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Unformatted text preview: Economics 3010
Fall 2009 Professor Daniel Benjamin
Cornell University Problem Set 7 Solutions 1. (Monopoly: Menu costs and price rigidity) [This question is drawn directly from the classic paper: Mankiw, N. Gregory (1985), “Small Menu Costs and Large Business Cycles: A Macroeconomic Model of Monopoly,” Quarterly Journal of Economics, 100, 529‐537. (Mankiw is a professor at Harvard and the author of the currently most popular introductory economics textbook. He writes an engaging economics blog, and he recently served as Chairman of President Bush’s Council of Economic Advisors.) If you are interested, feel free to take a look at the original paper: http://www.economics.harvard.edu/faculty/mankiw/files/Small_Menu_Costs.pdf ] Consider a monopolistically competitive firm that has a constant marginal cost of production of k. Therefore, the firm’s cost function is c(q) = kq, where q is quantity of output produced. Suppose the firm faces a standard downward‐sloping demand function, p(q) = f(q). where f’ < 0. We will think of the firm as choosing its price pm to maximize profit, and then producing however much output it takes to meet demand, qm =f‐1(pm). (Recall that because the demand curve is a one‐to‐one function, this is mathematically equivalent to considering the firm as choosing an output level qm and letting the price adjust to whatever level equates quantity supplied and quantity demanded, pm = f(qm).) 1 (a) The figure below shows (for a linear demand curve) consumer surplus and profit at the profit‐
maximizing pm and qm. a DWL
c pc= b qc Why is the marginal cost curve depicted as a horizontal line at k? Why is producer’s surplus exactly equal to profit in this case? What region in the figure corresponds to deadweight loss? From the cost function, c(q) = kq, marginal cost (MC) is ∂c(q)/ ∂q = k, which is constant. In Chapter 22, we’ve seen that producer’s surplus is equal to revenues minus variable costs. In this example with the cost function of c(q) = kq, revenue – variable cost = profit of the firm. Therefore, producer’s surplus is equal to profit. If the firm was under perfect competition with the same cost function, the price (denoted by pc in the picture above) would be determined at pc = MC which is equal to k and the output would be at qc. Hence, the total surplus under perfect competition (which is actually entirely the consumer surplus as the supplier’s profit is zero under perfect competition) is the area of the triangle [pkc]. However, under monopoly, the total surplus (consumer’s surplus plus producer’s surplus) is the area of the trapezoid [pabk]. Therefore, the deadweight loss (DWL) is the difference between [pabk] and [pkc], which is the area [abc]. 2 (b) Recall from our discussion of general equilibrium that supply and demand for goods (including both output goods and factors of production like labor and capital and land) determine price ratios, but the price level will depend on the amount of currency (i.e., dollars) in circulation. Let N represent the price level in the economy. Now we can write the nominal cost function (the cost function denominated in dollars) as C(q) = c(q) × N = kqN and the nominal demand function as P(q) = p(q) × N = f(q)N. Explain why the firm’s nominal profit is given by Π = q f(q)N kqN. Without actually maximizing this function, show that the profit‐maximizing level of output qm (and therefore the profit‐maximizing price pm) does not depend on N. Π = nominal revenue – nominal cost = q*P(q) – C(q) Substituting C(q) = c(q) × N = kqN and P(q) = p(q) × N = f(q)N, we have Π = q*P(q) – C(q) = q f(q)N – kqN The profit maximizing pm and qm is from MR = MC. From Π = q* f(q)N – kqN, MR = MC is q f’(q)N – kN = 0, i.e. regardless of N, the profit maximizing q satisfies q f’(q) – k = 0. (c) The firm must post its price in terms of dollars. That is, it sets the nominal price Pm but cannot directly set the real price pm (i.e., the price relative to other prices in the economy). Instead, the real price is determined indirectly as Pm / N. Hence the real price is partly under the control of the firm and partly depends on whatever N happens to be. If the firm knows what N is (and N remains constant over time), and if the profit‐maximizing real price is pm, explain why the firm will set its nominal price equal to pmN. Real price (pm) = Pm / N Hence, if the profit maximizing real price is pm and the firm knows exactly what N is in the economy, the nominal price (Pm) should be equal to pmN. (d) Suppose that the price level is initially at the level N’, and so the firm initially sets its nominal price at the profit maximizing level, pmN’. However, the price level unexpectedly falls to N0 < N’. (This would happen if there was a fall in “aggregate demand” in the economy as a whole. For example, this might occur if major banks failed, and firms and consumers all cut back their spending at the same time!) What is the firm’s new profit‐maximizing nominal price? If changing prices were costless, explain why the fall in the price level would not affect the level of output. (Economists call this the case of flexible prices.) 