Problem 3. (Technology) a) What can you say about the returns to scale (chose: IRS, CRS or DRS) and MPK (chose: increasing, constant, or decreasing). (2pt) The technology exhibits decreasing returns to scale (DRS) and the MPK is decreasing. b) Find the (variable) cost function c(y) given that the prices of inputs are wx = wL = 4.5 (give a function). (5pt) First, to find the optimal labor-to-capital ratio, set MAK = x MPK _LILI L MPL KIKI K WK _4 = 1 45 *= 1 = L =K Then, express the total cost as a function of one of the inputs by using L = K C = wxK + WIL = 4.5K + 4.5L = 4.5K + 4.5K = 9K (12) Now, express output y as a function of that same input and use equation (6) to derive the final answer: y = 3K LI = 3KIKI = 3k] = K= = = C(y) = 9K = y c) Find the supply function of the individual competitive firm and plot it in the graph (give the formula, in the graph mark the prices for which the market will not open). (4pt) To find its optimal output, the firm maximizes its profit function with respect to y: n(y) = TR - TC = py - 16 - yz ay on = p - 2y =0 = y= 5 To find the prices for which the firm would not produce because it makes negative profits, we need to find 1/MES. MC = ATC _ 16 2y = -+1 1/ 1MES =4 = ATCMES = 8 Then, the supply function is given by: y(p) = ifp 2 8 ifp and there are two states of the world which are equally likely. Find the certainty equivalent and the expected value of lottery (0, 2) (two numbers). Which of the two is bigger (choose one)? Explain why (one sentence). (4pt) The expected (von Neumann-Morgenstern) utility function is U(1, 12) = ]x + 13. The certainty equivalent (CE) of the lottery (0,2) solves the following equation: UI(CE, CE) = U(0,2) -CER + -CE? = =07 + -22 CEZ = 2Problem 1. (Consumer Choice) Jeremy's favorite flowers are tulips ri and daffodils 12. Suppose pi = 2, p2 = 4 and m = 40. a) Write down Jeremy's budget constraint (a formula) and plot all Jeremy's affordable bundles in the graph (his budget set). Find the slope of a budget line (number). Give an economic interpretation for the slope of the budget line (one sentence). b) Jeremy's utility function is given by U (n, 12) = \\(Inz1 + Inz2)? +7 Propose a simpler utility function that represents the same preferences (give a formula). Explain why your utility represents the same preferences (one sentence). c)' Plot Jeremy's indifference curve map (graph), find MRS analytically (give a formula) and find its value at bundle (2,4) (one number). Give economic interpretation of this number (one sentence). Mark its value in the graph. d) Write down two secrets of happiness (two equalities) that allow determining the optimal bundle. Pro- vide their geometric interpretation (one sentence for each). Find the optimal bundle (11, 12) (two numbers). Is your solution interior? (a yes -no answer) e) Hard: Find the optimal bundle given pi = 2, pa = 4 and m = 40 assuming U (21, 12) = 2x1 + 3x2 (two numbers). Is your solution interior? (a yes -no answer) Problem 2. (Producers) Consider production function given by F(K, L) = 3kit. a) Using the A argument demonstrate that production function exhibits decreasing returns to scale. b) Derive the cost function given wx = wy = 9. c) Derive a supply function of a competitive firm, assuming the cost function from b) and fixed cost F = 2 (give a formula for y(p)). Plot the supply function in a graph, marking the threshold price below which a firm chooses inaction. Problem 3. (Competitive Equilibrium) Consider an economy with apples and oranges. Dustin's endowment of two commodities is given by w = (8,2) and Kate's endowment is wh = (2,8). The utility functions of Dustin and Kate are the same and given by where i = D, K. a) Plot the Edgeworth box and mark the point corresponding to the initial endowments. b) Give a general definition of Pareto efficient allocation r (one sentence) and state its equivalent con- dition in terms of MRS (one sentence, you do not need to prove the equivalence). ) Using the "MRS" condition verify that the initial endowments are not Pareto efficient. d) Find a competitive equilibrium (six numbers). Provide an example of a competitive equilibrium with some other prices (six numbers). e) Using MRS condition verify that the competitive allocation is Pareto efficient. f) Hard: Find prices p1. py in a competitive equilibrium for identical preferences of two agents U (F1, T2) = 2x1 +3:2 (two numbers, no calculations). Explain why any two prices that give rise to a relative price higher than pa/p2 cannot be equilibrium prices (which condition of equilibrium fails?) 