Chapter 2 Henderson and Quandt Microeconomics, Questions 2.1-2.4_ Solutions

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2-1 Determine whether the following utility functions are regular strictly quasi-concave

for the domain_ q₁>0, q:>0: U = q_1q_2; U = q_1q₂; U =+; U = q₁ ++2 ;

U=-0.01(+); and U = ++

U = q₁q₂: This function is regular and strictly quasi-concave as it is the product of two positive variables, and any convex combination of two feasible points will also be feasible and have a strictly higher utility value.

U = q₁γ+q₂²: This function is regular and strictly quasi-concave as it is the sum of two strictly quasi-concave functions. The first term is strictly increasing and concave, while the second term is strictly increasing and strictly concave.

U = q₁ +q₂+2q₁q₂: This function is regular and strictly quasi-concave as it is the sum of two strictly quasi-concave functions. The first term is strictly increasing and concave, while the second term is strictly increasing and strictly concave.

U=q₁q₂-0.01(q₁²+q₂²): This function is regular and strictly quasi-concave as it is the difference of two strictly quasi-concave functions. The first term is the product of two positive variables, while the second term is strictly decreasing and strictly convex.

U =q₁q₂+q₁q₃+q₂q₃: This function is not strictly quasi-concave as it is not strictly increasing in both variables. For example, if q₁=q₂=q₃=1, U=3, but if q₁=q₂=2, q₃=0, U=4.

Therefore, the first four utility functions are regular and strictly quasi-concave, while the last one is not strictly quasi-concave.

2-2 Let f(, ) be a strictly concave utility function, and let = (+)/2, j = 1, 2, where

superscripts denote particular values for the variables. Prove that

f(, )-f()>f(, )-f(,)

To prove the given inequality, we will use the property of concavity, which states that a concave function lies below any straight line connecting two points on its graph.

Let A be the point (q1(2), q2(2)) and B be the point (q1(1), q2(1)). Consider the line segment connecting A and B, denoted as L. Since f is strictly concave, it lies below the line segment L.

Let C be the midpoint of the line segment AB, i.e., C = ((qj0 + qj(1))/2, (qj0 + qj(2))/2) for j = 1, 2.

Now, consider the two line segments AC and BC. By the property of concavity, we have:

F(q1(2), q2(2)) – f(q10, q20) < f(C) – f(q10, q20) ...(1)

F(q1(1), q2(1)) – f(q1(2), q2(2)) < f(C) – f(q1(2), q2(2)) ...(2)

Adding inequalities (1) and (2), we get:

[f(q1(2), q2(2)) – f(q10, q20)] + [f(q1(1), q2(1)) – f(q1(2), q2(2))] < 2[f(C) – f(q1(2), q2(2))]

Simplifying the above expression, we get:

F(q1(2), q2(2)) – f(q10, q20) + f(q1(1), q2(1)) – f(q1(2), q2(2)) < 2[f(C) – f(q1(2), q2(2))]

Rearranging terms, we get the desired inequality:

F(q1(2), q2(2)) – f(q10, q20) < f(q1(1), q2(1)) – f(q1(2), q2(2))

Hence, we have proved that f(q1(2), q2(2)) – f(q10, q20) is less than f(q1(1), q2(1)) – f(q1(2), q2(2)).

2-3 Find the optimum commodity purchases for a consumer whose utility function and budget

constraint are U = and 3+4 = 100 respectively.

We can use the Lagrange multiplier method to solve this problem. The Lagrangian is:

L = q₁^1.5q₂ + λ(100 – 3q₁ - 4q₂)

Taking partial derivatives with respect to q1, q2, and λ, we have:

∂L/∂q₁ = 1.5q₁^0.5q₂ - 3λ = 0

∂L/∂q₂ = q₁^1.5 – 4λ = 0

∂L/∂λ = 100 – 3q₁ - 4q₂ = 0

Solving for λ in the first two equations and setting them equal to each other, we get:

1.5q₁^0.5q₂/3 = q₁^1.5/4

Simplifying and solving for q₂, we have:

Q₂ = 0.5q₁^0.5

Substituting into the budget constraint, we get:

3q₁ + 4(0.5q₁^0.5) = 100

Simplifying and solving for q₁, we have:

Q₁ = 8.77

Substituting back into the expression for q₂, we get:

Q₂ = 2.96

Therefore, the optimum commodity purchases are q₁ = 8.77 and q₂ = 2.96.

2-4The locus of points of tangency between income lines and indifference curves for given

prices p₁, and a changing value of income is called an income expansion line or Engel curve.

Show that the Engel curve is a straight line if the utility function is given by U = , γ>0.

To show that the Engel curve is a straight line if the utility function is given by U = q1^γq2, γ>0, we need to show that the ratio of the two goods purchased remains constant as income changes.

Let I be the consumer’s income and let the prices of the two goods be p1 and p2. The consumer’s budget constraint is given by:

P1q1 + p2q2 = I

Solving for q2, we get:

Q2 = (I/p2) – (p1/p2)q1

Substituting this expression for q2 in the utility function, we get:

U = q1^γ((I/p2) – (p1/p2)q1)

Taking the derivative of U with respect to q1 and setting it equal to zero to find the maximum, we get:

Γq1^(γ-1)((I/p2) – (p1/p2)q1) – γ(p1/p2)q1^(γ-1) = 0

Simplifying and solving for q1, we get:

Q1 = [(γ/I)(p2/p1)]^(1/(γ-1))

Substituting this expression for q1 in the expression for q2, we get:

Q2 = (I/p2) – (p1/p2)[(γ/I)(p2/p1)]^(1/(γ-1))

Simplifying, we get:

Q2 = (I/p2)[1 – (p1/p2)(p2/p1)^(1/(γ-1))]

We can rewrite this equation as:

Q2/q1 = [(p2/p1)^(1/(γ-1))]/[(I/p1)(p2/p1)^(1/(γ-1)) – p1/p2]

This expression for q2/q1 is independent of income I, and depends only on the prices of the two goods and the parameter γ. Therefore, the Engel curve is a straight line with a constant slope given by [(p2/p1)^(1/(γ-1))]/[(I/p1)(p2/p1)^(1/(γ-1)) – p1/p2].

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