Microeconomics problem set (ECO601)

Microeconomic Theory Yong Wang (SITE, UIBE)

Microeconomic Theory, Problem Set 1 Due: Oct 18, 2022. Hand into frankwang@uibe.edu.cn

Question 1. Show that WARP holds if and only if it holds for compensated price change.

Question 2. ( B, c(·)

) is a choice structure. Show that WARP implies the following statement:

“For any sets B1, B2 ∈ B, if x ∈ B1, B1 ⊆ B2, and x ∈ c(B2), then x ∈ c(B1).”

In words, the statement says that if you choose x when faced with B2, then you should continue to choose x if some of the alternatives in B2 are removed.

Question 3. Consider a consumer with utility function,

u(x) =

(∑ l

αlx ρ l

)1/ρ

, or equivalently, û(x) = 1

ρ ln

(∑ l

αlx ρ l

)

where ρ ∈ (−∞, 1), ρ 6= 0, αl > 0 and ∑

l αl = 1.

Suppose for simplicity that inner solution exists, and the price of good l is pl > 0 and consumer’s wealth is w > 0.

1. Find the optimal demand for good l.

Hint: the answer is xl = ασl

pσl ( ∑ n α

σ np

1−σ n )

w where σ = 1/(1− ρ).

2. Verify that the demand obtained above is the same with that for the Cobb-Douglas utility function when ρ = 0.

3. Verify that as ρ → −∞, the demand converges to that of Leontief utility function, i.e., u(x) = min{xl}.

4. For the case L = 2, compute the elasticity of substitution between the two goods, defined as ε12 = −d(x2/x1)

d(p2/p1) p2/p1 x2/x1

.

Question 4. Suppose that consumer’s preference is strictly concave, and utility function u(x) is well-defined and homogeneous of degree 1.

1. Show that the Walrasian demand function x(p, w) and the indirect utility function v(p, w) are homogeneous of degree 1 in w, therefore can be rewritten in the form of x(p, w) = x̃(p)w and v(p, w) = ṽ(p)w.

2. Show that the Hicksian demand function h(p, u) and the expenditure function e(p, u) are homogeneous of degree 1 in u, therefore can be rewritten in the form of h(p, u) = h̃(p)u and e(p, u) = ẽ(p)u.

3. Show that the elasticity of demand for any good l is equal to 1, that is, dxl(p,w) dw

w xl(p,w)

= 1, ∀l.

Question 5. Consider a consumer with utility function u(x1, x2) = α lnx1 + (1 − α) lnx2, 0 < α < 1.

1. Solve the utility maximization problem to find Walrasian demand function x(p, w), and the indirect utility function v(p, w).

2. Verify that Roy’s lemma holds.

3. Solve the expenditure minimization problem to find Hicksian demand h(p, u), and the expenditure function e(p, u).

4. Verify that Shepards lemma holds.

5. Verify that h ( p, v(p, w)

) = x(p, w), and e

( p, v(p, w)

) = w.

6. Verify that Slutsky equation holds.

Question 6. Consider a consumer with Cobb-Douglas preferences u(x1, x2) = √ x1x2, where x1

and x2 denote the amounts consumed of goods 1 and 2, respectively. The prices of these goods are p1 > 0 and p2 > 0, respectively, and this individual’s wealth is w > 0. The government needs to collect a large amount of money to finance a new health care plan, and contemplates two options: (a) introduce an income tax equivalent to 40 percent of individuals wealth, or (b) charge a sales tax over the price of good 1 which would imply an increase in the price of good 1 from p1 to p1(1 + t), collecting the same dollar amount as with the income tax. Using the indirect utility function of this individual under option a (income tax) and option b (sales tax), explain which tax produces a smaller utility reduction.

2