Problem Set 4 : Due Wednesday September 25
September 18, 2024
Answers must be typed. Print out and bring to class on due date.
1. Consider the standard CD utility function u(x) = xα1x12−α
. Find the indirect utility function and use it to answer (and prove/disprove) the following
(a) Is the indirect utility function homogeneous degree 0?
(b) Is it strictly increasing in income and non-increasing in pℓ for any ℓ?
(c) Is the indirect utility function quasiconvex?
(d) Is it continuous in p and income?
2. A consumer has utility function u(x) = xα1x12−α + ln(x3). Assume α > 0 and p1 = p2 = p3 = 1.
(a) How does money spent on x3 depend on α? Sketch a graph.
(b) Referencing marginal utilities, say something that makes sense of why and how money spent on x3 depends on α in this way.
3. A consumer with locally non-satiated and strictly convex preferences has the following expenditure function:
e(p, u) = u(pα1 + pα2)α/1
(a) Find the consumer’s Marshallian demand
(check MWG 3.G and 3.H – this was only briefly mentioned on the slides but reading those sections can save you a lot of time on future problems/prelims).
(b) When α = 2 prove whether or not the expenditure function is a valid expenditure function.
(c) What does your answer to b) have to do with the substitution effect?
4. (potentially challenging) Consider a world with 3 commodities. Let the third good be a numeraire (p1 = 1). The demands are
x1(p, w) = a + bp1 + cp2
x2(p, w) = d + ep1 + gp2
(a) Utility maximization implies the following restrictions on parameters: c = e, b ≤ 0, g ≤ 0, bg − c2 ≥ 0. Show how to find these restrictions. (This may be difficult.)
(b) Estimate the equivalent variation for a change in prices from (p1, p2) = (1, 1) to (p1, p2) = (2, 2). Is there any problem if c ≠ e?