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17. Profit-Maximizing Quantity
Essay:
1. At what level of output are profits maximized? At what price? Carefully explain.
Problems
2. Given demand and cost information, be able to find the profit maximizing output and price. Be able to find
revenue, marginal revenue, total cost, fixed cost, variable cost, average variable cost, marginal cost, average cost,
and profit.
The Profit-Maximizing Price and Quantity
A firm that sets both the price and quantity maximizes profits by
producing a quantity at which the marginal revenue is equal to the marginal
cost. The marginal revenue is the increase in revenue a firm collects if it
produces and sells another unit of output. The marginal cost is the increase
in cost if a firm produces another unit of output. The firm charges what the
market will bear at the profit maximizing output. That is, it charges the
highest price it can and still sell all of the profit-maximizing output.
The argument is simple, though a bit long for a complete answer. One
must first explain what a firm would do if marginal revenue was greater than
marginal cost and then explain what it would do if marginal revenue was less
than marginal cost. It is also helpful why a firm will not expand or
contract output if marginal revenue is equal to marginal cost.
Suppose a firm is producing a level of output such that the marginal
revenue is greater than the marginal cost. If a firm increases its output by
one unit, it can sell one more unit and its revenues increase. The increase
in revenue is marginal revenue. But to produce more, the firm must purchase
more resources, which adds to its costs. The increase in costs is marginal
costs.
Since marginal revenue is greater than marginal cost, the firm’s
revenue rises by more than its costs. That implies that profits rise (or
perhaps losses fall) when the firm produces another unit of output.
Suppose a firm is producing an output such that marginal revenue is
less than marginal cost. If the firm produces one less unit, it will have
less to sell, so it will earn less revenue. The decrease in revenue is the
marginal revenue. When the firm sells one less unit, it can also cut its
production. It will no longer have to pay for the resources that were
necessary to produce that unit of output, so its total costs fall. Marginal
cost is the decrease in cost when output falls one unit.
With marginal
revenue being less than marginal cost, the reduction of output by one unit
causes revenue to fall less than costs. In other words, costs fall by more
than revenue. This will increase profit (or perhaps cut losses.)
Profit Maximizing Output: Algebra
Remember the simple demand function?
Our example was:
Qd = 65,060 – 20P
If you solve that function for price, you get:
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P = 65,060/20
-
1/20 Q
Or
P = 3253- .05Q
That is the price function.
The revenue function is found by multiplying the price function by Q.
That’s because revenue is price times quantity.
R = (3253 - .05Q) *Q
R = 3253Q - .05Q2
The marginal revenue function is the first derivative of the revenue
function.
With these simple linear demand functions, it will always be just
like the price function except that the coefficient on quantity (the number
quantity is multiplied by,) is doubled.
MR = 3253 – (2*.05)Q
MR = 3253 - .1Q
Remember the total cost function?
TC = 5,000,000 + .2Q2
The marginal cost function is the first derivative of the total cost
function. With this simple cost functions, this will involve dropping fixed
cost (the first term) and doubling the coefficient on quantity and dropping
the squared.
MC = (2*.2)Q
MC = .4Q
Since profit maximizing output is at the point where marginal revenue
equals marginal cost, the algebra is simple:
MR = MC
3253- .1Q = .4Q
3253 -.1Q + .1Q = .4Q + .1Q
3253 = .5Q
3253/.5 = .5Q/.5
6506 = Q
The profit maximizing quantity or output is 6506 units.
If you think about it, Q = 3253/(.4--.1)
(This will be useful on a spreadsheet.)
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Profit Maximizing Price
If you know the profit maximizing quantity, the price is easy to find.
Just find what the market will bear by using the price function:
P = 3253- .05Q
= 3253 - .05(6506)
= 2927.70
If you produce 6506 units and charge $2927.70 for each one, you will sell
them and make the most profit.
What is that profit?
First, find revenue.
That’s price times quantity.
R = P*Q
= 2927.7 * 6506
= 19,047,616.20
Next, use the cost function to find total cost.
TC = 5,000,000 + .2(6506)^2
TC = 13,465,607.20
And profit is revenue – Cost
Profit = R – TC
=
= 19,047,616.20 - 13,465,607.2
5,582,009.00
What’s marginal revenue?
MR = 3253 - .1(6506)
= 2602.40
And marginal cost?
MC = .4(6506)
= 2602.40
And that’s the key. To find any other information—variable cost, point price
elasticity of demand, etc., just use the functions developed in earlier
topics and substitute in the profit maximizing output.
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Some Interesting Levels of Output
Profit maximization
Unit Cost minimization
Revenue maximization
MR=MC
MC=AC
MR=0
Diagram for Profit Maximizing Output and Price
P
MC
AC
P*
AVC
D
MR
Q*
Q
Shutting Down in the Short Run
If a firm suffers losses and nothing is expected to change in its
situation in the future, the firm will shut down. (As capital goods
wear out, it becomes necessary to replace them. What were sunk costs
become opportunity costs and unless enough revenue will be generated to
cover them, the capital goods will not be replaced and the firm will
shut down.)
Firms, however, will operate at a loss in the short run. That is
because sunk costs are irrelevant. However, it is possible that a firm
will shut down even in the short run. That occurs if the revenues do
not cover the variable costs. Another way to say this, is that a firm
will shut down (or should do so) if the profit-maximizing price (what
the market will bear at the profit maximizing quantity) is less than
the average variable cost.
Variable costs are costs of that vary with output. For example,
the costs of materials and labor. If a firm is not earning enough
revenue to cover those costs, that means that it is foolish for the
firm to purchase those variable inputs. But if it doesn’t purchase
them, it produces nothing. It shuts down.
Using the example above, average variable cost will be AVC =.2Q
AVC = .2 (6506)
= 1301.20
Since the price at 2927.70 is greater than average cost, that means you
should stay open. Of course, in the example, the firm is making a
profit, so there really isn’t any need to check. If the firm is losing
money, however, this should be checked out.