Microeconomics II Teaching Materials This file contains a summary of notes for each lesson, and the exercises to be solved in the classes. Rafael Moner Colonques Departamento de Análisis Económico Universitat de València Year 2012-2013
Lesson 1. Technology Chapter 18 in Varian s textbook is the first chapter concerned with the basic theory of supply. Before proceeding it may be beneficial that you first review the derivation of demand. The review can be beneficial given the similarities between the theory of demand and the theory of supply. It is important to take the time to carefully go through the definitions, as this will be the foundation for what is done in the next two lessons. While the concept of a production function is not difficult, the mathematical and graphical representation can sometimes be confusing. Inputs, or factors of production (labour, capital, and materials) are combined to produce output using the current state of knowledge about technology. To maximize profits a firm must produce as efficiently as possible: it must get the maximum amount of output from the inputs it uses, given existing knowledge. A firm may have access to many efficient production processes that use different combinations of inputs to produce a given output (technical efficiency). A production function shows how much output can be produced efficiently from various levels of inputs. A firm can vary all its inputs in the long run but only some of them in the short run. Graphing the production function leads naturally to a discussion of marginal product and diminishing returns. Note that diminishing returns exist because some factors are fixed by definition, and that diminishing returns does not mean negative returns. Isoquants are defined and discussed. As with indifference curves, isoquants are a two-dimensional representation of a three-dimensional production function. Key concepts in this section of the chapter are the marginal rate of technical substitution and returns to scale. You should be aware of the most common production functions, which will be used throughout the course, as well as and the properties of the technology. Although all firms in an industry produce efficiently, some firms may be more productive than others. They can produce more output from a given combination of inputs. It may be due to innovations, such as technical progress or new means of organizing production; such innovations change the production function.
Problems. 1. Prunella raises peaches. Where L is the number of units of labour she uses and T is the number of units of land she uses, her output is f(l, T) = L 1/2 T 1/2 bushels of peaches. a) Plot some input combinations that give her an output of 4 bushels. Sketch a production isoquant that runs through these points. b) What type of returns to scale does the production function of peaches exhibit? c) In the short run, Prunella cannot vary the amount of land she uses. Consider T =1. Draw the short run output function curve. What is the marginal product of labour at the combination (1,1). Draw the marginal product of labour curve. d) Suppose that Prunella increases the size of her orchard to 4 units of land. Draw a new curve showing output as a function of labour. What happens to the marginal product of labour? 2. Consider the following production functions: 1/ 3 3 x 1/ 1/ 3 4 4 a) x 1/ 3/ 1 2x2 b) x1 2x2 c) x1 x2 d ) 1 x2 And say whether they exhibit constant, increasing or decreasing returns to scale. 3. Suppose that the production function is f(x 1, x 2 ) = (x a 1 + x a 2) b, where a and b are positive constants. For what positive values of a and b are there decreasing returns to scale? And increasing returns? 4. In a production process is it possible to have decreasing marginal product in an input and yet increasing returns to scale? 5. If each extra worker produces an extra unit of output, how do the total product of labour, average product of labour, and marginal product of labour vary with labour? 6. To produce a recorded CD, q=1, a firm uses one blank disk, D=1, and the services of a recording machine, M=1, for one hour. Draw an isoquant for this production process. Explain the reason for its shape.
Lesson 2. Profit maximization Chapter 19 in Varian s textbook describes how the firm chooses the amount of output to produce in order to maximize profits. In this chapter, we assume that prices of inputs and outputs faced by firms are given. Profits are defined as revenues minus costs; revenues are equal to price times the output produced and this is given by the production function, the topic explained in the previous lesson. Costs are equal to the sum of the costs associated to each input (price times the amount of the input employed). We will distinguish short-run and long-run profit maximization. An iso-profit line contains all the production plans that provide a certain profit level. Use of iso-profit lines will be made to illustrate the profit maximizing condition: the value of the marginal product of a factor should equal its price. In fact, such condition defines the short-run demand for an input. Comparative statics allow us to establish what happens when the input price changes and when the output price changes. Thus, we firstly observe that there s an inverse relationship between the amount of an input and its price. Secondly, that there s a positive relationship between the amount of output and its price hence, the supply function slopes upwards. When in the long-run, the firm is free to choose the level of all inputs. The solution to the long-run profit maximization problem leads to the factor demand curves. The chapter closes by relating returns to scale and profit maximization to show that increasing returns to scale are incompatible with firms being perfectly competitive.
