Note: The test has 110 points.

1 Matrices, Production Function and Derivatives

1.1 Matrices and Production Function (10 points, 2.5 each)

1. Provide an example of a production function that has constant Return to Scale for one of its

inputs.

2. Provide an example of a production function with two inputs that has constant Return to

Scale overall.

3. Write down a 2 by 2 matrix and its transpose (use letters like a,b,c,d)

4. Write down a 2 by 2 matrix, and a 2 by 3 matrix (with different letters for values in each

matrix cell), and multiply the matrixes together

1

1.2 Matrices and derivative (15 points)

1. (7p): Given X · β = Y , where X is a N by T matrix (where N > T), β is a T by 1 vector

and Y is a N by 1 vector. What does β equal to? (Assume the square matrix you obtain is

invertible)

2. (4p): derivative of

√

1

2πσ2

· e

(x−µ)

2

2·σ2

!

with respect to µ

3. (4p): derivatives of

A · (α · x

ρ + β · y

ρ + (1 − α − β) · z

ρ

)

1

ρ

with respect to x, to y, as well

as to z

2

2 Firm Maximization with K and L and Z (40 points)

2.1 A Firm with Three Inputs

A firm maximizes profit by choosing capital K, labor L and land Z. Firm output is:

Y = A · Kα

· L

β

· Z

ρ

Hiring each worker costs w dollars, renting each unit of capital costs r dollars, and renting each unit

of land costs q dollars. Output sell at price p dollars.

1. (4p) Write down the firm’s profit maximization problem

2. (5p) Derive the First Order Conditions

3. (4p) If we measured costs in cents rather than dollars, what happens to the first order conditions and optimal choices? Do units matter for optimal K, L and Z choices?

3

4. (4p) Identify MPL and MPK and MPZ, and their marginal costs

5. (4p) Log linearize first order conditions

6. (4p) Write the log linearized equations in matrix form as a system of linear equations.

4

2.2 A Firm with N Inputs

A firm maximizes profit by choosing N inputs: capital, X1, labor X2, land X3, … other inputs …,

XN−1, XN . The cost of each i input is wi

.

1. (5p) Write down the firm’s profit maximization problem with N inputs

2. (4p) Solve for the derivative of output with respect to the i’th input and the i + 1’th input

3. (6p) Log linearize the first order conditions and present the results in matrix form (use you

can dot dot dot to represent column and row details)

5

3 Utility Maximization with Apples and Bananas (15 points)

You are endowed with apples EA and bananas EB. You sell these at PA and PB to earn income.

You use the income to purchase apples A and bananas B given prices.

3.1 Solve (15 points)

You have Log utility over A and B: U(A, B) = log(A) + γ log(B)

1. Write down the maximization problem

2. Lagrangian and First Order Conditions

3. Marshallian demand for Apples and Bananas

6

4 Firm Maximization with K and L (30 points)

Firm output is constant returns to scale: Y = A · Kα · L

1−α. Hiring each worker costs w dollars,

renting each unit of capital costs r dollars.

4.1 Part One (15 points)

Solve the COO’s cost minimization problem given CRS

1. Write down the COO’s constrained problem

2. Lagrangian and First Order Conditions

3. Demand for Capital and Labor

4. If CEO wants 1 unit of output, how many worker and how much capital will COO hire?

7

4.2 Firm Maximization with constraint (15 points)

Suppose you have already solved the unconstrained firm’s optimal capital and labor choice problems,

with decreasing returns to scale. The solutions are: K∗,unc, L∗,unc. Now we have a labor constraint,

the total number of workers that the firm can possible hire is L¯.

Solve the constrained firm’s problem

1. Write down the constrained problem

2. What are the constrained optimal choices?

8

Typos modifications:

1. 1.2.1, wording adjusted “Y is a N by 1 vector”, said “T by 1 vector” prior.

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