Economics questions...

. Suppose that a market is composed of two firms that simultaneously choose quantities. Firm 1 has

22 a cost function C1(q1) = q1^2. Firm 2 has a cost function C2(q2) = (2q2)^2. The inverse demand equation

is

p(Q) = 100 ? Q Suppose first that firms are competing according to the Cournot model.

(a) Find the best response equation q1(q2) for firm 1. (5 points)

(b) Find the best response equation q2(q1) for firm 2. (5 points)

(c) Use your best response equations to mathematically solve for the equilibrium quantities q1?, q2?, Q?, pi1*, pi2*

equilibrium price p , and profit for each firm ?1 , ?2 . You may need to use the calculator for this part. Write the answers either as fractions or as numbers round to the second decimal place. (10 points)

Now assume the two firms are colluding. (d) Setup the joint profit maximization for the colluding firms (also known as the joint monopoly

or cartel problem). (6 points)

(e) Solve the collusion profit maximization problem for equilibrium quantities q1?, q2?, Q?, equilib

rium price p , and profit for each firm ?1 , ?2 . Suppose the firms split their joint profits in the following way: if ? is joint profits, firm 1 gets 0.65?, while firm 2 gets 0.35?. (10 points)

Suppose for simplicity that each firm can choose among only two levels of output: the Cournot level found in part (c), or the collusion level found in part (e). The objective is to write down a payoff matrix that represents the single period game in which each firm chooses the Cournot or the collusion quantity. To do that, you first need to:

(f) Find each firm's profits when firm 1 chooses the Cournot quantity while firm 2 chooses the collusion quantity. (7 points) (Hint: you need to first find aggregate quantity and price in order to derive profits).

(g) Find each firm's profits when firm 2 chooses the Cournot quantity while firm 1 chooses the collusion quantity. (7 points)

(h) Using the profits found in parts (c), (e), (f), and (g) as payoffs, write down a payoff matrix. Round profits to the closest integer. There should be two players, firm 1 and firm 2, and two possible actions for each player, namely producing the Cournot quantity and producing the collusion quantity. Solve for the Nash equilibrium in pure strategies. (10 points)

(i) Now assume that the firms play the previous game for infinitely many periods. In each period, they simultaneously choose quantities. They both discount future profits with a discount factor ?where0??

(i) Each firm produces the collusion quantity as long as the other firm does the same. (ii) Each firm produces the Cournot quantity in period t + 1 and thereafter if the other firm produces the Cournot output at time t. What is the value of ? such that collusion forever is an outcome of the repeated game? (12 points)

