Chapter 2 Henderson and Quandt Microeconomics, Questions 2.5-2.8_ Solutions

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2-5 Show that the utility functions U = Aand W = are monotonic transformations of

each other where A, α, and β are positive.

To show that U and W are monotonic transformations of each other, we need to demonstrate that one utility function can be transformed into the other by a strictly monotonic function. Let’s define a function g(q) such that:

G(q) = q^(1/β)

Then, we can rewrite W as:

W = q1^a * g(q2)^(-a)

W = q1^a * q2^(-a/β)

Comparing this to the form of U, we see that:

U = A * q1^a * q2^β

Thus, we have:

U = A * (q1^a * g(q2)^a/β)^β

U = A * (q1^a * q2^(-a/β))^β

So we have:

U = A * (W^β)

Since A and β are both positive constants, we have shown that U and W are monotonic transformations of each other, and thus represent the same underlying preferences.

2-6 Let a consumer's utility function be U = + 1.5 In q, + In and his budget constraint

3 +4 = 100. Show that his optimum commodity bundle is the same as in Exercise 2-3. Why is

this the case?

Exercise 2-3: Find the optimum commodity purchases for a consumer whose utility function and budget constraint are U = q1^1.5 q2 and 3q1 + 4q2 = 100 respectively.

To find the optimum commodity bundle for the given utility function and budget constraint, we need to maximize the utility function subject to the budget constraint. Using the Lagrangian method, we have:

L = q1^6q2^4 + 1.5 ln(q1) + ln(q2) – λ(3q1 + 4q2 – 100)

Taking partial derivatives with respect to q1, q2, and λ and setting them equal to zero, we get:

6q1^5q2^4/ q1 = 1.5/ q1 – 3λ

4q1^6q2^3/ q2 = 1/ q2 – 4λ

3q1 + 4q2 = 100

Simplifying the first two equations, we get:

Q2^4 = 0.25q1^6

Q1^6 = 4q2/3

Substituting these into the budget constraint, we get:

3(4q2/3) + 4q2 = 100

Solving for q2, we get q2 = 10. Substituting this back into the equation q1^6 = 4q2/3, we get q1 = 6.29.

Therefore, the consumer’s optimum commodity bundle is (6.29, 10), which is the same as in Exercise 2-3.

The reason why this is the case is that the two utility functions are equivalent up to a constant transformation. Specifically, we can write U = Aq1^αq2^β as ln(U) = ln(A) + α ln(q1) + β ln(q2). Therefore, the utility function U = q1^1.6q2^1.5 + 1.5 ln(q1) + ln(q2) can be written as U = Aq1^0.6q2^0.5, where A is a constant. Since the two utility functions have the same isoquants, they result in the same optimum commodity bundle.

2-7 Construct ordinary and compensated demand functions for Q₁ for the utility function

U=2+q₂. Construct expressions for , , and .

To find the ordinary and compensated demand functions for Q₁, we need to maximize the utility function subject to the budget constraint. The Lagrangian for the problem is:

L = 2q1q2 + q2 + λ(𝐼 – p₁q₁ − p₂q₂)

Taking the partial derivative of L with respect to q₁, q₂, and λ, we get:

∂L/∂q₁ = 2q2 – λp₁ = 0

∂L/∂q₂ = 2q1 + 1 – λp₂ = 0

∂L/∂λ = 𝐼 – p₁q₁ − p₂q₂ = 0

Solving for q₁ and q₂, we get:

Q₁ = (2/3)𝐼/p₁

Q₂ = (1/3)𝐼/p₂

These are the Marshallian demand functions for Q₁.

To find the compensated demand function for Q₁, we need to find the Hicksian demand function for Q₁. The Hicksian demand function for Q₁ is found by maximizing utility subject to the budget constraint, but holding utility constant. We can find the Hicksian demand function by solving the following system of equations:

2q2 + λp₁ = 𝑈₁

2q1 + 1 + λp₂ = 𝑈₂

𝐼 – p₁q₁ − p₂q₂ = 𝐵

Where 𝑈₁ and 𝑈₂ are the constant levels of utility, and 𝐵 is the budget.

Solving for q₁ and q₂, we get:

Q₁ = (2/3)𝐵/p₁

Q₂ = (1/3)𝐵/p₂ − (2/3)𝑈₁/p₂

These are the Hicksian demand functions for Q₁.

The elasticity of demand for Q₁ with respect to its own price (ε11) is:

Ε11 = (∂q₁/∂p₁) * (p₁/q₁) = -1

The cross-price elasticity of demand for Q₁ with respect to the price of Q₂ (ε12) is:

Ε12 = (∂q₁/∂p₂) * (p₂/q₁) = 0

The expenditure elasticity of demand for Q₁ (η1) is:

Η1 = (∂q₁/∂I) * (I/q₁) = 2

2-8 Derive the elasticity of supply of work with respect to the wage rate for the supply curve for

work given by the example in Sec. 2-4.

In section 2-4, the supply curve for work is given by:

W = H(1-τ)w

Where W is the wage income, H is the number of hours worked, τ is the tax rate, and w is the wage rate.

To find the elasticity of supply of work with respect to the wage rate (εw), we use the formula:

Εw = (∂H/∂w) * (w/H)

Taking the derivative of the supply curve with respect to w:

∂W/∂w = H(1-τ)

Using the chain rule, we find the derivative of H with respect to w:

∂H/∂w = (∂H/∂W) * (∂W/∂w)

= [(1-τ)/w] * [H(1-τ)]

= H(1-τ)²/w

Substituting into the elasticity of supply formula:

Εw = [H(1-τ)²/w] * (w/H)

Simplifying:

Εw = (1-τ)²

Therefore, the elasticity of supply of work with respect to the wage rate is (1-τ)².

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