Econ 101A Midterm 2 Th 8 April 2009.

Transcription

1 Econ A Midterm Th 8 Aril 9. You have aroximately hour and minutes to answer the questions in the midterm. I will collect the exams at. shar. Show your work, and good luck! Problem. Production (38 oints). Consider a farmer that roduces corn using labor. The labor cost in dollar to roduce y bushels of corn is c (y) =Cy, with C>. There are identical farms which all behave cometitively. Corn sells at a rice er bushel.. Derive the marginal cost c (y) and the average cost c (y) /y. Plot them. Derive grahically the suly curve. (Have the quantity y on the horizontal axis). (5 oints). Write the exression for the suly of corn y () for each firm, and derive the exression for the aggregate suly function Y S () (5 oints) 3. From now on, suose that the demand curve of corn is D () = 5 and assume C =. Derive the equilibrium rice and total quantity sold Y S. (5 oints) 4. Derive the rofits of each firm. (5 oints) 5. Now assume that land is not free, and that the farmers are renting the land from a monoolist (that is, there is only one land owner, and it is imossible to rent land somewhere else). How much will the land-owner charge each of the farms? Exlain. (8 oints) 6. Taking the rent of the land into account, how much are the rofits of each firm? (4 oints) 7. In what sense there is a arallel between the resence of rents in the land and the entry of firms in the long-run equilibrium? (6 oints) Solution of Problem.. c (y) =Cy and c (y) /y = Cy. Since marginal cost is always larger than average cost, the suly curve is the marginal cost curve. See grah in aendix.. The condition = c (y) imlies = Cy or y () = C The aggregate suly function, summing across all firms, is 3. The equilibrium is and y () = X j= C = 5 C y () = 5 = D () = 5 (5 + 5) = = Y S = y() = 5 =

2 4. Since Y S =,eachfirm roduces unit and earns rofit π = y c (y )= = 5. Since land does not enter the corn roduction function, the land rental rice (r) isafixed cost the farmer must ay to oerate. Fixed costs do not change the farmer s rofit-maximizing choice of y, but if rofit isnegativeatthaty, the farmer will choose not to roduce at all (exit the industry). In this case, given the rofits calculated above, no farmer will ay more than $ for land. Every farmer will ay any rice less than or equal to $, however, because oerating will still be rofitable. So the monoolist faces a land demand curve that is erfectly inelastic for r : ( if r, D(r) = if r>. Given this demand, the monoolist maximizes his own rofits by setting r =: the monoolist extracts all the farmers rofit as rent. 6. Each farmer s rofit will be zero. 7. Both drive farmers rofits to zero, though in the case of rents the surlus is taken by the land-owner while in the case of erfect cometition, it becomes consumer surlus.

3 Problem. Consumer Surlus (35 oints) We evaluate here the change in consumer surlus associated with a change in the rice of good from to for a consumer with Cobb-Douglas utility u (x,x )=x α x α, with <α<.. Exlain the intuition of why the change in consumer surlus is defined as CS = e(,,u) e (,,u), where e is the exenditure function. (6 oints). Define the exenditure function. (4 oints) 3. Derive an exression for the exenditure function e(,,u) for this Cobb-Douglas case given that the Hicksian demands in this case are h (,,u) = u ( α α ) α (5 oints) h (,,u) = u ( α α )α 4. Solve for the change in consumer surlus CS = e(,,u) e (,,u). (5 oints) 5. If you also substituted for u the exression for the indirect utility v (,,M), you would get that CS is roortional to µ α M () (Do not attemt to do this, it involves a fair amount of algebra, take it as given) Using exression (), show that the following statements are true, and rovide intuitive exlanations for each of them: (i) CS > if and only if < ; (ii) holding constant and with <, CS is increasing in α; (iii) holding constant and with <, CS is increasing in M. (5 oints) Solution of Problem.. e(,,u) is the minimum amount of money required to attain the initial utility level u at the new rice of good ( ). Subtracting this from the exenditure that is required before the rice change (e(,,u)) gives the maximum amount the consumer would be willing to ay to enact the rice change. If the consumer were to ay this amount (ayment may be ositive or negative), then she would reach an identical utility level after the rice change. In the absence of this comensating ayment, she will achieve either greater utility (if < ) or less utility (if > i ), but we can monetize the value of this gained/lost utility as the amount she would ay to get it.. The exenditure function is defined as: OR: e(,,u)= min h,h h + h s.t. u(x,x ) u e(,,u)= h + h where h and h are the quantities of goods and that minimize sending while attaining utility level u. 3

