Question

Schmutz Auto Wash provides car washes. Its production function is ? = 2?^1/2(? - 1)^1/2
where ? is cars washed per day, ? is daily hours of labor input, and ? is daily usage of capital inputs. The price of a unit of capital input is $48. The price of a unit of labor input is $16. In the short run, Schmutz has four units of capital input installed (so ? = 4).

a). Find Schmutz’s short run daily total cost function, short run daily variable cost function, and short run daily fixed costs.

b). Find Schmutz’s short run marginal cost function, short run average variable cost function, and short run total cost function.

c). Suppose that the market for car washes is perfectly competitive and the going market price of a car wash is ?. Find Schmutz’s short run daily supply function, including its shutdown price.

d). Find Schmutz’s profit in the short run if ? = 40.

Answer 1

Given:

The production function of the Schmutz Auto Wash:

where q is the number of cars washed per day. L is daily hours of labor input, and K is the daily usage of capital inputs.

The price of capital unit: r = $48

The price of labor unit: w = $16

In the short run, capital is fixed at 4. Therefore K = 4

a.) Short-run cost functions:

Short-run daily Total cost function:

Finding labor hours unit employed in the short as a function of output:

As K = 4

Therefore the number of daily hours of labor employed is this function of output.

Total cost is the sum of fixed cost and the variable cost. Here, the variable cost is the cost of labor units and the fixed cost is the cost of capital units.

Putting w = 16 and r = 48 and K = 4

Now, Putting the value of labor hours employed as a function of output q.

Therefore short-run daily total cost function:

Short-run Total variable cost (TVC) function:

The total variable cost function is the cost of the variable factor, here it is the cost of the labor hours unit employed.

Putting w = 16 and L as a function of output from the above conclusion.

That is,

Therefore short-run daily variable cost function;

Fixed cost:

Fixed cost is the cost of the fixed cost, here it is the cost of capital units employed.

here r = $48 and capital units are fixed at 4 (K=4)

Therefore the short-run daily fixed cost is $192.

b.)

Short-run marginal cost function:

Marginal cost is the additional cost that occurred on producing an additional unit. It is the derivative of the total cost function with respect to the output produced.

Therefore Schmutz's short-run Marginal cost function is:

Average variable cost function:

The average variable cost is the per unit variable cost. It is the ratio of total variable cost over the unit of output.

Therefore short run average variable cost functions:

Average total cost:

Short-run average total cost function: It is the per-unit total cost. It is the ratio of total cost over the quantity.

Therefore this is the short-run average total cost function.

c.) Short-run daily supply function:

The supply function is the portion of the MC curve where the price is greater than the average variable cost. Or in other words, it is the portion of MC curve aboce and beyond the AVC.

As in perfect competition, price is equal to the marginal cost. Therefore

Therefore, the supply function is the part of MC above and beyond AVC. Therefore Here, the supply function is the marginal cost function where q is greater than 4.

The direct supply function is the quantity as a function of price.

Shut down price:

Shut down price is where the Price is below the average variable cost. So it is the point of intersection of price line and average shut, below which every price is shut down price.

As price equal Marginal cost in the perfectly competitive market.

Putting q = 4 into the average variable cost to find the shutdown price.

Therefore shut down price is below the price of $8. At a price of $8, the firm is just covering its variable cost. So shut down price is:

d.) Schmutz'a profit in the short run if P = 40

Profit is the difference between total revenue and total cost.

In the perfect competition, the firm produced where the price is equal to the marginal cost. Therefore P = MC

MC = 40

Therefore in the short run, Schmutz's washes 20 cars per day.

Now total cost at this quantity.

q= 20

The total cost is $608

Now, total revenue is the price times quantity. here, P = 40 and q = 20

Total revenue is $800

Therefore profit in the short run is;

Therefore Schmutz'a profit in the short run is $192