Unformatted text preview: University of California, Davis Department of Economics Microeconomics Date: June 21, 2004 Time: 4 hours Reading Time: 20 minutes PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE
This exam is designed to be done in 3 hours. The extra 1 hour is intended to remove the time constraint, to make the process a little more relaxed. Please try to keep your answers brief and to the point; you are not expected to write a 4 hour exam! Good luck. Please answer four of the five questions 1. Inequality and luxuries It is sometimes asserted that an increase in the income inequality of a country, other things being equal, will increase the country's demand for luxuries. To scrutinize this claim, we consider a society with I individuals and two goods. The price vector (p1, p2) is fixed. All individuals have the same preferences, but their wealth levels may vary as long as they do not fall below a given minimum admissible wealth level 0 . We postulate that, for each consumer, given (p1, p2) and wi > , good 2 is a luxury,1 and both goods are normal. Denote by (x1(wi), x2(wi)) the (smooth) Walrasian demand function of a consumer with wealth wi (wi > ). (a). Using the above notation, define "luxury" in the sentence "for each consumer, good 2 is a luxury, given (p1, p2) and wi > ." (b). Given that good 2 is a luxury and that both goods are normal, what can be said about good 1? Justify your answer. 1 In the strict sense, i.e., the inequality in the definition is strict. 2
p1 x1 ( wi ) of wi (c). Suppose that consumer i's wealth increases. What happens to the share b1 ( wi ) good 1 in her budget? What happens to the share b2 ( wi ) p 2 x 2 ( wi ) of good 2 in her budget? wi (d). We define an equiproportional increase in the wealth vector as a move from (w1, ..., wI) to (w1 , ..., wI), where > 1. How does such an equiproportional increase in the wealth vector affect the aggregate budget share of good 2, defined as B2 ( w) answer. (e). We define an increase in inequality as a transfer of > 0 units of wealth from a person i to another one who is not poorer than i.2 Does the fact that good 2 is a luxury guarantee that an increase in inequality increases the aggregate demand
p 2 i =1 x 2 ( wi )
I w i =1 i I ? Justify your I i =1 x 2 ( wi ) for good 2 (and hence its budget share)? If not, what condition or conditions on the functions x1(wi) and/or x2(wi) guarantee it? In either case, prove your claim.
w 3 1 1 , x 2 ( wi ) = i  , 1 + wi 4 4 (1 + wi ) For (f) (i) below, refer to the following example: x1 ( wi ) = wi + considering only (p1, p2) = (3/4,1), and wi > = 3 / 2 .
(f). Classify the two goods as necessities, luxuries, or borderline. (g). Does an equiproportional increase in the wealth vector increase the budget share of good 2? Explain.
(h). Does the example satisfy the condition that you propose in (e) above? Explain. (i). In this example, does an increase in inequality increase the demand for a luxury? Comment
2 Of course, we still require that wi  > . 3 2. Labor supply under certainty and uncertainty We start with the certainty case. There are two goods: leisure (labor) and a consumption good. The wage rate (price of leisure) is p1 = p > 0, and the price of the consumption good is p2 = 1. The consumer is endowed with > 0 units of leisure, 0 units of the consumption good and m > 0 units of nonlabor wealth. The preferences of the consumer are represented by the utility function
aln x + (1 a)ln c , where x and c are, respectively, the amounts of leisure and consumption good that she enjoys. Writing L for the amount of labor supplied, we have that x =  L, whereas the budget equality implies that c = m + pL. Accordingly, we write the consumer's problem as the (unconstrained) maximization of the singlevariable function
aln (  L) + (1 a) ln(m + pL). ~ (a). Find the supplyoflabor function L ( p, m) . ~ L (b). Compute and interpret its sign. m ~ L (c). Compute and interpret its sign in terms of the substitution and wealth effects. p Now we move to a world of uncertainty. There are two states of the world: bad and good. In the bad state, which occurs with probability , (p, m) = (pB, mB). In the good state, which occurs with probability (1 ), (p, m) = (pG, mG). The consumer must commit to her labor supply L before the state of the world is known, and thus, she will get the amount cB = mB + pBL of the consumption good in the bad state, and cG = mG + pGL in the good state. Her preferences satisfy the expected utility hypothesis, with von NeumannMorgensternBernoulli utility function aln (  L) + (1 a)ln c.
(d). Argue that the consumer is strictly risk averse. (e). Write the expected utility maximization problem. (f). Write the first order condition for the expected utility maximization problem. (g). Show that the second order condition for the expected utility maximization problem is satisfied with strict inequality. 4 Find the signs of the derivatives requested below by applying the implicit function theorem to the appropriate version of (f) above, taking (g) into account. All epsilons below are positive.
(h). Uncertainty about m: the effect of increased mB and mG. Suppose that pB = pG = p, but mG > mB. Let mB and mG increase to mB + h and mG + h, respectively. What is the sign of in words, and compare with (b) above. dL ? Interpret d h (i). Uncertainty about m: the effect of increased dispersion at unchanged mean. Suppose again that pB = pG = p, and that mG > mB. Let mB decrease to mB  i, while mG increases to mG +
What is the sign of i. 1 dL ? Interpret in words, and comment. d i (j). Uncertainty about p: the effect of increased pB and pG. Suppose that mB = mG = m, but pG > pB. Let pB and pG increase to pB + j and pG + j, respectively. What is the sign of words, and compare with (c) above.
(k). dL ? Interpret in d j Uncertainty about p: the effect of increased dispersion at unchanged mean. Suppose again that mB = mG = m, and that pG > pB. Let pB decrease to pB  k, while pG increases to pG + What is the sign of k. 1 dL ? Interpret in words, and comment. d k ݽ ݾ ݽ ݾ ݺ ݽ ݾ ݵ ݺ ݸ ݸ ݽ ݾ к Ҵ ֺ ݸ Խ Ծ ظ ݾ ָ Խ ݽ Ծ ݾ Թѵ Խ Ծ ݽ ݾ Ӭظ Һ Ӭ ݽ ݾ Ӭ ָ ָ Խ Ծ ݽ ݾ Ӭ ظ Ӭ ݽ ݾ Ӭع Ӭ ֳ ڴ س Ӭ ĵ ظ ٹ ӫ ̴ մ յ մ س س ظ մ ӫ մ ̴ س Ӭ س ...
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 Winter '06
 PONTUSRENDAHL
 Microeconomics

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