1. Given Q = 560 – 10P and TC = 6000 + 5Q for an oligopolistic...

1. Given Q = 560 – 10P and TC = 6000 + 5Q for an oligopolistic firm, determine mathematically the price and output at which the firm:

a. Maximizes its total profits and calculate those profits

b. Maximizes its total revenues and calculate the profits are that price and quantity

c. Maximizes its total revenue in the presence of a $480 profit constraint (20 points)

• Reference Figure 10-6 on p. 442 •

See also Sales Maximization Model on pp. 467-468

• Part (a) is the standard MR = MC procedure.

• For part (b) you are looking for the turning point of the TR function. Note that the derivative of a function measures its rate of change and when a that derivative (the Marginal Revenue) equals 0 it is at its turning point (in this case, maximum)

• The profit constraint for the sales maximizer means that he must earn at least $480 and cannot maximize sales (he will come as close as he can given the constraint). To do this write a total profit equation (TR minus TC) and set it equal to $480

• Solve the equation by turning it into a quadratic equation and use the quadratic formula to find the quantities that satisfy the equation. Since the firm is a sales maximizer, the larger of the two will be the one chosen.

• Hint: Although the question doesn’t require that you find the profit for part (c), you can check to see if you’ve done it correctly by calculating the profit. You set it equal to $480 so it should be $480

Solutions

Expert Solution

(a) Here we apply the standard MR-MC approach where we quate MR=MC to get the profit maximising values of P and Q and profit comes out to be $502.5

(b) Here we have to get the revenue maximising values of P and Q which we get by equating MR=0. At these values, only reevnue gets maximised from $7777.5 to $7840 but profits are lower than at the equilibrium level.

(c) Here we are given a constraint of profit to be maximum of $480. So, we equate the profit equation to $480 i.e., TR-TC=480. we solve the quadractic equation and get 2 values of Q as 354.12 and 155.88 units. we will choose the higher of the two because our motive is to maximise revenue which is possible only by maximising sales. At this level of Q, TR= $7292.4 approx

Rahul Sunny answered 5 months ago

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