3 In (c), we showed that the profit maximizing nominal price is profit maximizing real price times the corresponding N. Hence, when N changes to N0, the new profit maximizing nominal price is pmN0. Since N0 < N’, pmN0 < pm N’, i.e. new nominal price should be lower. If changing prices were costless, the firm would change its nominal price, but its real price – and hence its output level – would remain constant. (e) Unfortunately, the firm must pay a so‐called menu cost to change its price. In the case of a restaurant, this could literally be the cost of printing up new menus (hence the term “menu cost”). More generally, the menu cost could represent the difficulty of figuring out the new demand curve, calculating the new optimal price, or (in the case of price increases) dealing with the anger of customers. Suppose the firm chooses not to incur the menu cost and hence does not change its price when the price level falls. Explain why the real price is now p0 = pmN’ / N0. Is this larger or smaller than the profit‐maximizing real price? Due to the menu cost, when the price level unexpectedly falls to N0 < N’, the nominal price does not immediately adjust to pmN0 (the profit maximizing nominal price we got in part d). Instead, the nominal price tends to remain at pmN’ in the short‐run. Since the real price is nominal price divided by N in the economy, while this is the case, the real price (p0) becomes pmN’ / N0. Since N0 < N’, pmN’ / N0 is greater than the profit‐maximizing real price (p0 = pmN’ / N0 and p0 > pm). (f) The below figure illustrates how the firm’s situation changes when the price level changes. a c b d 4 Suppose the menu cost is z > 0. Explain why it is socially efficient for the firm to change its price whenever B + C > z. Explain why the firm will actually change its price whenever B – A > z. Conclude that if the fall in the price level is such that B + C > z > B – A, then the firm does not change its price, even though doing so would be socially optimal. (Formally, when the change in N is small, it can be proven that the private benefit to the firm of changing its price is “second‐order” (i.e., small), while the social benefit is “first‐order” (i.e., large). In that sense, even if menu costs are “small,” they can have a “large” negative effect on social welfare. This excessive tendency for firms to keep their prices constant in the face of changing economic conditions is called price rigidity, also called price stickiness.) In (e), we’ve seen that when the price level unexpectedly falls to N0 < N’, the real price will increase to p0. If the firm pays the “menu cost”, it can set its real price back to pm. With this price change, the additional total surplus is B + C (difference between [pcdk] and [pabk]). Hence, it is socially efficient if the firm changes its price whenever B + C > z. However, this adjustment changes the firm’s profit by B – A (difference between the area [pmcdk] and the area [p0abk]). Therefore, the firm will actually change its price only if B – A > z. Hence, if it happens to be B + C > z > B – A, the firm will not change its price even though doing so is socially efficient. [ Note: Price rigidity plays a major role in explaining business cycles in macroeconomic analysis. If prices were flexible, then you have shown that there would be no “real” effects (i.e., no effects on production or consumption) of changes in the price level. But if prices are sticky, then a negative shock to the price level causes a decline in output. In addition to the welfare losses in partial equilibrium you worked out above, there are important general equilibrium consequences. When the price level falls but a firm does not cut its price, the relative price of other goods in the economy is higher than they otherwise would be, which reduces demand for their products (and hence also reduces demand for inputs to production, like labor and capital). This is called an aggregate demand externality because the firm does not take this effect on the rest of the economy into account when it decides whether or not to cut its price. ] 5 ...
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