# Chapter 3 Henderson and Quandt Microeconomics, Questions 3.4-3.7_ Solutions

**3-4 Construct an indirect utility function that corresponds to the direct function U = α In q₁ +.**

**Use Roy's identity to construct demand functions for the two goods. Are these the same as the**

**demand functions derived from the direct utility function?**

Ans. To find the indirect utility function v(p₁, p₂, y) corresponding to the direct utility function U(q₁, q₂) = α ln(q₁) + q₂, we can use the expenditure minimization problem:

minimize p₁q₁ + p₂q₂

subject to α ln(q₁) + q₂ = y

Solving this problem for q₁, we get:

q₁(p₁, p₂, y) = e^{(y-p₂)/α} / p₁

Substituting this expression for q₁ into the budget constraint, we get:

U(q₁(p₁, p₂, y), q₂(p₁, p₂, y)) = α ln(e^{(y-p₂)/α} / p₁) + q₂(p₁, p₂, y) = y

Solving for q₂, we get:

q₂(p₁, p₂, y) = y - α ln(e^{(y-p₂)/α} / p₁) = y - (y-p₂) ln(p₁/α)

Thus, the indirect utility function is:

v(p₁, p₂, y) = U(q₁(p₁, p₂, y), q₂(p₁, p₂, y)) = α ln(e^{(y-p₂)/α} / p₁) + (y - (y-p₂) ln(p₁/α))

Now, we can use Roy's identity to find the demand functions for q₁ and q₂:

∂v/∂p₁ = -q₁(p₁, p₂, y)/p₁

∂v/∂p₂ = -q₂(p₁, p₂, y)

Substituting the expressions for q₁ and q₂, we get:

∂v/∂p₁ = - e^{(y-p₂)/α} / p₁^2

∂v/∂p₂ = - (y-p₂) / p₁

These are not the same as the demand functions derived from the direct utility function, which are:

q₁(p₁, p₂, y) = α p₂ / (p₁ e^{y/α + p₂/α})

q₂(p₁, p₂, y) = y - α ln(p₁/p₂)

However, these demand functions do satisfy the law of demand, and they also satisfy the symmetry and homogeneity properties required by a well-behaved demand system.

**3-5 A consumer is observed to purchase q₁ = 20, q2 = 10 at prices p₁=2, = 6. She is also**

**observed to purchase q₁ = 18, =4 at the prices p₁ = 3, P₂-5. Is her behavior consistent with**

**the axioms of the theory of revealed preference?**

Ans. To determine if the consumer's behavior is consistent with the axioms of the theory of revealed preference, we can use the weak axiom of revealed preference (WARP). WARP states that if a consumer chooses bundle A when it is available and also chooses bundle B when it is available, and A is not more expensive than B, then A must be at least as preferred as B.

In this case, the consumer purchases bundle A = (20, 10) at prices (2, 6) and bundle B = (18, 4) at prices (3, 5). We can calculate the cost of each bundle at each set of prices:

A at (2, 6): 2(20) + 6(10) = 80

B at (2, 6): 2(18) + 6(4) = 60

A at (3, 5): 3(20) + 5(10) = 110

B at (3, 5): 3(18) + 5(4) = 74

We can see that the cost of bundle B is always lower than the cost of bundle A, so we cannot apply WARP directly. However, we can look at the ratio of the prices:

(2/6) < (3/5)

Since the ratio of the prices is different for the two observations, we can conclude that the consumer's behavior is not consistent with the axioms of the theory of revealed preference. Specifically, the consumer violates the weak axiom of revealed preference, which requires that if A is chosen over B at one set of prices, it cannot be the case that B is chosen over A at another set of prices with a different price ratio.

**3-6 Let the consumer's utility function be f()=, and her budget constraint**

**y = ++. Consider q₁ + (p₂/P₁)q2 = as a composite good. Formulate the consumer's optimization problem in terms of , and find the demand function for.**

Ans. To formulate the consumer's optimization problem in terms of the composite good q_c, we can substitute q₂ = (P₁/P₂)(q_c - q₁) into the budget constraint, yielding:

y = p₁q₁ + p₂(q_c - q₁) + p₃q₃

Simplifying, we get:

y = (p₂/P₁)q_c + (p₁ - p₂/P₁)q₁ + p₃q₃

Since the utility function is f(q₁, q₂, q₃) = q₁q₂q₃, we can rewrite it in terms of the composite good q_c as:

f(q₁, q₂, q₃) = q₁(q_c - q₁)q₃ = q_cq₁q₃ - q₁²q₃

Now we can formulate the consumer's optimization problem as:

maximize q_cq₁q₃ - q₁²q₃ subject to y = (p₂/P₁)q_c + (p₁ - p₂/P₁)q₁ + p₃q₃

To find the demand function for q_c, we take the first-order condition and solve for q_c:

∂f/∂q₁ = q_cq₃ - 2q₁q₃ = 0 q_c = 2q₁

Substituting this into the budget constraint, we get:

y = 2p₂q₁/P₁ + (p₁ - p₂/P₁)q₁ + p₃q₃

Solving for q₁, we get:

q₁ = [(P₁y - 2P₃p₂)/((P₁P₃ - 2P₂)P₁)]

Substituting q₁ back into the expression for q_c, we get the demand function for the composite good:

q_c = 2[(P₁y - 2P₃p₂)/((P₁P₃ - 2P₂)P₁)]

Therefore, the demand function for q_c is q_c = 2[(P₁y - 2P₃p₂)/((P₁P₃ - 2P₂)P₁)].

**3-7 Let the consumer's inverse demand curve be p = a-bq with a, b>0, and assume that a sales**

**tax of 100t percent is imposed so that the unit price she pays is increased to p(1+t). Prove that her**

**loss of consumer's surplus will always exceed the revenue raised by the government through the**

**imposition of the tax.**

Ans. To prove this statement, we can use the concept of the deadweight loss of a tax. The deadweight loss is the loss in consumer and producer surplus due to a tax that causes a reduction in the quantity traded.

Let's assume that the original equilibrium quantity and price are q* and p*. After the tax is imposed, the new equilibrium quantity and price are q_t and p_t, where p_t = (a-bq*) (1+t) and q_t is the new quantity demanded.

The government revenue raised by the tax is equal to t(p_t*q_t) = t(a-bq*)q_t(1+t).

The loss in consumer surplus is equal to the area between the demand curve and the new price line, from 0 to q_t, minus the revenue raised by the government. This can be expressed as:

∫[0,q_t] [(a-bq*) - (a-bq*) (1+t)] dq - t(a-bq*)q_t(1+t)

Simplifying, we get:

∫[0,q_t] btq* dq - t(a-bq*)q_t(1+t)

= (1/2)btq_t^2 - (1/2)btq*^2 - t(aq_t - bq_t^2) - t(a-bq*)q_t

The first two terms are the original consumer surplus, the third term is the loss in producer surplus, and the fourth term is the additional loss in consumer surplus due to the tax.

To prove that the loss in consumer surplus exceeds the government revenue raised by the tax, we need to show that the last term is greater than the government revenue:

t(a-bq*)q_t > t(a-bq*)q_t(1+t)

Simplifying, we get:

q_t > q*

This means that the quantity traded after the tax is imposed is less than the original equilibrium quantity. Since the deadweight loss is a function of the reduction in quantity traded, we can conclude that the loss in consumer surplus will always exceed the revenue raised by the government through the imposition of the tax.

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