profit opportunities and enter the market. But this entry will cause the supply curve to shift right, pushing down the price. As the price falls, profit for all the firms in the market goes down. This process continues until P = M C = AC and profits equal zero. On the other hand, if firms were making negative profit, they would start to exit the market. But firm exit would shift the market supply curve left, pushing up the price and increasing profit for all firms in the market. Once again, this process would continue until P = M C = AC and profits equal zero. For example, imagine we had the same market demand curve as above 150 XD = px But now assume firms producing good x all have a cost curve given by C(Qx ) = Q2x + 25 We know each firm producing good x will want to set px = M C = 2Q, so Q = 12 px for each firm. Then we could write the market supply curve as N XS = px 2 where N is the number of firms in the market. Solving for the equilibrium price 1/2 N 150 2 300 ∗ 300 XS = XD =⇒ px = =⇒ px = =⇒ px = 2 px N N We can see that the equilibrium price is decreasing as the number of firms increases. Now let’s look at an individual firm’s profit. π = px ∗ Qx − Q2x − 25 We have argued that the cutoff point for a firm to stay in the market occurs at 25 px = M C = AC =⇒ 2Qx = Qx + =⇒ Qx = 5 =⇒ px = 10 Qx The equilibrium price above will be equal to this cutoff point when 1/2 300 = 10 =⇒ N = 3 N So this market would have a long run equilibrium with 3 firms, each producing 5 units and selling at a price of 10. 4 12.5 The Long Run Supply Curve In the long run, we argued above that profits will always be equal to zero in a competitive market. However, the shape of the long run supply curve (the quantity produced given a price in the long run) depends on the characteristics of the market. Even when marginal costs (and therefore the short run supply curve) are upward sloping, the long run supply curve can be flat, increasing or decreasing. Its shape will depend on the effect of firm entry on average costs for the industry as a whole. Long Run Supply with Constant Cost In the example we did on the previous page, we implicitly assumed the average cost curve does not change as firms enter the market. This assumption means that no matter how many firms enter or exit the market, the equilibrium price in the long run will always be equal to 10. The number of firms will change in response to demand shifts to keep the long run price at a constant level. In that example, let’s say demand shifted to become 1/2 500 px = N This change would cause firms to enter the market. If this entry didn’t affect the average cost function, we still have a cutoff price px = 10. So then the new equilibrium would have 5 firms, still each producing a quantity of 5 at a price of 10. The total quantity produced goes up, but the price doesn’t change. Therefore, the supply curve in the long run is just a horizontal line. Long Run Supply with Increasing Cost In some industries, we expect that firm entry would cause average costs to increase. For example, imagine some industry requires a lot of natural resources to operate (agriculture for example). As more firms enter and use up the scarce resources available, the price of these resources goes up, increasing costs for all firms in the market. As the average cost curve shifts up, the cutoff price for firms to earn 0 profit at any given quantity also increases. Therefore, as demand shifts right, the equilibrium price and quantity both increase together, marking an upward sloping long run supply curve. Long Run Supply with Decreasing Cost On the other side, there might be some industries where having more firms in the market actually pushes down average costs. One reason for decreasing costs could be that as an industry gets large, firms might have better access to various markets as transportation and communication networks improve for that industry. In this case, we will have the cutoff zero-profit price falling as quantity increases, leading to a downward sloping long run supply curve. 5