Eco 600E: Advanced Microeconomics I (Spring, 1st, 2011)
Final Exam: June 5
1. True or False (15 points)
Answer whether each of the following statements is true (T) or false (F). You do NOT need to explain the reason.
(a) Suppose % is represented by utility function u( ). Then, u( ) is concave ONLY IF % is convex.
(b) Marshallian demand of some good CAN be increasing in its own price.
(c) It can NEVER happen that a production function shows both (strictly) de- creasing and increasing returns to scale.
(d) If the relative risk aversion of some risk averse decision maker is independent of her wealth, then her absolute risk aversion MUST be decreasing in wealth. (e) The competitive equilibrium in exchange economy ALWAYS exists when all
agents have monotone preferences. 2. Consumer Theory (20 points)
Consider a consumer who has monotone preferences. Let v(p; !) and e(p; u) denote her indirect utility function and expenditure function, respectively.
(a) Show that v(p; !) is strictly increasing in !.
(b) State the de…nition of normal goods, either verbally or mathematically. (c) Show that good j is a normal good if and only if @p@2e
j@u >0.
Hint: You can use the duality, xj(p; !) = xhj(p; v(p; !)). 3. Producer Theory (25 points)
A …rm has a production function given by f (x1; x2; x3; x4) = minfx
1 3
1x
2 3
2; x3+ 2x4g.
Let w = (w1; w2; w3; w4) be factor prices and y be an (target) output.
(a) Does the production function exhibit increasing, constant or decreasing returns to scale? Explain.
(b) Calculate the conditional input demand function for factors 1 and 2. (c) Suppose w3 > w24. Then, derive the cost function c(w; y).
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4. Uncertainty (20 points)
Consider the following three lotteries, L1; L2 and L3: L1 :
50 dollars with probability 12
150 dollars with probability 12 L2 :
100 dollars with probability 23 200 dollars with probability 13
L3 :
8
< :
50 dollars with probability 13 150 dollars with probability 59 300 dollars with probability 19
Answer the following questions:
(a) Suppose that a decision maker prefers for sure return of 90 dollars rather than L1. Then, can we conclude that she is (i) risk averse or (ii) not risk loving? Explain.
(b) Show that any risk averse decision maker whose preference satis…es indepen- dence axiom must prefer L2 to L3.
5. General Equilibrium (30 points)
Ann and Bob are stranded on a desert island. Each has some slices of ham (H) and cheese (C). More precisely, suppose that Ann initially had 40H and 80C and Bob had 60H and 120C. The utility function for Ann and Bob are given by uA= min(H;C2) and uB = 4H + 3C.
(a) Draw the Edgeworth box diagram and indicate the contract curve in the box. (b) Derive the competitive equilibrium (both price ratio and allocation). Who does become strictly better o¤ by the transaction in the competitive market? (c) Show that competitive equilibrium price (ratio) is independent of initial en-
dowments.
The allocation is called “envy-free” if ui(xi) ui(xj) holds for every pair of people fi; jg where xi is i’s consumption bundle and xj is j’s consumption bundle.
(d) Derive an envy-free and Pareto e¢cient allocation in our problem. Show that it is unique.
(e) Now suppose that the government tries to achieve the allocation in (d) through the transfer of initial endowment of ham (H), combined with the market trans- action afterwards. How much slices of ham should the government transfer from Bob to Ann?
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