Chapter 3 Henderson and Quandt Microeconomics, Questions 3.1-3.3_ Solutions

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3-1 Which of the following utility functions are (a) strongly separable, or (b) additive with

respect to all variables: U= (+); U = +; U = β₁ In (q₁-γ₁) + β₂ In (-);

U=(q₁ +2q₂+3 )^1/4. Show for each strongly separable or additive function what the F and

functions are.

Ans. (a) The utility function U= (q_1^(1/4)+q_2^(1/2)) is not strongly separable or additive because it cannot be expressed as a sum of functions that depend only on a single variable.

(b) The utility function U = q_1 q_2 +q_3 q_4 is additive with respect to all variables because it can be expressed as the sum of two functions, one that depends only on q_1 and q_2, and another that depends only on q_3 and q_4. Therefore, F(q) = q_1 q_2 +q_3 q_4, and f_1(q) = q_2, f_2(q) = q_1, f_3(q) = q_4, and f_4(q) = q_3.

(c) The utility function U = β₁ In (q₁-γ₁) + β₂ In (q_2-γ_1) is not additive with respect to all variables. Therefore, it is not strongly separable.

(d) The utility function U=(q₁ +2q₂+3 q_3)^1/4 is strongly separable because it can be expressed as the sum of three functions, each depending only on a single variable. Therefore, F(q) = (q₁ +2q₂+3 q_3)^1/4, f_1(q) = (q₁)^1/4, f_2(q) = (2q₂)^1/4, and f_3(q) = (3q_3)^1/4.

3-2 Prove that if the consumer is indifferent between commodity bundles (....) and

(......,) and has a homothetic utility function, she will also be indifferent between the

bundles (.....,) and (.....,).

Ans . Suppose that the consumer is indifferent between two commodity bundles (q_1^0,.....q_n^0) and (q_1^((1)),......,q_n^((1))) and has a homothetic utility function. That is, the consumer's utility function can be expressed as U(q) = v(p) * f(q), where v(p) is a homogeneous function of degree zero in prices, and f(q) is a homogeneous function of degree k in quantities, where k is a constant.

Now, consider the bundles (〖tq〗_1^0,.....,〖tq〗_n^0) and (〖tq〗_1^((1)),.....,〖tq〗_n^((1))). These bundles are simply the original bundles multiplied by a positive scalar t. Let U_0 = U(q^0) and U_1 = U(q^1) be the utility levels associated with the two original bundles.

Then, the utility levels associated with the new bundles are U_0' = U(tq^0) = v(p) * f(tq^0) and U_1' = U(tq^1) = v(p) * f(tq^1).

Since v(p) is homogeneous of degree zero in prices, it follows that v(p) is constant when prices are multiplied by t. That is, v(tp) = v(p) for all t > 0.

Using this fact and the homogeneity of f(q) of degree k, we have:

U_0' = v(p) * f(tq^0) = v(tp) * f(tq^0) = v(p) * f(q^0) = U_0

U_1' = v(p) * f(tq^1) = v(tp) * f(tq^1) = v(p) * f(q^1) = U_1

Therefore, the consumer is also indifferent between the bundles (〖tq〗_1^0,.....,〖tq〗_n^0) and (〖tq〗_1^((1)),.....,〖tq〗_n^((1))), since they yield the same utility levels as the original bundles.

3-3 Prove that an additive, strictly quasi-concave utility function is concave.

Ans. Suppose that the utility function u is additive and strictly quasi-concave. Then, for any two consumption bundles x and y, we have:

u(x + λ(y-x)) > u(x) for 0 < λ < 1.

Let z = (x+y)/2 be the average of x and y. Since u is additive, we have:

u(z) = u((1/2)x + (1/2)y) = u(1/2x) + u(1/2y)

Since u is strictly quasi-concave, it follows that u(λx + (1-λ)y) is strictly decreasing in λ for any fixed x and y. Therefore, for any λ > 1/2, we have:

u(λx + (1-λ)z) > u(z) and u(λy + (1-λ)z) > u(z)

On the other hand, for any λ < 1/2, we have:

u(λx + (1-λ)z) > u(x) and u(λy + (1-λ)z) > u(y)

Using the additivity of u, we can write:

u(λx + (1-λ)y) = u(λx + (1-λ)z + λz + (1-λ)y) ≤ u(λx + (1-λ)z) + u(λy + (1-λ)z)

Therefore, for any λ < 1/2, we have:

u(λx + (1-λ)y) < u(x) + u(y) - u(z)

For λ = 1/2, we have:

u(z) = u(1/2x) + u(1/2y) < u(x) + u(y)

Thus, we have shown that for any two bundles x and y, and any λ ∈ (0,1), we have:

u(λx + (1-λ)y) < λ u(x) + (1-λ) u(y)

Therefore, u is concave.

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