Ps2 Micro 1 M2 2020 PDF

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TSE Micro 1 Master 2 Problem Set 2

Prof. Renato Gomes TD Paul-Henri Moisson Please try to solve the exercises before the TD session. It will dramatically improve your performance.

Exercise 1: Solve the following problem from MWG: 3.D.4 Exercise 2: Let the consumption set be X = RL+ with L  1. The Stone-Geary utility function has the form ⇢ QL bl if xl > a l 8l l=1 (xl  al ) u(x) = 0 otherwise, PL bl = 1. The terms al are often interpreted where al , bl  0 for all l, and l=1 as the subsistence levels of the respective commodities. Define the discretionary PL income under prices p and income w according to D(p, w) ⌘ w  l=1 al · pl , i.e., D(p, w) is the amount of income in excess of what is necessary to purchase subsistence levels of all commodities. Assume that D(p, w)  0. 1. Derive the associated expenditure function e(p, u) and show that e(p, u) = A(p) + B(p) · u for suitable functions A(p) and B(p). 2. Derive the associated indirect utility function v(p, w). Show that v(p, w) = α · D(p, w) for some α  0. 3. Show that bl measures the share of discretionary income that will be spent on discretionary purchases of good l, i.e., purchases in excess of the subsistence level al .

Exercise 3: A consumer of two goods has utility function u(x, y) = max{ax, ay} + min{x, y }, with a 2 (0, 1). 1

1. Draw the indifference curves for these preferences. 2. Show that the preferences represented by u(x, y) are convex but not strictly convex. 3. Derive the Walrasian demand function. 4. Derive the indirect utility function.

Exercise 4: An infinitely lived agent owns 1 unit of a commodity that he consumes over his life time. The commodity is perfectly storable and he will receive no more than he has now. Consumption of the commodity in period t is denoted by xt , and his lifetime utility function is given by u(x0 , x 1 , x2 , . . .) =

∞ X

β t · ln(xt ),

t=0

where β 2 (0, 1). Compute his optimal level consumption in each period. Exercise 5: Let the consumption set be X = RL+ with L  2. The constant elasticity of substitution (or CES) utility function has the form u(x) =

L X

αl ·

µ xl

l=1

! 1µ

,

for strictly positive constants αl , l = 1, . . . , L, and µ < 1, µ 6= 0. 1. Derive the Walrasian demand function at prices p and income w. 2. What happens to the indifference curves as µ ! 1? 3. Show that as µ ! 0, this utility function comes to represent the same preferences as the generalized Cobb-Douglas utility function u(x) =

L Y

xαl l .

l=1

4. Show that as µ ! 1, this utility function comes to represent the same preferences as the Leontief utility function u(x) = min {x1 , . . . , x L } . Exercise 6: Let the consumption set be X = RL+ with L  1 and consider a consumer whose preferences are described by a utility function which is homogenous of degree one. Denote by p = RL++ the price vector faced by the consumer and by w the consumer’s wealth level. 2

1. Show that the expenditure function can be written in the form: e(p, u) = e(p, 1)u. 2. Let v(p, w) be the consumer’s indirect utility function. Show that the ∂v marginal utility of income, ∂w (p, w), depends on p but is independent of w.

Exercise 7: Let the consumption set be X = RL+ with L  1. The Stone-Geary utility function has the form ⇢ QL βl if xl > α l 8l l=1 (xl  αl ) u(x) = 0 otherwise, PL where αl , βl  0 for all l, and l=1 βl = 1. 1. Verify that the Stone-Geary utility function leads to an indirect utility satisfying the Gorman form: v(p, w) = a(p) + b(p)w,

(1)

and obtain the functions a(p) and b(p). Restrict attention to interior solutions. 2. Show that, if a consumer has an indirect utility satisfying (1), then his expenditure function takes the form e(p, u) = c(p)u + d(p), and express the functions c(p) and d(p) in terms of a(p) and b(p). 3. Show that, if a consumer has an indirect utility satisfying (1), his income elasticity of (Walrasian) demand for every good approaches zero as w ! 0 and approaches unity as w ! 1. Exercise 8: Let the consumption set be X = RL+ , with L  2, and consider a consumer endowed with preferences ⌫ over X, which are rational, continuous, locally non-satiated, and strictly convex. The consumer’s Walrasian demand is x(p, w), where w > 0 is her income and p 2 RL++ are the prices. Assume x(p, w) is differentiable. 1. Show that, for l = 1 . . . , L, L X ∂xl ∂xl (p, w)pk + (p, w)w = 0. ∂pk ∂w k=1

2. Show that, for k = 1 . . . , L, L X ∂xl l=1

∂pk

(p, w)pl + xk (p, w) = 0.

3...