Problems 1. If a firm had everywhere increasing returns to scale, what would happen to its profits if prices remained fixed and if it doubled its scale of operation? 2. The short-run production function of a competitive firm is given by f(l) = 6L 2/3, where L is the amount of labour it uses. The cost per unit of labour is w = 6 and the price per unit of output is p = 3. a) Plot the production function. Draw the iso-profit lines that pass through points (0,12), (0,8) and (0,4). b) How many units of labour will the firm hire? How much output will it produce? If the firm has no other costs, how much will its total profits be? c) Suppose that the wage of labour falls to 4, and the price of output remains at p. On the graph, draw the new iso-profit line that passes through the old choice of input and output. Will the firm increase its output at the new price? Why? 3. A Valencia firm uses a single input to produce a recreational commodity according to a production function f(x) = 4 x, where x is the number of units of input. The commodity sells for 100 per unit. The input costs 50 per unit. (a) Write down a function that states the firm s profit as a function of the amount of input. b) What is the profit-maximizing amount of input? And output? How much profits does the firm make? c) Suppose that the firm is taxed 20 per unit of its output and the price of its input is subsidized by 10. What is its new input level? And output? And profits? (d) Repeat the computations supposing instead that the firm is taxed at 50% of its profits. 4. A firm s production function is given by q = 3L + 4L 2 L 3. The wage w is equal to 1. a) Obtain the expressions for the average and the marginal product of labour. b) Which is choice of L that maximizes profits? Do the computations for p = 1/7 and for p = 1/3. Represent graphically. 5. A firm only employs labour to produce output q according to the following technology L = (37/3)q 4 q 2 + q 3. Labour is hired in a competitive market at price w = 1. Characterize the firm s profit maximizing behaviour. Calculate the output produced when market price p is equal to 8 and when it equals 10.
Lesson 3. Cost minimization and cost curves Key topics in Varian s chapters 20-21 are: cost minimization, graphically and mathematically, and the derivation of conditional factor demand curves, definitions of total, average, and marginal cost in the short run and long run, a graphical representation of total, average, and marginal cost, and the relationship between returns to scale and the cost function. It is important to spend time on the cost curve definitions and graph because they form the foundation for what will be covered in the coming lessons. The cost minimization problem is useful for explaining which inputs the firm should use to produce a given quantity of output, and this discussion draws on the discussion of isoquants from chapter 18. It is also important to understand the basic concept of hiring input until the input price is equal to the marginal product of the input, and then obtain the conditional demand for the input (what quantity of the inputs should the firm use to produce a given level of output). At this point, you should be able to distinguish demand for an input and conditional demand for an input. While the definitions of total cost, fixed cost, average cost, and marginal cost and the graphical relationships between them can seem tedious and/or uninteresting to the student, both are important in terms of understanding the derivation of the firm s supply curve and the industry supply curve in lesson 4 (chapters 22-23). Each firm has a unique set of cost curves based on its own particular production function and resulting total cost function. Note the importance of returns to scale and diminishing returns in explaining the shapes of the cost curves. Average cost tends to be u-shaped in the short run and marginal cost will hit average cost and average variable cost at their respective minimum points. See the problems solved in lesson 2 about finding the profit maximizing level of output, noting that the firm may well not produce at all. Further note that the necessary condition for cost minimization, where the ratio of the marginal products is equal to the ratio of the input costs, is very similar to the necessary condition for profit maximization. A clear understanding of short-run cost and cost minimization (long-run) is necessary for the derivation of long-run average cost. With long-run costs, you must understand that firms are operating on short-run cost curves at each level of the fixed factor and that long-run costs do not exist separately from short-run costs. Observe the connection between the shape of a long-run cost curve and returns to scale.
