Math for Economists Midterm

Fan Wang

March 10, 2021

Note: The test has 115 points.

1 Derivatives

1.1 Production Functions and Elasticity (10 points, 2 each)

1. What is a linear equation? give one example below:

2. Firm A has Cobb-Douglas production function with K and L inputs, write down the production function:

3. Firm B has Cobb-Douglas production function with K as input and the production function

has constant returns to scale, write down the production function:

4. Firm C has a linear production function with input K. Output when K = 0 is a, and each

additional unit of capital leads to b units of increase in output, write down the production

function:

5. For firm C, what is the elasticities of the input with respect to output?

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1.2 Derivatives (15 points)

Take derivative below:.

1. (2p): derivative of (a + b · x) with respect to a

2. (3p): derivative of

a + b · c

−d

with respect to d

3. (3p): derivative of

(a+b+c+d)·(x+y)

(E+F +G)

(h+i+g)

with respect to (x + y)

4. (3p): derivative of (log (a + b · x)) with respect to x

5. (2p): derivative of

√

1

2πσ2

· e

(x−µ)

2

2·σ2

!

with respect to σ

6. (2p): derivate of

A · (α · x

ρ + β · y

ρ + (1 − α − β) · z

ρ

)

1

ρ

with respect to x

1.3 First Order Taylor Approximation (15 points)

1. (3p) Write the definition/formula for first order Taylor approximation

2. (9p) We have a supply curve for credit which is a function of interest rate r:

Supply(R) = Qs = A −

B

(1 + r)

We have a demand curve for credit which is a function of interest rate r:

Demand(r) = Qd =

C

rD

Provide the First Order Taylor approximation of Qs and Qd around some interest rate level

r0:

3. (3p) Given the First Order Taylor approximations, which are linear, solve for the equilibrium

price and quantity.

3

2 Log and Exponential (10 points, 2.5 each)

1. Expand (linearize) the function below by taking log.

Y =

exp(A + ) · a

α · b

β

c

θ

· d

φ

2. What is the marginal effect of a change in A on the marginal effect of b on y?

3. Given the equation below, what is the impact of an additional unit of education on log wage?

log(wage) = a + b · edu + c · edu2 + d · exp +

4. Given the equation above, what is the impact of an additional unit of education on wage?

4

3 Household Problem (25 points)

1. (5p) Write down the household utility maximization problem, where the household chooses

how much to save/borrow given endowment in period 1 and period 2.

2. (10p) Solve this problem analytically and derive the optimal savings/borrowing equation.

3. (4p) Under what condition does the household borrow?

4. (3p) Analyze the marginal effect of the discount factor on optimal borrowing/savings.

5. (3p) Analyze the elasticity of first period endowment on optimal borrowing/savings.

5

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

4.1 A Firm with Three Inputs

A firm maximizes profit by choosing land Z. There are fixed labor L and fixed capital K levels that

the firm does not choose. 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 as well as the second derivatives. Is the profit function concave or convex?

Is the production function concave or convex?

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 Z choices?

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4. (4p) Identify marginal costs, marginal revenue and marginal product of Z. Even though K and

L are fixed, what would be the marginal impact of K on the marginal cost/revenue/product

of Z?

5. (4p) Solve for the optimal Z choice. What is the elasticity of the demand for Z with respect

to prices w, r, and q?

6. (4p) Is it possible that our firm might lose money, even though it is solving a profit maximization problem? (Note that K and L are fixed.)

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4.2 A Firm with N Inputs

A firm with Cobb-Douglas production function maximizes profit by choosing the first input of N

inputs: capital, X1, labor X2, land X3, … other inputs …, XN−1, XN . The cost of each i input is

wi

. All inputs except for the first input is fixed.

1. (5p) Write down the firm’s profit maximization problem (note you should use the product

symbol Q

: x1 · x2 · x3 =

Q3

i=1 xi)

2. (4p) Solve for the derivative of output with respect to the 1st input.

3. (3p) What is the marginal product of the 1st input? What must this be equal to?

4. (3p) Log linearize the first order condition for the 1st input.

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