'If you do not know the answer to b), to get partial credit in points c)-e) your can assume U (21, 22) = Problem 4. (Short Questions) a) Uncertainty: Find the certainty equivalent of a lottery which, in two equally likely states, pays (0, 9). Bernoulli utility function is u(c) = vc (one number). Is the certainty equivalent smaller or bigger than the expected value of a lottery 4.5. Why? (one sentence) b) Market for lemons: In a market for racing horses one can find two types of animals: champions (Plums) and ordinary recreational horses (Lemons). Buyers can distinguish between the two types only long after they buy a horse. The values of the two types of horses for buyers and sellers are summarized in the table Lemon Plum Seller Buyer 5 Are champions (Plums) going to be traded if probability of Lemons is - (yes-no). Why? (a one sentence argument that involves the expected value of a horse to a buyer) ) Signaling: The productivity of high ability workers (and hence the competitive wage rate) is 1000 while productivity of low ability workers is only 400. To determine the type, employer can, first offer an internship program with the length of a months, during which a worker has to demonstrate her high productivity. A low ability worker by putting extra effort can mimic high ability performance, which costs him c(2) = 200z. Find the minimal length z for which the internship becomes a credible signal of high ability. (one number) d) PV of Perpetuity: You can rent an apartment paying 1000 per month (starting next month, till the 'end of the world") or you can buy the apartment for 100.000. Which option are you going to chose if monthly interest rate is r = 2%? (find the PV of rent and compare two numbers) Problem 5. (Market Power) Consider a monopoly facing the inverse demand p (y) = 25 -y, and with total cost TC(y) = 5y. a) Find the marginal revenue of a monopoly, MR (y) and depict it in a graph together with the demand (formula +graph). Which is bigger: price or marginal revenue? Why? (one sentence) b) Find the optimal level of production and price (two numbers). Illustrate the optimal choice in a graph, depicting Consumer and Producer Surplus, and DWL (three numbers +graph). c) Find equilibrium markup (one number). d) First Degree Price Discrimination: Find Total Surplus, Consumer, Producer Surplus and DWL if monopoly can perfectly discriminate among buyers and quantities. (four numbers +graph) e) Hard: find the individual level of production and price in a Cournot-Nash equilibrium with N identical firms with cost TO(y) = 5y. both as a function of N (two formulas). Argue that the equilibrium price converges to the marginal cost as NV goes to infinity.Problem 1. Ace consumes bananas I, and kiwis 12. The prices of both goods are p1 = 4, p2 = 10 and Ace's income is m = 120. His utility function is U (21, 12) = (11)"(12)"0 a) Find analytically Ace's MRS as a function of (21, 12) (give a function) and find its value for the consumption bundle (21, 12) = (20, 20). Give its economic and geometric interpretation (one sentence and find MRS on the graph) b) Give two secrets of happiness that determine Ace's optimal choice of fruits (give two equation). Explain why violation of any of them implies that the bundle is not optimal (one sentence for each condition). c) Using magic forumula find the optimal bundle of Ace (two numbers), and show geometrically the . Problem 2. Adria collects two types of rare coins: Jefferson Nickels 21 and Seated Half Dimes 22. Her utility from a collection (21, 12) is U (21, 12) = 21 + 12 a) Propose a utility function that gives a higher level of utility for any (21, 12), but represents the same preferences (give utility function). b) Suppose the prices of the two types of coins are p1= 4 and p2= 2 for $1, 12 respectively and the Adria's income is m = $20. Plot her budget set and find the optimal collection (21, 72) and mark it in your graph (give two numbers) c) Are the coins Giffen goods (yes or no and one sentence explaining why)? d) Harder: Suppose Adria's provider of coins currently has only six Seated Half Dimes 12 in stock (hence 12 5 6). Plot a budget set with the extra constraint and find (geometrically) an optimal collection given the constraint. Problem 3. (Equilibrium) There are two commodities traded on the market: umbrellas r, and swimming suits r2. Abigail has ten umbrellas and twenty swimming suits (w= (10, 20) ). Gabriel has forty umbrellas and twenty swimming suits (w= (40, 20)). Abigail and Gabriel have identical utility functions given by U' (21, 12) =5 In (x1) +5 In(12) a) Plot an Edgeworth box and mark the point corresponding to endowments of Abigail and Gabriel. b) Give a definition of a Pareto efficient allocation (one sentence) and the equivalent condition in terms of MRS (equation). Verify whether endowment is Pareto efficient (two numbers tone sentence). c) Find prices and an allocation of umbrellas and swimming suits in a competitive equilibrium and mark it in your graph. d) Harder: Plot a contract curve in the Edgeworth box assuming utilities for two agents U. (21, 12) = min($1, 12) - Problem 4.