Question

The lamp industry is perfectly competitive. The price of a lamp is $50. A representative firm’s cost function is TC=1000+20Q+5Q².

h. What output for the representative firm will minimize ATC? (hint: use solver or find by solving MC=ATC, or if you are calculus whiz find ATC’=0)
i. Is the lamp industry in equilibrium? Why or why not?
j. Suppose total industry demand at P=$50 is 3000. How many firms are operating in the industry?
k. What would you expect to happen to future firm population in the lamp industry?
l. What would you expect to happen to future price in the lamp industry?
m. What industry price will result in LR industry equilibrium?

Answer 1

(h) Using solver, you may solve for the problem as follows. The left hand side table shows the solver solution, where as the right hand side table shows the formula view of the calculation to set up the solver problem. The ttal cost is given which has been applied in cell B2. The formula for AC and MC are applied in cells C2 and D2 respectively.

Once you set up the spreadsheet, you may set up the solver window as shown below. Note that the objective is to minimize AC (cell $C$2). Add a constraint such as AC = MC, i.e., $C$2 = $D$2 as shown in the screenshot below.

Solution by calculus can also be found as follows:

So, second order condition for minimization is satisfied and hence, the average cost is minimized at Q = 14.14 (same as solver solution).

Minimized average cost = (1000/14.14) + 20 + 5 (14.14) = 161.42

(i) For a longrun equilibrium, price = minimum average cost. Since price = 50 is less than average cost = 161.42, the longrun equilibrium is not achieved.

(j) Each firm produces 14.14 and total industry demand = 3000, then number of firms = 3000/14.14 = 212 (rounded).

(k) Since price < minimum average cost, some firm will go out of market in future so that output will fall and price will rise to match the minimum average cost. Or there may be some innovation which reduces the average cost so that it may match the price.

(l) Assuming there won't be any innovation, price will rise in future to match the average cost.

(m) An industry price of P = min AC = 161.42 will result in longrun equilibrium in the industry.