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Chapter 2 Henderson and Quandt Microeconomics, Questions 2.9-2.12_ Solutions







2-9 Prove that Q₁ and Q₂ cannot both be inferior goods.


We can prove that Q₁ and Q₂ cannot both be inferior goods by contradiction.


Suppose that both Q₁ and Q₂ are inferior goods. By definition, an inferior good is a good whose demand decreases as income increases.


Now, consider an increase in income. As income increases, the demand for Q₁ and Q₂ both decrease, since they are both inferior goods. However, if the demand for both goods decreases, then the total expenditure on both goods must also decrease. This implies that the total expenditure on goods other than Q₁ and Q₂ must increase, since total income has increased.


But this contradicts the assumption that both Q₁ and Q₂ are inferior goods. If total expenditure on other goods increases as income increases, then those goods must be normal goods. Therefore, at least one of Q₁ and Q₂ must be a normal good, since both cannot be inferior goods.


Hence, we have proved that Q₁ and Q₂ cannot both be inferior goods.



2-10 Verify that p₁+ =0 for the utility function U =


To verify that S11p1 + S12p2 = 0 for the utility function U = q1^γq2, we need to find the demand functions for q1 and q2 and then calculate the partial derivatives of the demand functions with respect to the prices p1 and p2.


From the utility function U = q1^γq2, we get the following indirect utility function:


V(p1, p2, y) = y/(p1^γp2)


Taking the partial derivative of v with respect to p1, we get:


∂v/∂p1 = -γy/(p1^(γ+1)p2)


This is the compensated (Hicksian) demand function for q1, denoted by x1(p1, p2, u).


Taking the partial derivative of x1 with respect to p1, we get:


∂x1/∂p1 = -γx1(p1,p2,u)/p1


This is the elasticity of demand for q1 with respect to p1.


Similarly, taking the partial derivative of v with respect to p2, we get:


∂v/∂p2 = -yγ/(p2^(γ+1)p1)


This is the compensated demand function for q2, denoted by x2(p1, p2, u).


Taking the partial derivative of x2 with respect to p1, we get:


∂x2/∂p1 = 0


Now, we can find the Slutsky matrix, S, which relates the changes in the Marshallian demand functions to the changes in the compensated demand functions:


S = [∂x1/∂p1 ∂x1/∂p2; ∂x2/∂p1 ∂x2/∂p2] + diag[x1(p1,p2,u)/p1, x2(p1,p2,u)/p2]


Substituting the partial derivatives we have derived, we get:


S = [-γx1(p1,p2,u)/p1 0; 0 0] + diag[x1(p1,p2,u)/p1, 0]


Simplifying, we get:


S11p1 + S12p2 = -γx1(p1,p2,u) + x1(p1,p2,u) = (1-γ)x1(p1,p2,u)/p1


Since γ is positive, (1-γ) is negative, and x1 is positive, we can conclude that S11p1 + S12p2 < 0, which means that q1 is a normal good. Therefore, q2 cannot be an inferior good.


2-11 Let U = f(q, H) be a utility function the arguments of which are the quantity of a

commodity (q) and the time taken to consume it (H). The marginal utilities of both arguments are

positive. Let W be the amount of work performed, W+H = 24 (hours), r be the wage, and p be

the price of q. Formulate the appropriate constrained utility maximization problem. Find an

expression for dH/dr. Is its sign determined unambiguously?


The constrained utility maximization problem can be formulated as:


Max U(q, H) subject to pq = rW, H + W = 24


Using Lagrange multiplier method, the Lagrangian function is:


L(q, H, λ, μ) = f(q, H) + λ(pq – rW) + μ(H + W – 24)


Taking partial derivatives with respect to q, H, λ, and μ and setting them equal to zero, we get:


∂L/∂q = ∂f/∂q + λp = 0


∂L/∂H = ∂f/∂H + μ = 0


∂L/∂λ = pq – rW = 0


∂L/∂μ = H + W – 24 = 0


From the first equation, we have λ = -∂f/∂q*p, substituting this into the third equation, we get:


Q = (r/p)W(∂f/∂q)^-1


Differentiating this equation with respect to r, we get:


