Problem 1. Expenditure Minimization. In this exercise, we consider an expenditure minimization problem with a utility function over three goods. The utility function is u(x₁, 2, 3)=√√123. The prices of goods x = (1, 2, 3) are p = (P1, P2, P3). We denote income by M, as usual, with M > 0. Assume x E R+ (i.e., z₁ > 0, 2 > 0, 3 > 0), unless otherwise stated.
Problem 1. Expenditure Minimization. In this exercise, we consider an expenditure minimization problem with a utility function over three goods. The utility function is u(x₁, 2, 3)=√√123. The prices of goods x = (1, 2, 3) are p = (P1, P2, P3). We denote income by M, as usual, with M > 0. Assume x E R+ (i.e., z₁ > 0, 2 > 0, 3 > 0), unless otherwise stated.
Chapter1: Making Economics Decisions
Section: Chapter Questions
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Transcribed Image Text:8. Derive the indirect utility function (p. M) by substituting u = v(p, M) in the expen-
diture function, setting the expenditure function equal to M, and solving for v(p, M).
9. Derive the Marshallian demand functions ri, 2, and r as functions of the prices
p= (P₁, P2, P3) and income M. To do this, use Roy's identity.
(p, M) =
Əv(p, M)/api
Əv(p, M)/OM
10. Are goods ₁, 2, and a normal goods? Compute the appropriate partial derivatives
and answer.
11. What are the signs of Oz (p, M)/Op,, for i=1,2,3? Answer and back up your answer
without directly computing the partial derivatives.
![Problem 1. Expenditure Minimization.
In this exercise, we consider an expenditure minimization problem with a utility function
over three goods. The utility function is
u(x₁, x2, 3)=√√x1x₂x3.
The prices of goods x = (1, 22, 23) are p = (P₁, P2, P3). We denote income by M, as usual,
with M > 0. Assume x R (i.e., ₁ > 0, ₂ > 0, 3 > 0), unless otherwise stated.
1. Compute du/or and ou/or. Is the utility function increasing in ₁? Is the utility
function concave in a₁?
2. The consumer minimizes expenditure subject to a utility constraint. Let u represent
the minimum level of utility the consumer requires. Write down the expenditure min-
imization problem of the consumer with respect to x = (₁, 2, 3). Explain briefly
why the utility constraint is satisfied with equality.
3. Write down the Lagrangian function (use À for the Lagrange multiplier).
4. Write down the first order conditions with respect to x = (₁, 2, 3) and A.
5. Solve for the Hicksian demand functions hi, h₂, and ha as functions of the prices
p= (P₁, P2, P3) and the minimum required utility u. [Hint: combine the first and
second first-order conditions, then combine the first and third first-order conditions,
and finally plug into the utility constraint.]
6. Find 0h (p, ü)/ap2 and 0h; (p. u)/Ops. Based on this, are these goods (net) substitutes
or (net) complements?
7. Write down the expenditure function as a function of p= (P₁, P2, P3) and u, i.e., what
is e(p, ū)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03112f96-24a2-45f9-8478-3c225e33c54f%2Fd77aaefb-cc7f-4e9a-be98-630cf55b82a0%2F38q8y2j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 1. Expenditure Minimization.
In this exercise, we consider an expenditure minimization problem with a utility function
over three goods. The utility function is
u(x₁, x2, 3)=√√x1x₂x3.
The prices of goods x = (1, 22, 23) are p = (P₁, P2, P3). We denote income by M, as usual,
with M > 0. Assume x R (i.e., ₁ > 0, ₂ > 0, 3 > 0), unless otherwise stated.
1. Compute du/or and ou/or. Is the utility function increasing in ₁? Is the utility
function concave in a₁?
2. The consumer minimizes expenditure subject to a utility constraint. Let u represent
the minimum level of utility the consumer requires. Write down the expenditure min-
imization problem of the consumer with respect to x = (₁, 2, 3). Explain briefly
why the utility constraint is satisfied with equality.
3. Write down the Lagrangian function (use À for the Lagrange multiplier).
4. Write down the first order conditions with respect to x = (₁, 2, 3) and A.
5. Solve for the Hicksian demand functions hi, h₂, and ha as functions of the prices
p= (P₁, P2, P3) and the minimum required utility u. [Hint: combine the first and
second first-order conditions, then combine the first and third first-order conditions,
and finally plug into the utility constraint.]
6. Find 0h (p, ü)/ap2 and 0h; (p. u)/Ops. Based on this, are these goods (net) substitutes
or (net) complements?
7. Write down the expenditure function as a function of p= (P₁, P2, P3) and u, i.e., what
is e(p, ū)?
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