Question

Christine has exactly 4 units of labor and 1 unit of capital to produce 1 door.

a)What is the production function using D as the amount of doors , K as the amount of capital and L as the amount of labor? Derive the input demand function for K and L.
b) what is total (long run) cost function of output (D), price of labor (w) and price of capital (r)?
c) What is her MC, ATC and AVC?
d)What is the lowest price that she will produce at in the long run?
e) What is the breakeven and shutdown in the short run if she has already bought 8 units of capital priced at $50 and each labor unit cost $10

Answer 1

a) Input Demand Functions: The demand for labor L is
`\((K)/(4)\)`, and the demand for capital K is
`\((L)/(4)\)` in the production of doors. b) Total Long-Run Cost Function: The total long-run cost is given by C = 10L + 50K, considering labor costing $10 per unit and 8 units of capital priced at $50 each. c) MC, ATC, and AVC: These can be calculated using the relevant derivatives and the given cost functions. d) Lowest Price to Produce: The lowest price is where Marginal Cost (MC) equals the minimum Average Total Cost (ATC). e) Breakeven and Shutdown: At breakeven, door revenue equals total cost; shutdown occurs if total revenue is less than total variable cost.

a) Production Function and Input Demand Function:

The production function is given by D = f(K, L), where D is the quantity of doors, K is the amount of capital, and L is the amount of labor.

Given that Christine has 4 units of labor (L) and 1 unit of capital (K) to produce 1 door (D), the production function is
`\(D = f(K, L) = (L)/(4) \cdot (K)/(1) = (KL)/(4)\).`

The input demand functions are obtained by partial differentiation:

- For labor (L):
`\((\partial D)/(\partial L) = (K)/(4)\).`

- For capital (K):
`\((\partial D)/(\partial K) = (L)/(4)\).`

b) Total Long Run Cost Function:

The total long-run cost function is given by C = wL + rK, where w is the price of labor, L is the amount of labor, r is the price of capital, and K is the amount of capital.

c) Marginal Cost (MC), Average Total Cost (ATC), and Average Variable Cost (AVC):

- Marginal Cost (MC) is the derivative of the total cost function with respect to quantity (D).

- Average Total Cost (ATC) is the total cost divided by the quantity (D).

- Average Variable Cost (AVC) is the variable cost divided by the quantity (D).

d) Lowest Price to Produce in the Long Run:

The lowest price to produce in the long run is where MC equals the minimum ATC.

e) Breakeven and Shutdown in the Short Run:

- Breakeven occurs when total revenue equals total cost.

- Shutdown occurs when total revenue is less than total variable cost.

- Each unit of labor (L) costs $10.

- Christine has bought 8 units of capital (K) priced at $50 each.

Breakeven:

Breakeven occurs when total revenue (TR) equals total cost (TC).

TR = D · P

TC = wL + rK

Since TR = TC at breakeven, we can equate the expressions:

D · P = wL + rK

Shutdown:

Shutdown occurs when total revenue (TR) is less than total variable cost (TVC).

TVC = wL

If TR < TVC, then Christine should consider shutting down.