Assignment 2

1. For the production function

y = 3x₁ + 2x₂,

Find

a. The MPP of x₁.

b. The MPP of x₂.

c. The marginal rate of substitution of x₁ for x₂.

2. Suppose that the production function is given by

y = x₁^0.5x₂^0.3

a. Set up a Lagrangean optimization problem using this production function. Derive firstorder conditions.

b. Suppose that the output, y, sells for $4.00 per unit and that x₁ and x₂ both sell for$0.10 per unit? How much x₁and x₂would the farmer purchase in order to maximize profit?

3.Suppose that the production function is

y = x₁x₂

The input x₁sells for $1 per unit and input x₂sells for $2 per unit. The farmer has $200 tospend on x₁and x₂. How much of each input will the farmer purchase in order to be at a pointof constrained output maximization?

4. Assume that the production function is

y = x₁^0.5x₂^0.33

x₁costs $1 per unit; x₂costs $2 per unit. Find the corresponding total cost function with totalcost expressed as a function of output (y), the input prices, and the production functionparameters.

5. Assume that the production function is

y = x₁^0.5x₂^0.5

The price of y is $10 per unit, and the price of x₁ and x₂are each $2 per unit. How much ofeach of x₁ and x₂ would the manager demand if he or she had but $100 to spend on x1 and x₂?Now suppose that the price of x₁increases to $10 per unit, and the manager has the same$100 to spend. How much of x₁and x₂would the manager demand?

6. A market demand curve for a monopoly firm is P = 120 - Q, andmarginal cost curve is MC = 10 + Q.

a. what is the firm's total revenue curve?

b. what is the firm's marginal revenue curve?

c. what is the monopoly quantity?

d. what is the monopoly price?

e. What is price elasticity of demand at the monopoly price and quantity?

7.Assume that market demand for a perfectly competitive market in the short run is and market supply is P=Q_s. Denoting firm level quantity by q, assume TC=50+4q+2q² so that MC=4+4q.

a) What is the market equilibrium price and quantity?

b) How many firms are in the industry in the short run?

c) Do firms make a profit or loss in the short run, and how much are these profits/losses?

d) What is the equilibrium price in the long run?What will be equilibrium profit in the long run?How many firms will there be in the long run?

Hint, for the last part of the question, assume that there can be fractional firms, if necessary - if the numbers of firms are in units of 10,000, for example, the answer will be fine.Moreover, assume the entry or exit in the industry will cause the supply curve to shift, while the demand curve does not shift.Therefore, industry output can be found by taking the long-run price and plugging it into the demand curve.

8. An industry with two firms (called a duopoly) faces a market demand described by P=240-Q

where Q is the sum of the individual firms' outputs. Each firm faces costs TC(q)=100+q².

a. find the profit maximizing output assuming that the two firms cooperate

b. assume the firms cannot cooperate to maximize their profits, find the profit maximizing output for each firm.