Additional Instructions (read carefully!)

Hand in only one document per group. You will all receive the same grade.

Max 4 students per group with names entered below (except with prior approval).

Name

Student ID

1.Suppose you run a firm that produces output y with the Cobb-Douglas production function: y = .

a.Demonstrate the returns to scale of this technology.

The sum of the power is 1 .5 and .5 is so it can be called the it has constant return of technology

b.If x₂ is fixed at 1 unit, derive the marginal product of input 1.

c.If both the price of x₁ (w₁) and the output price (p) are equal to 1, how much of input 1 (x₁) should the firm produce in the short run to maximize profits?(hint: use the profit maximizing "rule of thumb")

d.What is the profit maximizing firm's output level?

e.What are the firm's revenues?

f.If w₂ = 1, what are the firm's profits?

g.Should the firm shut down?

h.Now suppose we are analysts outside the firm trying to estimate its technology.In the simple one variable input case (x₁), what variables must we observe to estimate the production function based on the Revealed Profitability technique discussed in class and in the text? How many observations of these variables do we need to get a good estimate?

Variables:

Number of observations?

2.Now switch perspective to the firm's cost side assuming the same technology: y = .

a.Again assuming, in the short-run, x₂ is fixed at 1 unit, derive the equations and graph of the firm's short-run cost curves:Total Cost (C(y)), ATC, AVC, and MC.

Derivation:

Graph:

b.If the firm again faces an output price of 1, use the MR=MC rule to compute the firm's profit maximizing output level.

c.Compare the solution (y*) form the profit maximizing approach in question 1 to the solution using the MR = MC rule.

d.What is the firm's supply curve?

e.What is the firm's shut down price?

f.Suppose there are Z firms in this market.Write the equation for market supply.

Econ 3033 - Professor Horowitz - Spring 2020- HW 4

Distributed: 4/16 on Blackboard.

Due: 4/23: As email attachment

Additional Instructions (read carefully!)

Hand in only one document per group. You will all receive the same grade.

Max 4 students per group with names entered below (except with prior approval).

Name

Student ID

1.Suppose you run a firm that produces output y with the Cobb-Douglas production function: y = .

a.Demonstrate the returns to scale of this technology.

The sum of the power is 1 .5 and .5 is so it can be called the it has constant return of technology

b.If x₂ is fixed at 1 unit, derive the marginal product of input 1.

c.If both the price of x₁ (w₁) and the output price (p) are equal to 1, how much of input 1 (x₁) should the firm produce in the short run to maximize profits?(hint: use the profit maximizing "rule of thumb")

d.What is the profit maximizing firm's output level?

e.What are the firm's revenues?

f.If w₂ = 1, what are the firm's profits?

g.Should the firm shut down?

h.Now suppose we are analysts outside the firm trying to estimate its technology.In the simple one variable input case (x₁), what variables must we observe to estimate the production function based on the Revealed Profitability technique discussed in class and in the text? How many observations of these variables do we need to get a good estimate?

Variables:

Number of observations?

2.Now switch perspective to the firm's cost side assuming the same technology: y = .

a.Again assuming, in the short-run, x₂ is fixed at 1 unit, derive the equations and graph of the firm's short-run cost curves:Total Cost (C(y)), ATC, AVC, and MC.

Derivation:

Graph:

b.If the firm again faces an output price of 1, use the MR=MC rule to compute the firm's profit maximizing output level.

c.Compare the solution (y*) form the profit maximizing approach in question 1 to the solution using the MR = MC rule.

d.What is the firm's supply curve?

e.What is the firm's shut down price?

f.Suppose there are Z firms in this market.Write the equation for market supply.