What is the lowest price at which each firm would sell its output in the long run? Is profit positive, negative, or zero at this price? Explain.
In the long run the firm will not sell for a price that is below minimum average cost. At any price below minimum average cost, profit is negative and the firm is better off selling its fixed resources and producing zero output. To find the minimum average cost, set marginal cost equal to average cost and solve for q:
Therefore, the firm will not sell for any price less than 3.8 in the long run.
What is the lowest price at which each firm would sell its output in the short run? Is profit positive, negative, or zero at this price? Explain.
The firm will sell for any positive price, because at any positive price marginal cost will be above average variable cost (AVC=q/2000). Profit is negative as long as price is below minimum average cost, or as long as price is below 3.8.
11. Suppose that a competitive firm has a total
cost function
and a marginal cost function
.
If the market price is P=$115 per unit, find the level of output
produced by the firm. Find the level of profit and the level of
producer surplus.
The firm should produce where price is equal to marginal cost so that P=115=15+4q=MC and q=25. Profit is
.
Producer surplus is profit plus fixed cost, which is 1250. Note
that producer surplus can also be found graphically by calculating
the area below the price line and above the marginal cost (supply)
curve, so that PS=0.5*(115-15)*25=1250.
12. A number of stores offer film developing as a
service to their customers. Suppose that each store that offers this
service has a cost function
and a marginal cost
.
If the going rate for developing a roll of film is $8.50, is the industry in long run equilibrium? If not find the price associated with long run equilibrium.
First find the profit maximizing quantity associated with a price of $8.50 by setting price equal to marginal cost so that MC=0.5+0.16q=8.5=P, or q=50. Profit is then 8.5*50-(50+0.5*50+0.08*50*50)=$150. The industry is not in long run equilibrium because profit is greater than zero. In a long run equilibrium, firms produce where price is equal to minimum average cost and there is no incentive for entry or exit. To find the minimum average cost point, set marginal cost equal to average cost and solve for q:
To find the long run price in the market, substitute q=25 into either marginal cost or average cost to get P=$4.50.
Suppose now that a new technology is developed which will reduce the cost of film developing by 25%. Assuming that the industry is in long run equilibrium, how much would any one store be willing to pay to purchase this new technology?
The new total cost function and marginal cost function can be found by multiplying the old functions by 0.75 (or 75%) and are as follows:
The firm will set marginal cost equal to price, which is $4.50 in the long run equilibrium. Solve for q to find that the firm will develop approximately 34 rolls of film (rounding down). If q=34 then profit is $33.39. This is the most the firm would be willing to pay for the new technology. Note that if all firms adopt the new technology and produce more output, then price in the market will fall and profit for each firm will be reduced to zero.
13. Consider a city that has a number of hot dogs stands operating throughout the downtown area. Suppose that each vendor has a marginal cost of $1.50 per hot dog sold, and no fixed cost. Suppose the maximum number of hot dogs any one vendor can sell in a day is 100.
If the price of a hot dog is $2, how may hot dogs does each vendor want to sell?
Since marginal cost is equal to 1.5 and the price is equal to 2, the hot dog vendor will want to sell as many hot dogs as possible, or in other words, 100 hot dogs.
If the industry is perfectly competitive will the price remain at $2 for a hot dog? If not, what will the price be?
The price should fall to $1.50 so that price is equal to marginal cost. Each hot dog vendor will have an incentive to lower the price of a hot dog below $2 so they can sell more hot dogs than their competitors. No hot dog vendor will sell a hot dog for a price below marginal cost, so the price will fall until it reaches $1.50.
If each vendor sells exactly 100 hot dogs a day and the demand for hot dogs from vendors in the city is Q=4400-1200P, how many vendors are there?
If price is 1.50 then Q=4400-1200*1.5=2600 in total. If each vendor sells 100 hot dogs then there are 26 vendors.
Suppose the city decides to regulate hot dog vendors by issuing permits. If the city issues only 20 permits, and if each vendor continues to sell 100 hot dogs a day, what price will a hot dog sell for?
If there are 20 vendors selling 100 hot dogs each then the total number sold is 2000. If Q=2000 then P=$2, from the demand curve.