Question

A number of stores offer film developing as a service to their customers. Suppose that each store offering this service has a cost functionC(q) = 50 + 0.5q+ 0.08q² and a marginal costMC= 0.5 + 0.16q.

a. 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.

b. Suppose now that a new technology is developed which will reduce the cost of film developing by 25 percent. Assuming that the industry is in long-run equilibrium, how much would any one store be willing to pay to purchase this new technology?

Short answer

1. The industry is not in long-run equilibrium. The price associated with long-run equilibrium is $4.5.
2. A store owner would be willing to pay the price of $28.13 to purchase the new technology.

Step-by-step solution

Finding whether the economy is in a long-run equilibrium position or not

The firm will choose an output level where it can maximize its profit. The optimal output is determined by the profit-maximizing firms by equating marginal cost (MC) with marginal revenue (MR).

$$\begin{gathered} MR=\frac{d\left(TC\right)}{dq} \\ =\frac{d\left(p\times q\right)}{dq} \\ =\frac{d\left(8.50q\right)}{dq} \\ =8.50 \\ MC=MR \\ 0.5+0.16q=8.50 \\ 0.16q=8 \\ q=50 \end{gathered}$$

The equilibrium quantity for price $8.50 is 50 units.

The LAC and LMC for the firm at price level $8.50 and output 50 are calculated below:

$$\begin{gathered} LMC(\$)=0.5+0.16q \\ =0.5+0.16\left(50\right) \\ =8.50 \\ LAC(\$)=\frac{TC}{q} \\ =\frac{50+0.5\left(50\right)+0.08\left({50}^2\right)}{50} \\ =1+0.5+4 \\ =5.5 \end{gathered}$$

For a firm in long-run equilibrium, its long-run average cost (LAC) and long-run marginal costs (LAC) should be equal. But the values of LMC and LAC are not equal for this firm at the price level of $8.50. Hence, the firm is not in a long-run equilibrium position.

Calculating the long-run equilibrium price for the firm

The firm would be in a long-run equilibrium position when its long-run average cost (LAC) equals long-run marginal costs (LAC).

$$\begin{gathered} LAC=LMC \\ \frac{50+0.5q+0.08q^2}{q}=0.5+0.16q \\ 0.08q=\frac{50}{q} \\ q=25 \end{gathered}$$

The optimal level of output for the firm in the long-run position is 25 units.Since the firm is in a long-run equilibrium position, it must be earning zero profit. Thus, the firm's total revenue (TR) would be equal to the total costs (TC). Applying this concept and putting the value of q =25, the long-run equilibrium price is calculated below:

TR = TC

25p = 50 + 0.5(25) + 0.08(25)²

25p=112.5

p=$4.5

The equilibrium price of the firm, in the long run, is $4.5.

Determining the price a store owner is willing to pay for the purchase of new technology

The value of the total cost is $112.5. The new technology will bring down the cost by 25 percent. Since the market is in a long-run equilibrium position, the price and output will remain unaffected.Hence, the price a store would be willing to pay for the new technology would be equal to the amount reduced in total cost because of the technology.

The reduction in the amount of total cost because of technology is calculated below:

$$\begin{gathered} ReductioninTC=25\%ofTC \\ =\frac{25}{100}\times 112.5 \\ =28.13 \end{gathered}$$

The price the store would be willing to pay on the purchase of new technology is $28.13.