1. (27 points) For each of the following production functions, sketch a representative isoquant (2 points). Calculate the marginal product for each input, and indicate whether each marginal product is dimin- ishing, constant, or increasing (3 points). Calculate the marginal rate of technical substitution for each function (2 points). Also indicate whether the function exhibits constant, increasing, or diminishing returns to scale (2 points)- (a) F(L, K) = LK3 (b) F(L. K) = L+3K (c) F(L. K) = (min ( L. K)): 2. (21 points) Consider the production function /(L. K) = 214x1. (a) (15 points) Find the associated (long run) total, average, and marginal cost curves. (b) (6 points) Sketch the total, average, and marginal cost curves. 3. Problem removed due to copyright restrictions. This content is presented in audio form in the Solution Video for Problem Set 4, Problem 3. 4. (28 points) You run a cost-minimizing firm with production function /(L, K) = [min(L. A]], where L is labor and K is capital. Assume that you are a price-taker in the input markets: you pay w for each unit of labor you hire and r for each unit of capital (where w and r are set exogenously), and face no costs other than those from labor and capital. (a) (15 points) Assuming that you can freely choose both labor and capital (ie., the "long run prob- lem"), derive expressions for your cost-minimizing conditional input demands, L'(r, w, Q) and K* (r.w, Q). Confirm that the conditional input demand functions are "homogeneous of degree zero" in w and r; that is, L' (tr, fur, Q) = L'(r, IF, Q) (K* (tr, tu, Q) = K* (r, w.Q) for all t > 0 (b) (8 points) What will happen to your conditional demand for labor if there is an increase in the wage rate, assuming that r and @ remain the same? Explain in one sentence why your answer makes intuitive sense. (c) (5 points) Use your answers from (a) to write down an expression for your total cost function TC(r, w. Q). Is this function "homogeneous of degree one" in w and r; that is, does TC(tr. tu, () = . TC(r, w. Q)?2. Biased Growth and Trade There are two countries, Home and Foreign. There are two goods: Grains and Textiles. The preferences of the Home and Foreign consumers are described by the utility function: logxc + log.Xr. Each consumer in the Home country has a fixed endowment of 2 units of G and 1 unit of T. The consumers in the Foreign country have 1 unit of G and 2 units of 7. That is, the endowments are: eG = 2, er = 1 ed = 1,e; = 2 1. Show, by deriving from utility maximization, that the demand functions for the two goods by the Home consumer are given by: 1-(poec + prer) XT = 1 - (poec + prer) 2. Find the relative prices in autarky in the two countries, ( )", (" )". Show that 3. Now suppose there is free trade. Find the equilibrium relative price in the world market. What good is country Home exporting (importing)? Why? What are total exports of country Home? 4. Suppose there is growth in the textile sector in the Home country and the endowment of textiles becomes er = 4. All other endowments remain the same and there is still free trade. Find the new equilibrium price ( , )" in the world market with the new levels of endowment. 5. What does this mean for the terms of trade of country Home? What are the total exports of country Home after growth has taken place? Was growth import-biased or export-biased? Who benefits from growth? 6. Can you show that country Foreign is worse-off after growth (hint: show that after growth takes place country Foreign goes back to autarky consumption). Use a graphical illustration to make your point.1. Comparative Advantage Two countries, Home and Foreign, use one factor, labor, to produce two goods, Shoes and Computers. The Home country can produce Shoes with one unit of labor and Computers with two units of labor. The Foreign country can produce Shoes with 3 units of labor and Computers with 4 units of labor. Home country is endowed with a labor force of 200 units, while Foreign country is endowed with a labor force of 100 units. Preferences are the same in the two countries and are described by the following utility function: U(S, C) = InS+ InC 1. Which country has an absolute advantage in producing Shoes? Which country has an absolute advantage in producing Computers? 2. Which country has a comparative advantage in producing Shoes? Explain in words why this is the relevant information to determine the pattern of production in an economy where the two countries are allowed to trade. 3. What is the pattern of production and consumption in the Home country when in autarky? 4. What is the pattern of production and consumption in the Foreign country when in autarky? 5. Draw for both countries a graph with the production possibility frontier and the graphical solution to the maximization problem. 6. Now imagine the two countries are allowed to trade. Draw the world relative supply curve. For which range of prices will both countries specialize? What happens if the price is not in this range? 7. Now draw the world relative demand curve and find the world equilibrium price. Do both countries specialize? What is the pattern of trade? Does your answer correspond to your explanation for question 2.? 8. Explain what is the source of gains from trade in this problem.In this economy two countries. Home and Foreign. produce two goods. Ii'iomputel'tlt and lZ'tesks+ using three factors of production. skilled labor {H}. unskilled labor {L} and IIpitai (K). Skilled workers are self-employed. They rent capital at the rental rate rand produce computers according to the production function: 9.: = Hiya their income is given by the prots as = Page rK.-:. tinskilled workers are also self-employed. They rent capital and use it to produce desks according to the production function: 9;, = stag their income is given by the prots RD = PDQ rKD. The Home country is endowed with the following factor amounts: H = 9.1: = 2t]. L = 4. The Foreign country is endowed with the following factors: h" = t. a" = 2|], L' = 4. 1. Preferences are identical in the two countries and are described by the following utility function: J. a \"(Coral = CE CE Remember that in the economy all the factors are always fully employed. Assume for now that the two countries are in autarky. 1. Derive the production possibility frontier for both countries using the 4-quadrant graph seen in class. 2. Given prices of Computers and Desks. pr: and po. consider the Home country and nd the autaricyr equilibrium return to capital r; using the graph showing the "v'alue Marginal Product of Capital {p s MPK} for both goods. What is the allocation of I{Itapi'tal to the production of the two goods? 3. Now consider the Foreign country anti+ for given prices p.33- and pi) nd the retum to Ilpitai r1- using the same method as in part 2. 1li'ir'hat is the allocation of Capital to the production of the two goods? 1t. Take the Home country and consider an increase in the relative price g. What is the effect ofthis price change on the income of the skilled {R3} and unskilled workers [Fly]. 5. From parts 2 and 3 you are able to derive the relative supply of the two goods in both countries. Knowing preferences you can also derive the relative demand for each country. Using relative demand and relative supply nd for each country the equilibrium autarky relative price: (gid and {j}; . You should use [task as the numeraire good by setting its price to one. 1. Pure Exchange Economy In this problem we will go through the model scen in class using an example with specic assumptions about tastes and endowments. Hume and Foreign are endowed with different amounts of two goods, Apples and Bananas. The two countries are characterized by the following preferences. Home country's aggregate utility function is: J. J. U=rjr while Foreign cuuntIy's aggregate utility mctiun is: .1. ,1 Lu" =xfx3' The two countries' endowments are the following: Home has lllf'.I Apples and 't' Bananas while Foreign has 5ft Apples and El] Bananas. Let's start by looking at what happens if the two countries are in autarky and then we will analyze what happens if the two countries are allowed to trade. As for all general equilibrium problems you have to think in terms of relative prices, therefore 'ocm now on we are interested only in the relative price of Apples relative to Bananas p = g which is equivalent to assuming that we are taking Bananas as our numeraire.p3 = l. 1. For the Home country draw a graph with the quantity of the two goods on the two axes and explain why at the consumption point the MRS must be equal to the autarky relative price. 1tIti'hat is the meaning of autarky relative price since there is no exchange allowed with other countries? The autarky prices are in fact determined by two factors: what are those factors? Find the autarky price p in the Home country. 2. For the Foreign country do the same analysis and draw a graph showing what determines relative prices in autarky. Find the autarky price p' in the Foreign country. 3. Now compare autarky prices in the two countries. In which country is the price of Apples higher relative to the price of Bananas? What factors determine the different autarky prices? In particular in this example why is p E p'? 4. Now imagine the two countries are allowed to trade. Similarly to what we saw in class, draw an Edgeworth box. where the indifference curves for the two countries are drawn in the same graph. Show where the endowment point is located. Show also the "lens" area that cointains points which are preferred to the original endowment by both countries. 5. Derive the demand for both goods for both countries and nd the equilibrium relative price at which the two countries exchange Apples for Bananas. pr. What are MR3 and MM' equal to in equilibrium