(Short questions) a) You are going to pay taxes of $200 every year, forever. Find the Present Value of your taxes if the yearly interest rate is r = 10% b) Consider a lottery that pays 0 with probability , and 16 with probability , and a Bernoulli utility function is u (r) = VT. Give a corresponding von Neuman-Morgenstern utility function. Find the certainty equivalent of the lottery. Is it bigger or smaller than the expected value of the lottery? Why? (give a utility function, two numbers and one sentence.) c) Give an example of a Cobb-Douglass production function that is associated with increasing returns to scale, decreasing MPK and decreasing MPL (give a function). Without any calculations, sketch the average total cost function (ATC) associated with your production function. d) Let the variable cost be c(y) = y' and fixed cost F = 4. Find ATCOMES and youES (two numbers). Given demand D (p) = 8 - p determine a number of firms in the industry assuming free entry (and price taking). Is the industry monopolistic, duopolistic, oligopolistic or perfectly competitive? Find Herfindahl- Hirschman Index (HHI) of this industry (one number)- e) In a market for second-hand vehicles two types of cars can be traded: lemons (bad quality cars) and plums (good quality ones). The value of a car depends on its type and is given by Lemon Plum Seller 0 20 Buyer 10 26 Will we observe plums traded on the market if the probability of a lemon is equal to ? (compare two relevant numbers). Is the equilibrium outcome Pareto efficient (yes-no answer+ one sentence)? Give a threshold probability for which we might observe pooling equilibrium (number). Problem 5.(Market Power) Consider an industry with the inverse demand equal to p (y) =6-y, and suppose that the total cost function is TC = 2y. a) What are the total gains to trade in this industry? (give one number) b) Find the level of production and the price if there is only one firm in the industry (i.e., we have a monopoly) charging a uniform price (give two numbers). Find demand elasticity at optimum. (give on number) Illustrate the choice using a graph. Mark a DWL. c) Find the profit of the monopoly and a DWL given that monopoly uses the first degree price discrimi nation. d) Find the individual and aggregate production and the price in a Cournot-Nash equilibrium given that there are two firms (give three numbers). Show DWL in the graph. e) In which of the three cases, (b.c or d) the outcome is Pareto efficient? (chose one+ one sentence)Problem 4.(Short questions) a) You are going to pay taxes of $20 every year, forever. Find the Present Value of your taxes if the yearly interest rate is r = 10%. b) Consider a lottery that pays 0 with probability , and 4 with probability - and a Bernoulli utility function is u (x) = r-. Give a corresponding von Neuman-Morgenstern utility function. Find the certainty equivalent of the lottery. Is it bigger or smaller than the expected value of the lottery? Why? (give a utility function, two numbers and one sentence.) c) Give an example of a Cobb-Douglass production function that is associated with increasing returns to scale, increasing MPK and decreasing MPL (give a function). Without any calculations, sketch the average total cost function (ATC') associated with your production function. d) Suppose the cost function is such that ATCES= 2 and yofES= 1 and the demand is D (p) = 4 - p. Determine a number of firms in the industry given the free entry (and price taking). Is the industry monopolistic, duopolistic, oligopolistic or perfectly competitive? Find Herfindahl-Hirschman Index (HHI) of this industry (one number). e) In a market for second-hand vehicles two types of cars can be traded: lemons (bad quality cars) and plums (good quality ones). The value of a car depends on its type and is given by Lemon Plum Seller 0 20 Buyer 10 26 Will we observe plums traded on the market if the probability of a lemon is equal to ? (compare two relevant numbers). Is the equilibrium outcome Pareto efficient (yes-no answer+ one sentence)? Give a threshold probability for which we might observe pooling equilibrium (number). Problem 5.(Market Power) Consider an industry with the inverse demand equal to p (y) =6 -y, and suppose that the total cost function is TO = 0. a) What are the total gains to trade in this industry? (give one number) b) Find the level of production and the price if there is only one firm in the industry (i.e., we have a monopoly) charging a uniform price (give two numbers). Find demand elasticity at optimum. (give on number) Illustrate the choice using a graph. Mark a DWL. c) Find the profit of the monopoly and a DWL given that monopoly uses the first degree price discrimi nation. d) Find the individual and aggregate production and the price in a Cournot-Nash equilibrium given that there are two firms (give three numbers). Show DWL in the graph. e) In which of the three cases, (b.c or d) the outcome is Pareto efficient? (chose one+ one sentence) Problem 6.(Externality) A bee keeper chooses the number of hives h. Each hive produces ten pounds of honey which sells at the price of $2 per pound. The cost of holding h hives is TC (h) = th. Consequently the profit of bee keeper is equal to The (h) = 2h- -h2 The hives are located next to an apple tree orchard. The bees pollinate the trees and hence the total production of apples y = h + t is increasing in number of trees and bees. Apples sell for $5 and the cost of t trees is TC (t) = t'. Therefore the profit of an orchard grower is *1 (t ) = 5 (t + h) -5+2 a) Market outcome: Find the level of hives h that maximizes the profit of a beekeeper and the number of trees that maximizes the profit of an orchard owner (assuming h optimal for a bee keeper) (two numbers) b) Find the Pareto efficient level of h and t. Are the two values higher or smaller then the ones in a)? Why? (two numbers + one sentence)