Dq/dr = (1/p)(∂f/∂q)^-1 * W – (r/p^2)W(∂f/∂q)^-2 * (∂^2f/∂q^2)(dq/dr)


Simplifying, we get:


Dq/dr = (1/p)(∂f/∂q)^-1 * (1 – r(∂^2f/∂q^2)(∂f/∂q)^-2 * W)


Since the marginal utilities of both arguments are positive, we know that (∂^2f/∂q^2) and (∂f/∂q) are positive. Therefore, the sign of dq/dr is determined by (1 – r(∂^2f/∂q^2)(∂f/∂q)^-2 * W). This term could be positive or negative depending on the specific values of r, (∂^2f/∂q^2), (∂f/∂q), and W. So, the sign of dq/dr is not determined unambiguously.


2-12 Imagine that coupon rationing is in effect so that each commodity has two prices: a dollar

price and a ration-coupon price. Assume that there are three commodities and that the consumer

has a dollar income y and a ration-coupon allotment z. Also assume that this allotment is not so

liberal that any commodity combination that he can afford to purchase with his dollar income can

also be purchased with his coupons. Formulate his constrained-utility-maximization problem

assuming a strictly concave utility function. Derive conditions for a maximum. Interpret the

conditions from an economic point of view. Find a sufficient condition which guarantees that the

imposition of rationing does not alter the consumer's purchases.


Let the prices of the three commodities be denoted by (p₁, p₂, p₃) in dollar terms, and by (q₁, q₂, q₃) in ration-coupon terms. The consumer’s utility function is denoted by U(q₁, q₂, q₃), and his dollar income and ration-coupon allotment are denoted by y and z, respectively.


The consumer’s problem can be formulated as follows:

Maximize U(q₁, q₂, q₃) subject to p₁q₁ + q₁z = y, p₂q₂ + q₂z = y, p₃q₃ + q₃z = y.


Using the Lagrange multiplier method, the Lagrangian can be written as:

L = U(q₁, q₂, q₃) + λ₁(y – p₁q₁ - q₁z) + λ₂(y – p₂q₂ - q₂z) + λ₃(y – p₃q₃ - q₃z)


Taking the first-order conditions with respect to q₁, q₂, q₃, and z, we get:

∂L/∂q₁ = 0 ⇒ λ₁p₁ = U₁(q₁, q₂, q₃)

∂L/∂q₂ = 0 ⇒ λ₂p₂ = U₂(q₁, q₂, q₃)

∂L/∂q₃ = 0 ⇒ λ₃p₃ = U₃(q₁, q₂, q₃)

∂L/∂z = 0 ⇒ λ₁q₁ + λ₂q₂ + λ₃q₃ = 0


From the first three conditions, we can derive the following equalities:

Λ₁p₁/p₂ = λ₂p₂/p₁

Λ₂p₂/p₃ = λ₃p₃/p₂


Multiplying these equalities together, we get:

(λ₁p₁/p₂)(λ₂p₂/p₃)(λ₃p₃/p₁) = λ₁λ₂λ₃


Since the utility function is strictly concave, we have the Inada conditions, which imply that the marginal utilities are positive and approach infinity as the arguments approach zero. This implies that λ₁, λ₂, λ₃ are all positive. Therefore, we can cancel out the λ’s and rearrange the equalities to obtain:

P₁q₁/p₂q₂ = p₂q₂/p₃q₃ = p₁q₁/p₃q₃


This is the condition for the consumer to maximize his utility subject to his budget constraint. It states that the ratios of the dollar prices to the coupon prices must be equal for all pairs of commodities.


To interpret this condition, note that it implies that the consumer is indifferent between different combinations of commodities that have the same dollar-to-coupon price ratios. This is because he can use his fixed coupon allotment to purchase any combination of commodities that satisfies this condition, and then use his remaining dollar income to purchase more of any commodity that he desires. Thus, his purchases are determined only by the relative prices of the commodities, and not by the absolute values of the prices.


A sufficient condition for the imposition of rationing not to alter the consumer’s purchases is that the consumer’s initial purchases satisfy the condition above. This implies that the imposition of rationing simply reduces the consumer’s dollar income and forces him to adjust his purchases to satisfy the same condition, without changing his relative valuations of the commodities.



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