Christopher Makler
Stanford University Department of Economics
Econ 50 | Lectures 20 & 21
What is an agent's optimal behavior for a fixed set of circumstances?
Utility-maximizing bundle for a consumer
Profit-maximizing quantity for a firm
Profit-maximizing input choice for a firm
How does an agent's optimal behavior change when circumstances change?
Utility-maximizing bundle for a consumer
Profit-maximizing quantity for a firm
Profit-maximizing input choice for a firm
We will be analyzing a
competitive (price-taking) firm
in the short run
[ MOVEMENT
ALONG CURVES]
[ SHIFTS OF
CURVES]
We'll stick with our same function
1. Costs and Revenues
2. Profit = total revenues minus total costs
3. Take derivative of profit function, set =0
1. Costs and revenues
2. Profit = total revenues minus total costs
3. Take derivative of profit function, set =0
1. Costs and Revenues
2. Profit = total revenues minus total costs
3. Take derivative of profit function, set =0
NUMBER
FUNCTION
TR
TC
MR
MC
Take derivative and set = 0:
Solve for \(q^*\):
SUPPLY FUNCTION
When \(p = 4\), this function says that the firm should produce \(q = 8\).
If it does this...
pollev.com/chrismakler
We just showed that for a firm with the cost function 𝑐(𝑞)=64+14𝑞2c(q)=64+41q2, the profit-maximizing choice when 𝑝=4p=4 is to produce 𝑞=8q=8. If the firm does this:
Text
1. Costs and Revenues
2. Profit = total revenues minus total costs
3. Take derivative of profit function, set =0
NUMBER
FUNCTION
TR
TC
MRPL
MC
Take derivative and set = 0:
Solve for \(L^*\):
LABOR DEMAND FUNCTION
(over to powerpoint for a moment)
1. Costs for general \(w\) and revenue for general \(p\)
2. Profit = total revenues minus total costs
3. Take derivative of profit function, set =0
[cost of labor required for \(q\) units of output]
[revenue of output produced by \(L\) hours of labor]
MARGINAL COST (MC)
MARGINAL REVENUE PRODUCT OF LABOR (MRPL)
"Keep producing output as long as the marginal revenue from the last unit produced is at least as great as the marginal cost of producing it."
"Keep hiring workers as long as the marginal revenue from the output of the last worker is at least as great as the cost of hiring them."
LABOR DEMAND FUNCTION
LABOR DEMAND FUNCTION
SUPPLY FUNCTION
the conditional labor demand
for the profit-maximizing supply:
The profit-maximizing labor demand is
Edge Case 1:
Multiple quantities where P = MC
Edge Case 2:
Corner solution at \(q = 0\)
"The supply curve is the portion of the MC curve above minimum average variable cost"
A competitive firm takes input prices \(w\) and \(r\), and the output price \(p\), as given.
We can therefore characterize its optimal choices of inputs and outputs
as functions of those prices: the supply of output \(q^*(p\ |\ w)\),
and the demand for inputs (e.g. \(L^*(w\ |\ p)\)).
We can find the optimal input-output combination either by finding the optimal quantity of output and determining the inputs required to produce it, or to find the profit-maximizing inputs and determine the resulting output. These two methods are equivalent.
Profit is increasing when marginal revenue is greater than marginal cost, and vice versa.
In most cases, the profit-maximizing choice occurs where \(MR = MC\).
If \(p\) is below the minimum value of AVC, the profit-maximizing choice is \(q = 0\).
In which MR or MC is discontinuous, logic must be applied. (There is an old exam question on the homework that explores this...and this kind of thing often shows up on exams...)