Question

1. (a) A firm is a monopoly for the good it produces. It has average variable cost function Cvar = 2q2 + 1 (where q represents the production level) and it has fixed costs of 8. If a maximum profit of 36 is achieved for a production level of 5, determine the law of demand for this good as a linear relation of the type p = a q + b where p is the price and a and b constants. [7 Marks]

Answer 1

Solution:

Profit = total revenue - total cost

Furthermore, total cost = total variable cost + total fixed cost

Total variable cost = average variable cost*quantity, so, TVC= (2q² + 1)*q = 2q³ + q

And, TC = 2q³ + q + 8

Total revenue = price*quantity

TR = (aq + b)*q = aq² + bq

Profit is maximized where marginal revenue = marginal cost

Marginal revenue, MR = dTR/dq = 2aq + b

Marginal cost, MC = dTC/dq = 3*2q² + 1 + 0 = 6q² + 1

So, at profit maximization, 2aq + b = 6q² + 1

As this condition is satisfied at production level of 5, that is q = 5, we have: 2a*5 + b = 6*5² + 1

10a + b = 151 ... (1)

Also, profit at this level is given as 36, so, 36 = (a*5² + b*5) - (2*5³ + 5 + 8)

36 = 25a + 5b - 263

So, 25a + 5b = 299 ... (2)

Now, solving (1) and (2) simultaneously, we can solve for values of a and b (2 linear equations in 2 unknowns will give exactly one solution)

b = 151 - 10a

So, 25a + 5*(151 - 10a) = 299

25a + 755 - 50a = 299

25a = 755 - 299

a = (755 - 299)/25 = 18.24

And, b = 151 - 10*18.24 = -31.4

So, demand function is p = 18.24*q - 31.4

Note that the demand function in this case does not indicate a negative relation between the price charged and quantity demanded (a is positive), so law of demand does not hold here.