Question

A firm faces the following demand function: p = 1100 − 30q. As usual, p and q are the price and the quantity of the good. The cost function for this firm is defined by C = 20q2 + 100q + 1500.

(a) Is it economically reasonable to have q = 40? Explain.

(b) Write down the profit function for this firm.

(c) Find the optimal level of production q∗ and check whether it pertains to a maximum or to a minimum.

Answer 1

Ans. Demand function, p = 1100 - 30q

and Cost function, C = 20q^2 + 100q + 1500

a) at q = 40, from the demand fuction,

p = 1100 - 30*40

=> p = -100

As negative price level is not possible, so, it is not economically reasonable to have q = 40 units.

b) Total Revenue, TR = p*q = 1100q - 30q^2

The profit function, G = TR - C

=> G = 1100q - 30q^2 - (20q^2 +100q + 1500)

=> G = 1000q - 50q^2 -1500

c) To find the optmal quantity of output which maximises profit, differentiate the profit function with respect to q and equate the result with 0

=> dG/dq = 1000 - 100q = 0

=> q = 10 units

To check whether this is maximum or not, differentiate the profit function again with respect to q and substitute the above determined value of q,

d^2G/dq^2 = -100 < 0

As second order derivative is negative, so, at q = 10 units level of output, the firm is maximizing its profit. Thus, optimal level of production is q* = 10 units

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