Question #114969

QNO2 Crystal Ball Corp. has estimated its demand and cost functions as follows:

Q = 80 - 5P

C = 30 + 2Q + 0.5Q 2

where P is in $, Q is in thousands of units and C is in $,000.

a. Calculate the profit-maximizing price and output.

b. Calculate the size of the above profit.

c. Calculate the price elasticity of demand at the above output; is demand elastic or inelastic here? What should it be?

d. Calculate the marginal cost at the above output.

e. If unit costs rise by $2 at all levels of output and the firm raises its price by the same amount, what profit is made?

f. What is the profit-maximizing strategy given the above rise in costs?

g. How much profit is the firm forgoing by raising its price $2?

Q = 80 - 5P

C = 30 + 2Q + 0.5Q 2

where P is in $, Q is in thousands of units and C is in $,000.

a. Calculate the profit-maximizing price and output.

b. Calculate the size of the above profit.

c. Calculate the price elasticity of demand at the above output; is demand elastic or inelastic here? What should it be?

d. Calculate the marginal cost at the above output.

e. If unit costs rise by $2 at all levels of output and the firm raises its price by the same amount, what profit is made?

f. What is the profit-maximizing strategy given the above rise in costs?

g. How much profit is the firm forgoing by raising its price $2?

Expert's answer

a. The profit-maximizing price and output are:

MR = MC,

Q = 80 - 5P, so P = 16 - 0.2Q, MR = TR'(Q) = 16 - 0.4Q,

MC = C'(Q) = 2 + Q,

16 - 0.4Q = 2 + Q,

Q = 10 units,

P = 16 - 0.2*10 = 14.

b. The size of the above profit is:

"TP = TR - C = 14*10 - (30 + 2*10 + 0.5*10^2) = 40."

c. The price elasticity of demand at the above output is:

"Ed = -b\\times P\/Q = -5\\times14\/10 = -7," so demand is elastic.

d. The marginal cost at the above output is:

MC = 2 + 10 = 12.

e. If unit costs rise by $2 at all levels of output and the firm raises its price by the same amount, its profit will decrease.

f. The profit-maximizing strategy given the above rise in costs is to decrease price to increase total revenue.

g. The firm is forgoing a huge portion of profit by raising its price $2.

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