Problems 1/3 2/3 1. A firm s Cobb-Douglas production function is y f( x1, x2) x1 x2 ; input prices are w 1 and w 2. What are the firm s conditional input demand functions? What is the optimal expansion path? What is the firm s total cost function? 2. If a firm is producing where MP 1 /w 1 > MP 2 /w 2, what can it do to reduce costs but maintain the same output? 3. Nadine sells user-friendly software. Her firm s production function is f(x 1, x 2 ) = x 1 + 2x 2, where x 1 is the amount of unskilled labour and x 2 is the amount of skilled labour that she employs. a) Draw the isoquant for output y = 20. What kind of returns to scale does the production function exhibit? b) If Nadine faces factor prices (w 1 =1, w 2 =1), what is the cheapest way for her to produce 20 units of output? What would happen if (w 1 =1, w 2 =3)? c) If Nadine faces factor prices (w 1, w 2 ), what will be the minimal cost of producing y units of output? 4. The Ontario Brassworks produces brazen effronteries. As you know brass is an alloy of copper and zinc, used in fixed proportions. The production function is given by: f(x 1, x 2 ) = min{x 1, 2x 2 }, where x 1 is the amount of copper it uses and x 2 is the amount of zinc that it uses in production. a) Draw an isoquant and tell the type of returns to scale this technology has. b) If the firm faces factor prices (1, 1), what is the cheapest way for it to produce 10 effronteries? How much will this cost? c) If the firm faces factor prices (w 1,w 2 ), what will be the minimal cost of producing y effronteries? 5. Consider the cost function c(y) = y 2 +1. Write down the expressions for the variable costs function, fixed costs function, AVC, AFC, AC and MC. Draw in the same diagram the AVC curve, the AC curve and the MC curve.
Lesson 4. Firm supply and industry supply. We begin the analysis of firm conduct in settings where firms do not have market power. Chapters 22 and 23 identify the behaviour of the firm and the industry in a competitive market, respectively. It is also important to distinguish both the short-run and the long-run analyses. The basic assumptions of perfect competition are: there is a large number of sellers, homogeneous product, information is perfect, and there are no entry or exit barriers. The first three assumptions imply price-taking behaviour: firms act as if they can sell or buy as much or as little as they want without affecting the price. Note that the perfectly competitive firm chooses quantity and not price in order to maximize profit. To establish how the firm should choose the optimal quantity to produce, you will struggle with the cost curve diagrams presented in the previous lesson. In particular, you must understand the implications of i) the price equal marginal cost condition (first order condition), ii) the upward-sloping part of the marginal cost curve (second order condition), and iii) that the firm finds it profitable to produce something (shutdown condition). The price equal marginal cost condition determines the supply function of the firm. Then, and given a certain market price, it is easy to assess whether the competitive firm makes positive profits, or losses but produces a positive amount of output, or it shuts down. The situation where the firm remains in business despite making losses can only occur in the short run. The first half of this lesson ends with the derivation of the long run supply curve: the portion of the long run marginal cost curve above the long run average cost curve. We then move on the analysis of industry equilibrium. The summation of firm supply curves into a market supply curve is straightforward, but the analysis of long-run competitive equilibrium is not easy. Difficult concepts include: why it may be optimal for the firm to incur losses in the short run but not the long run. why free entry and exit will reduce economic profit to zero in the long run. why price is equal to minimum average cost in the long run. how the long-run supply curve is constructed. The long-run industry equilibrium is given by a triplet (q, p, n), the amount of output produced by each firm, the market price, and the number of firms in the industry.
Problems 1. A firm has a cost function given by c(y) = 10y 2 + 1000. a) What is its supply curve? What is its inverse supply curve? b) At what output is average cost minimized? 2. Dent Carr, who is in the auto repair business, found that the total cost of repairing s cars is c(s) =2s 2 + 100. a) Write average total cost, average variable cost and marginal cost. Plot them in a graph. What is Dent s long-run supply curve? b) If the market price is 20, how many cars will Dent be willing to repair? And if price is 40? c) Suppose the market price is 40 and Dent maximizes his profits. On the graph, shade in and label the following areas: total costs, total revenue, and total profits. 3. A competitive firm has the following short-run cost function: c(y) = y 3 8y 2 + 30y + 5. Write the marginal cost function, the average variable cost function, and plot them in a graph. How much output will the firm produce if price is equal to 12? At what price would the firm supply exactly 6 units of output? 4. Mr. McGregor owns a 5-acre cabbage patch. His wife, Flopsy, and his son, Peter, work in the cabbage patch without wages. The only input that Mr. McGregor pays for is fertilizer. If he uses x sacks of fertilizer, the amount of cabbages that he gets is10 x. Fertilizer costs 1 per sack. a) What is the total cost of the fertilizer needed to produce 100 cabbages? What is the total cost of the amount of fertilizer needed to produce y cabbages? Write an expression for his marginal cost, b) If the price of cabbages is 2 each, how many cabbages will Mr. Mc-Gregor produce? How many sacks of fertilizer will he buy? How much profit will he make? 5. Consider a competitive industry with a large number of firms, all of which have identical cost functions c(y) = y 2 + 1 for y > 0 and c(0) = 0. Suppose that initially the demand curve for this industry is given by D(p) = 52 p. a) What is the supply curve of an individual firm? If there are n firms in the industry, what will be the industry supply curve? What is the smallest price at which the product can be sold?
