Propylene is used to make plastic. The propylene industry is perfectly competiti

Question

Propylene is used to make plastic. The propylene industry is perfectly competitive and each producer has a long run total cost function given by LTC = 1/3 Q^3 - 6Q^2+40Q Where Q denotes the output of the individual firm. The market demand for propylene is X = 2200 - 100P Where X and P denote the market output and price respectively. Calculate the optimal output produced by each firm at the long run competitive equilibrium (LRCE). Calculate the market price and market output at the LRCE. Calculate the number of firms at the LRCE. Suppose the demand curve shifts to X = A - 100P Where A is a positive number. Calculate how large A would have to be so that in the new LRCE, the number of firms is twice what it was in the initial equilibrium.

Explanation / Answer

In perfect competetive market, a firm is price taker and the industry is price maker. Each firm sells same product (with no differentiation) at same price. Therefore, its demand curve equals average revenue and marginal revenue. And we know, the optimal output level is the level at which a firm's marginal cost curve cuts marginal revenue curve from below in short run. But in long run a firm in perfect competitive market earns no super normal profit which means all its cost equals all its revenue i.e TC = TR.

Or MC = ATC

ATC = (1/3) - 6Q + 40

and MC = 1/3 -12Q + 40

(1/3) - 6Q + 40 =1/3 -12Q + 40

6Q = 12Q

Note : the given total cost function does nt hold good as it is impossible to find Q from the above equation but the method is correct. The asker can use the method by applying the correct total cost function. The resultant Q will be each firms' output.

(b) In long run ATC equals Price.

In long run equilibrium, there must be zero profits. Therefore, rewriting the profit function,

p = TR - TC = P*q - ATC*q = (P - ATC) *q

We can see that zero profit requires that P = ATC.

ATC = (1/3)Q-6Q2+40Q / Q = 1/3 - 6Q + 40

Putting Q (each firms' output) = 5 (assuming for explanation)

ATC = 10.33 = 10 (approx)

And we will get Market Output by putting the price into the demand function

Q = 2200-100P

Q = 2200 - 1000 = 1200 units

Thus the market output is 1200 units.

(c) We have total market output = 1200 units and each firms' output = 5 units

Thus No. of firms = Market output / each firm output = 1200/5 = 240 firms.

(d) If new demand function is Q = A - 100P

A should be greater than 100*P to give positive demand. Thus A should be greater that 100* 10 i.e 1000.

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