4 3. e(,,u) = h + h µ α µ α = u + u α α α α " µ α µ # α α α = u α α + α α = u α α α α ( α) α 4. Using the exression found above, CS = e(,,u) e (,,u) = u α α α ( α) α ( α α ) By the fundamental theorem of calculus, the same exression can be found by integrating e(,,u) h over the range α to. 5. If we take the revious exression and substitute µ α µ α α α u = v (,,M)= M, we obtain CS = µ α µ α α = α M ( α α )=M α M α µ α α ( α) α α (You were not required to show this) (i) Proof of CS > < : ( α α )= Given that CS is roortional to the exression in (), its sign is the sign of ³ α quantity is ositive if and only if <. < /α <. = h ³ αi. This Intuition: Given Cobb-Douglas utility, the consumer is buying a ositive quantity of x at the original rice. After a rice decrease, s/he will be able to urchase the same bundle for less money and use the leftover money to buy more goods: the increase in utility that follows corresonds to an increase in CS. (ii) Proof that CS is increasing in α: d CS dα h ³ αi d M dα ³ α d = M = M dα µ α µ ln ³ Given that <, ln <. Therefore, the quantity above is ositive and d CS dα >. Intuition: α is the share of income sent on x.whenmorex is urchased, the consumer has more to gain from a decrease in. (iii) Proof that CS is increasing in M: d CS dm h ³ αi d M dm = µ α From (i), this quantity is ositive when >. Intuition: A consumer with higher M buys a larger quantity of x at, and thus gets more extra" income when falls to. 4

5 Problem 3. Uncertainty (45 oints). A first consumer has an exected utility function of the form u (w) = w. She initially has a wealth of $4. She has a lottery ticket that will be worth $4 with robability / and will be worth $ with robability /. What is her exected utility? What is the lowest rice she is willing to accet to sell her ticket? ( oints). A second consumer has an exected utility function of the form u (w) =ln(w). Afriendthatisfond of gambling offers him the oortunity to bet on the fli of a coin that has robability π of coming u heads. If he bets $x, hewillhavew + x if head comes u and w x is tails comes u. Notice that x has to be non-negative (x ). Provide a definition of risk-aversion and show that the agent is risk-averse (or not). (6 oints) What is his exected utility? (4 oints) Solvefortheotimalx as a function of π. Discuss the qualitative features of the solution (8 oints) In articular, discuss the otimal x for π</ and for π>/. Draw a arallel with the case of investment in risky asset that we saw in class. (8 oints) True or not true. Exlain as recisely as you can. Given that the agent is risk-averse, he will not bet in the lottery unless the lottery has a substantially ositive exected value (9 oints) Solution of Problem 3.. Exected utility with lottery ticket: E[U] = u(4 + 4) + u(4) = = 3 + = 3 + Utility of selling at rice is: u(4 + ) = 4+. She will sell the ticket if this utility is greater than the exected utility of keeing the ticket or (squaring both sides) = +3. Risk aversion means E[U(x)] < U(E(x)) for any robabilistic outcome x, meaning a risk-averse agent will always reject a fair bet. Risk-aversion results from diminishing marginal utility: the utility that is gained from winning $y is less than the utility that is lost when losing $y. We can show an agent is risk averse by showing that her utility function is concave. Check for u(x) =ln(x): u (x) = x > for all x. u (x) = x < for all x. u(x) is concave, so the agent is risk-averse. E[U] =π ln(w + x)+( π)ln(w x) 5

6 Solving for x (π): max π x ln(w + x)+( π)ln(w x) s.t.x f.o.c. π w + x π w x = x =max{w(π ), } s.o.c. π (w + x ) π (w x ) < for all x, sox is a maximum when x > (when w(π ) <, a negative bet would be utility-maximizing, but this is not allowed). x is roortional to w, so the amount a consumer wants to ut at risk is a constant fraction of her wealth. It is also increasing in π conditional on π>/, which means she will take larger bets as the robability of winning (and thus, the exected value of the bet) increases. x is strictly ositive whenever π>/, otherwise it is. Parallel to the examle we saw in class on risky investments: In allocating savings between a risky and safe investment, any risk-averse erson will ut a strictly ositive (albeit small) amount in the risky investment as long as its exected value is strictly greater than the certain value of the safe investment. Here, a risk averse agent will always articiate in the bet if π>/, whichis equivalent to the bet having strictly ositive exected value. The statement is not true. It is shown above that even if π is only very slightly greater than /, bringing the exected value of the lottery arbitrarily close to zero, the agent will still bet a ositive amount of money. The amount will certainly get smaller as the exected value of the lottery aroaches zero, but it will remain ositive. This result does not rely on the articular utility function, because any continuous function is locally linear; thus, for small enough changes in wealth, a risk-averse agent will behave as if he is risk-neutral. 6