b) What will be the equilibrium number of firms in the industry? What will be the equilibrium price? What will be the equilibrium output of each firm? What will be the equilibrium output of the industry? c) Now suppose that the demand curve shifts to D(p) = 52.5 p. What will be the equilibrium number of firms? d) Now suppose that the demand curve shifts to D(p) = 53 p. What will be the equilibrium number of firms? 6. Suppose you are given the following information about a particular industry: Q D 6500 100P Market demand Q S 1200P Market supply C(q) 722 q2 Firm total cost function 200 MC(q) 2q Firm marginal cost function. 200 Assume that all firms are identical, and that the market is characterized by pure competition. a) Find the equilibrium price, the equilibrium quantity, the output supplied by the firm, and the profit of the firm. b) Would you expect to see entry into or exit from the industry in the longrun? Explain. What effect will entry or exit have on market equilibrium? c) What is the lowest price at which each firm would sell its output in the long run? Is profit positive, negative, or zero at this price? Explain.
Lesson 5. Monopoly. This lesson examines market power. We begin with a discussion of monopoly. In previous lessons the general rule of profit maximization has been presented. Still, it would be useful that you refresh the concepts of marginal revenue and elasticity of demand. The equality of marginal revenue to marginal cost (first order condition) determines the monopolist choice of output and price. We can present this condition to illustrate mark-up pricing and to note how the exercise of market power varies with the elasticity of demand. The linear case is developed, as well as the effect of an ad-valorem and a per unit tax. We conclude that the monopolist operates where the demand curve is elastic (because marginal revenue is positive at the profit maximizing choice), and that the more price inelastic the demand the higher the mark-up. For example, if the elasticity is large (e.g., because of close substitutes), then (1) the demand curve is relatively flat, (2) the marginal revenue curve is relatively flat (although steeper than the demand curve), and (3) the monopolist has little power to raise price above marginal cost. Using the equality of marginal revenue to marginal costs allows us to discuss how much of a tax is passed on to consumers. The comparison of perfect competition with monopoly establishes the inefficiency of monopoly. It permits to see the conflict of interests between consumers and firms. The deadweight loss of a monopolized market is a measure of the inefficiency associated with market power. However, there are certain goods supplied by only one firm, like public utilities. We then have a natural monopoly: it arises when the firm s technology has economies-of-scale large enough for it to supply the whole market at a lower average total production cost than is possible with more than one firm in the market. If governments forced the monopolist to price at marginal costs then it would make losses and would leave the market. There is the alternative of imposing price equal to average cost. The objective of every pricing strategy is to capture as much consumer surplus as possible. The lesson discusses first, second, and third degree price discrimination. Note the requirements for profitable price discrimination: (1) supply-side market power, (2) the ability to separate customers, and (3) differing demand elasticities for different classes of customers. Make sure that you fully understand third degree price discrimination: price paid by buyers in a given group is the same for all units purchased. But price may differ across buyer groups. The market with the higher price must have the lower elasticity of demand. This lesson finishes with a brief discussion of monopolistic competition, a market structure where firms have some market power but profitable entry drives profits to zero.
