91Ó°ÊÓ

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).

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

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.

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.

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:

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

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