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Practice Questions for Final

Subject: Advanced Microeconomics I (ECO600E) Professor: Yosuke YASUDA

1. True or False

Answer whether each of the following statements is true (T) or false (F). You do NOT need to explain the reason.

(a) Marshallian demand of some good is ALWAYS decreasing in its own price. (b) Gi¤en’s paradox occurs ONLY for inferior goods.

(c) Expected utility theory sais that each decision maker chooses a lotteries with the highest expected monetary return.

(d) vNM utility unique up to monotone increasing transformation. 2. Homothetic Function

A function f (x) is homothetic if f (x) = g(h(x)) where g is a strictly increasing function and h is a function which is homogeneous of degree 1. Suppose preferences can be represented by a homothetic utility function. Then, show the followings.

(a) The marginal rate of substitution between any two goods depends only on the ratio of the demands consumed. That is M RSij is identical whenever xxi

j takes

the same value.

(b) The cross price derivatives of Marshallian demands are identical,

@xi(p; I)

@pj =

@xj(p; I)

@pi . 3. Consumer Theory

A consumer’s utility function is given as

u(x; y) = min( x; y)

where ; > 0. Let p; q > 0 be the prices for good x and y respectively. Then, answer the following questions.

(a) Derive the Marshallian demand functions. (b) What is the indirect utility function?

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4. Duality

Consider the indirect utility function given by v(p1; p2; !) =

! p1+ p2

. (a) What are the Marshallian demand functions? (b) What is the expenditure function?

(c) What are the Hicksian demand functions? 5. Properties of Value Functions

(a) Suppose the utility function is continuous and strictly increasing. Then, show that the associated indirect utility function v(p; !) is quasi-convex in (p; !). (b) Show that the (minimum) expenditure function e(p; u) is concave in p.

(c) A real-valued function f ( ) is called superaddittive if f(x1+ x2) f(x1) + f (x2).

Show that every cost function is superadditive in prices. Use this property to prove that the cost function is nondecreasing in input prices.

6. Cost Minimization Problem

A …m can rent capital (K) at a rental price r and hire labor (L) at a wage w. To produce anything at all requires one unit of capital, i.e. r 1 = r is a …xed cost; this is sunk in the short run, but not sunk in the long run. If in a unit of time the …rm employs L units of labor, and rents K units of capital (in addition to the one unit needed as a …xed cost), its output Q is given by one of the following production function:

Q= K1=4L1=8 Then, answer the following questions.

(a) In the short run, the …rm is committed to hire a …xed amount of capital K(+1), and can vary its output Q only by employing an appropriate amount of labor L. Derive the …rm’s short-run total, average, and marginal cost functions. (b) In the long run, the …rm can vary both capital and labor. Derive the …rm’s

long-run total, average, and marginal cost functions. 7. Expected Utility

Suppose that an individual can either exert e¤ort or not. The cost of e¤ort is c. Her initial wealth is 100. Her probability of facing a loss 75 (that is, her wealth becomes 25) is 13 if she exerts e¤orts and 23 if she does not. Her wealth will not

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change with the rest of probability in each scenario. Let u(x) be her vNM utility function.

(a) Express her expected utilities in each scenario, i.e., exerting e¤ort or not, by using u(x). You can assume that her expected utility is additively separable between e¤ort cost and (probabilistic) monetary outcome, i.e., E[u(x)] c. (b) Assume u(x) = px. For what values of c will she exert e¤ort?

8. Risk Aversion

Consider the following vNM utility function, u(x) = + ln(x).

(a) What restrictions must be placed on parameters and for this function to express risk aversion?

(b) Given the restrictions derived in (a), show that u(x) displays decreasing ab- solute risk aversion.

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