ANSWERS TO 4010 MIDTERM
Short Answers:
1. False. DMRS only ensures convex indifference curves and, therefore, downward
sloping compensated demand. It follows from Slutsky equation that if income
effect is sufficiently negative (as in the case of Giffen goods), ordinary
(Marshallian) demand can be upward sloping.
2. True. One is a positive monotonic transformation of the other. Check that the
MRS is identical.
3. Trure if the goods are consumed in the ratio of 1 x to 1 y and the utility function is
given by U = min{x, y}.
4. True. Do the tax case (Fig 4.5) in reverse.
5. False. Expenditure function is homogenous of degree zero:
E = P x + P y , where x and y is the commodity bundle that minimizes the cost
x y
of achieving a given level of utility. Clearly, if both prices double, minimum cost
of retaining the same level of utility must double as well, since, at unchanged
relative prices cost minimizing bundle remains the same.
But, the indirect utility can not be homogenous of degree one in prices since,
doubling of both prices lowers utility with fixed income.
6. False. The sum is zero. Because any proportional increase in all prices and
income leave quantity demanded unchanged, the net sum of all price elasticities
together with income elasticity for a particular good must sum to zero.
7. Suppose the consumer prefers gamble 1 to gamble 2. Therefore,
expected utility (1) = q · U(x2) + (1-q) · U(x3) >expected utility (2) = t · U(x5) +
(1-t) · U(x6). But by VNM index U(Xi) = Πi, where, Πi is the probability of
winning the best prize, in a lottery involving the best and worst prize, that makes
Xi equivalent to the lottery. Therefore substituting U(Xi) = Πi, q · π2 + (1-q) · π3
> t · π5 + (1-t) · π6. Because {q · π2 + (1-q) · π3} and {t · π5 + (1-t) · π6} are
respectively the probabilities of winning the best prize through lotteries equivalent
to gambles 1 and 2, the inequality must hold.
Problems:
1. a. Setting MRS = fx/fy =Px/Py, 1/x = Px/Py. Therefore, Marshallian demand for
x = Py/Px and, by substitution in the budget constraint, Marshallian demand for y
is given by y = (I – Py)/Py.
b. Indirect Ut

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