Problems 1. There is only one producer in the market for masks, which can produce at constant average cost AC = MC =10. In the beginning, the firm faces the following demand curve Q = 60 p. a) What are the profit maximizing price and quantity for this firm? Calculate firm s profits. b) Suppose now that the demand curve becomes steeper, and is given by Q = 45 0.5p. What are now the profit maximizing levels of output and price? Calculate firm s profits. c) Repeat your calculations if the demand curve becomes Q = 100 2p. d) Graph parts a-c and discuss. 2. Consider a perfectly competitive industry that produces aromatic candles at a constant marginal cost of 10 per unit. Later the industry has been monopolized and marginal costs increase to 12 per unit (because politicians require 2 per nit for the monopolist to get a license). Suppose market demand for aromatic candles is given by Q=1000 50p. a) Calculate equilibrium price and output both under perfect competition and monopoly. b) Calculate the loss in consumer surplus due to the monopolization in aromatic candles. c) Represent graphically and discuss. 3. Suppose that the demand function for Japanese cars in the United States is such that annual sales of cars (in thousands of cars) will be 250 2P, where P is the price of Japanese cars in thousands of dollars. (a) If the supply schedule is horizontal at a price of $5,000, what will be the equilibrium number of Japanese cars sold in the United States? How much money will Americans spend in total on Japanese cars? b) Suppose that in response to pressure from American car manufacturers, the United States imposes an import duty on Japanese cars in such a way that for every car exported to the United States the Japanese manufacturers must pay a tax to the U.S. government of $2,000. How many Japanese automobiles will now be sold in the United States? At what price will they be sold? (c) How much revenue will the U.S. government collect with this tariff? Represent graphically. d) Suppose that instead of imposing an import duty, the U.S. government persuades the Japanese government to impose voluntary export restrictions on their exports of cars to the United States. Suppose that the Japanese agree to restrain their exports by requiring that every car exported to the United States must have an export license. Suppose further that the Japanese government agrees to issue only 236,000 export licenses and sells these licenses to the Japanese firms. If the Japanese firms know the American
demand curve and if they know that only 236,000 Japanese cars will be sold in America, what price will they be able to charge in America for their cars? e) How much will a Japanese firm be willing to pay the Japanese government for an export license?) f) How much will be the Japanese government s total revenue from the sale of export licenses? How much money will Americans spend on Japanese cars? g) Why might the Japanese voluntarily submit to export controls? 4. a) A monopolist faces a demand curve given by D(p)=10p -3. Its cost function is c(y)=2y. What is its optimal level of output and price? b) What is the answer if D(p)=100/p? 5. Professor Margin has just written a new book, Microeconomics. His publisher estimates that the demand for this book in Spain Q 1 = 50, 000 2,000p 1, where p 1 is the price in Spain measured in euros. The demand for Margin s opus in England is Q 2 = 10, 000 500p 2, where p 2 is its price in England measured in euros. His publisher has a cost function, in, C(Q) = 50, 000 + 2Q, where Q is the total number of copies of Microeconomics that it produces. (a) If the publisher must charge the same price in both countries, how many copies should it sell? What price should it charge to maximize its profits? How much will those profits be? (b) If the publisher can charge a different price in each country and wants to maximize profits, how many copies should it sell in Spain? What price should it charge in Spain? How many copies should it sell in England? What price should it charge in England? How much will its total profits be? c) At the profit-maximizing price and quantity, what is the price elasticity of demand in the Spanish market? And in the English market? 6. A firm in monopolistic competition has the following short-run cost function TC=4y 2 10y + 576, and the inverse demand function it faces is given by p=278 32y. a) Calculate a firm s short-run equilibrium. Will there be entry in the long run? b) What s the long-run equilibrium? Is a firm operating with excess capacity? Represent graphically.
