5. Define Q to be the level of output produced and sold, and assume that the fir
ID: 1252261 • Letter: 5
Question
5.Define Q to be the level of output produced and sold, and assume that the firm’s
cost function is given by the relationship
TC = 20 + 5Q + Q2
Furthermore,assume that the demand for the output ofthe firm is a function of
price P given by the relationship
Q = 25 -P
a. Define total profit as the difference between total revenue and total cost,
express in terms ofQthe total profit function for the firm.(Note: Total
revenue equals price per unit times the number ofunits sold.)
I don't understand how to put together the profit function.
b. Determine the output level where total profits are maximized.
c. Calculate total profits and selling price at the profit-maximizing output level.
d. If fixed costs increase from $20 to $25 in the total cost relationship,determine
the effects ofsuch an increase on the profit-maximizing output level and total
profits.
6.
Using the cost and demand functions in Exercise 5:
a. Determine the marginal revenue and marginal cost functions.
b. Show that,at the profit-maximizing output level determined in part (b)
of Exercise 5,marginal revenue equals marginal cost.This illustrates the
economic principle that profits are maximized at the output level where
marginal revenue equals marginal cost.
Explanation / Answer
5.
Define Q to be the level of output produced and sold, and assume that the firm’s
cost function is given by the relationship
TC = 20 + 5Q + Q2
Furthermore,assume that the demand for the output ofthe firm is a function of
price P given by the relationship
Q = 25 - P meaning that P = 25 - Q
REVENUE = (25 - Q)(Q)
a. Define total profit as the difference between total revenue and total cost,
express in terms ofQthe total profit function for the firm.(Note: Total
revenue equals price per unit times the number ofunits sold.)
Profit = REVENUE minus COST
Profit = (Q)(25 - Q) - (20 + 5Q + Q2) =
(note that the highlighted portion equals price)
25Q - Q2 - 20 - 5Q - Q2
PROFIT = 20Q - 20 - 2Q2
I don't understand how to put together the profit function. Notice that total revenue is the same thing as price times quantity. Then just subtract total costs. :)
b. Determine the output level where total profits are maximized
Take derivative and set it equal to zero.
dP/dQ = 20 - 4Q
4Q = 20
Q = 5
c. Calculate total profits and selling price at the profit-maximizing output level. Plug in Q = 10 into PROFIT = 20Q - 20 - Q^2
PROFIT = 20(5) - 20 - 25 =
100 - 45
$55
d. If fixed costs increase from $20 to $25 in the total cost relationship,determine
the effects ofsuch an increase on the profit-maximizing output level and total
profits.
NEW PROFIT FUNCTION =
20Q - 25 - Q^2 = PROFIT
However, the DERIVATIVE of profit, dP/dQ, still equals
20 - 2Q
Thus a change in fixed cost has no effect on the output level.
Total profits simply decrease by $5 to new profit = $50. That's all.
6.
Using the cost and demand functions in Exercise 5:
a. Determine the marginal revenue and marginal cost functions.
MR = derivative of revenue = derivative of (Q)(25 - Q) = 25 - 2Q
MC = derivative of total cost = 5 + 2Q
b. Show that,at the profit-maximizing output level determined in part (b)
of Exercise 5,marginal revenue equals marginal cost.This illustrates the
economic principle that profits are maximized at the output level where
marginal revenue equals marginal cost.
MC = MR
25 - 2Q = 5 + 2Q
20 = 4Q
Q = 5
this matches our original finding, that Q = 5
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