Lesson 6. Oligopoly This is probably the first time you see such a number of models presented in one single lesson. Lesson 6 discusses five models, and you should be able not only to tell the difference between them but also to compare them with the competitive and the monopoly market structures presented in the two previous lessons. We consider the classic models of oligopolistic competition and these are static models. Under oligopoly, firms have market power. When maximizing their profits, firms may choose quantity or price. We will then say that their strategic variables are either quantities or prices. Furthermore, firms may make their choice either simultaneously or sequentially. This leads to four combinations, to four models of non-cooperative behaviour. The Cournot model corresponds with simultaneous quantity competition whereas the Stackelberg model corresponds with sequential quantity competition. On the other hand, we have the Bertrand model where oligopolists choose prices simultaneously, whereas the price leadership model assumes that one firm plays before the others do and chooses price. It is important that you understand the concept of reaction function (or best response function). This will allow you to see the Cournot-Nash equilibrium on a graph. We will then follow and study the Stackelberg model. Emphasis will be put on the linear case example and symmetric firms. We will conclude that a firm prefers (higher profits) to be a Stackelberg leader, then a Cournot oligopolist, and then a Stackelberg follower; consumers are better off in a sequential model of quantity competition. If we consider that firms simultaneously compete in prices then we have the Bertrand model concluding that, in equilibrium, price equals marginal cost. Thus, though output is positive firms do not make any profits thus reaching the so called Bertrand paradox: firms have market power but we obtain the competitive equilibrium. This setting is obviously the best from consumers perspective. The variation where one of the firms sets price in advance, and such price is accepted by competitive followers is the price leadership model. But firms might do better if they collude. We will study cooperative behaviour of firms that choose quantity to maximize joint profits. Rather naturally, we get back to the results under monopoly but you must make sure to understand that collusive behaviour (cartels) are not stable, in the sense that a firm has an incentive to cheat and deviate from the agreement.
Problems 1. The inverse market demand curve for bean sprouts is given by P(Y ) = 100 2Y, and the total cost function for any firm in the industry is given by TC(y) = 4y. a) If the bean sprout industry were perfectly competitive, what would industry output and price be? b) Suppose that two Cournot firms operated in the market. How much does each firm produce in equilibrium? What is industry price and profits? Graph the Cournot equilibrium in reaction-function space. c) If the two firms decided to collude, what would industry output and market price be? d) Suppose both of the colluding firms are producing equal amounts of output. If one of the colluding firms assumes that the other firm would not react to a change in industry output, what would happen to a firm s own profits if it increased its output by one unit? e) Suppose one firm acts as a Stackelberg leader and the other firm behaves as a follower. How much output would the leader produce in equilibrium? And the follower? What is equilibrium market price? And profits? 2. Two Cournot duopolists produce in a market with demand p=100-q. The marginal cost for firm is constant and equals 10. The marginal cost for firm 2 is also equal and it equals 25. The two firms want to merge. They argue for the merger on the grounds that marginal production costs would fall to 10 for all units of output after the merger. Given this information, would you recommend the merger? Explain why calculating the benefits and costs from the merger? 3. Consider an industry with the following structure. There are 50 firms that behave in a competitive manner and have identical cost functions given by c(y) = y 2 /2. There is one monopolist that has zero marginal costs. The demand curve for the product is given by D(p) = 1, 000 50p. The monopolist is a leader in setting the price. What is the supply of the competitive sector? What is the monopolist s profit maximizing price? How much does it produce? How much does the competitive sector produce? 4. Demand for light bulbs can be characterized by Q = 100 - P, where Q is in millions of lights sold, and P is the price per box. There are two producers of lights: Everglow and Dimlit. They have identical cost functions: C 1 2 2 i 10Qi Qi for i=e,d, and Q = Q E + Q D.
a) Unable to recognize the potential for collusion, the two firms act as shortrun perfect competitors. What are the equilibrium values of Q E, Q D, and P? What are each firm s profits? b) Top management in both firms is replaced. Each new manager independently recognizes the oligopolistic nature of the light bulb industry and plays Cournot. What are the equilibrium values of Q E, Q D, and P? What are each firm s profits? c) Suppose the Everglow manager guesses correctly that Dimlit has a Cournot conjectural variation, so Everglow plays Stackelberg. What are the equilibrium values of Q E, Q D, and P? What are each firm s profits? d) If the managers of the two companies collude, what are the equilibrium values of Q E, Q D, and P? What are each firm s profits? 5. Suppose two firms that produce cars which face the inverse demand curve given by p = 82 4Y. Their cost functions are the following: C ( y ) 60 16 y and C ( y ) 90 16y 1 1 1 2 2 2 a) Calculate the Cournot equilibrium (individual output, price and profits). Graph the equilibrium quantities in reaction function space. b) Calculate the Stackelberg equilibrium (individual output, price and profits) if firm 1 acts as the leader. Indicate the equilibrium in the previous graph. c) Calculate the equilibrium when both firms can collude. Will firms always agree?