TC(q)=wL^c(q|\overline K) + r \overline K
pollev.com/chrismakler
In the expression for the short-run cost function
what are the units of \(w\)?
Characteristics of
Cost Functions
Christopher Makler
Stanford University Department of Economics
Econ 50 : Lecture 16
Today's Agenda
- Review of Econ 1 Treatment of Production and Cost
- Review of Last Time: LR and SR Cost Functions
- Total, Marginal, and Average Costs
- Curvature of Cost Functions
- Elasticity? Probably Friday.
Long-Run and Short Run Costs (from Last Time)
f(L,K) = \sqrt{LK}
Long Run (can vary both labor and capital)
L^c(q) = \sqrt{\frac{r}{w}}q
K^c(q) = \sqrt{\frac{w}{r}}q
TC^{LR}(q) = wL^c(q) + rK^c(q)
=2\sqrt{wr}q
Short Run with Capital Fixed at \(\overline K \)
L^c(q | \overline K) = {q^2 \over \overline K}
TC(q) = wL^c(q | \overline K) + r \overline K
={wq^2 \over \overline K} + r \overline K
f(L,K) = \sqrt{LK}
Long Run (can vary both labor and capital)
TC^{LR}(q)=2\sqrt{wr}q
Short Run with Capital Fixed at \(\overline K \)
TC(q)={wq^2 \over \overline K} + r \overline K
Let's fix \(w= 8\), \(r = 2\), and \(\overline K =32\)
TC^{LR}(q)=8q
TC(q)=64 + {1 \over 4}q^2
TC(q)=64 + {1 \over 4}q^2
F = 64
VC(q) = {1 \over 4}q^2
Relationship between
Short-Run and Long-Run Costs
TC(q | \overline{K}) = \text{Cost if capital is fixed at }\overline{K}
TC(q,K^c(q)) = \text{Cost if capital is fixed at optimal }K \text{ for producing }q
What conclusions can we draw from this?
TC(q,\overline{K}) \ge TC^{LR}(q)
= TC^{LR}(q)
TC^{LR}(q) \text{ is the lower envelope of the family of }TC(q)\text{ curves}
\text{Total Cost}: TC(q) = F + VC(q)
\text{Average Cost}: ATC(q) = \frac{TC(q)}{q} = \frac{F}{q} + \frac{VC(q)}{q}
Fixed Costs
Variable Costs
Average Fixed Costs (AFC)
Average Variable Costs (AVC)
Average Costs
\text{Total Cost}: TC(q) = F + VC(q)
\text{Average Cost}: ATC(q) = \frac{F}{q} + \frac{VC(q)}{q}
Average Costs
TC(q) = 64 + {1 \over 4}q^2
ATC(q) = {64 \over q} + {1 \over 4}q
\text{Total Cost}: TC(q) = F + VC(q)
\text{Marginal Cost}: MC(q) = TC'(q) = 0 + VC'(q)
Fixed Costs
Variable Costs
Marginal Cost
(marginal cost is the marginal variable cost)
\text{Total Cost}: TC(q) = F + VC(q)
\text{Marginal Cost}: MC(q) = TC'(q)
Marginal Cost
TC(q) = 64 + {1 \over 4}q^2
MC(q) = {1 \over 2}q
pollev.com/chrismakler
Suppose q* is the quantity
for which ATC is lowest.
Which of the following must be true?
(Assume that ATC and MC are continuous functions of q.)
(a) MC also reaches its minimum at q*
(b) MC reaches its maximum at q*
(c) MC and ATC are equal at q*
Marginal cost tends to "pull" average cost toward it:
MC > AC \Rightarrow AC \text{ increasing}
MC = AC \Rightarrow AC \text{ constant}
MC < AC \Rightarrow AC \text{ decreasing}
Marginal grade = grade on last test, average grade = GPA
Relationship between Average and Marginal Costs
Relationship between Marginal Cost and Marginal Product of Labor
TC(q) = wL^c(q | \overline K) + r \overline K
{dTC(q) \over dq} = w \times {dL^c(q) \over dq}
= w \times {1 \over MP_L}
Scaling and Curvature
How does the nature
of the production function
affect the shape of the cost curves?
Returns to a Single Input
- Increasing marginal product: MPL is increasing in L
- Constant marginal product: MPL is constant in L
- Diminishing marginal product: MPL is decreasing in L
Returns to Scale (Scaling all inputs.)
- Increasing returns to scale: doubling all inputs more than doubles output.
- Constant returns to scale: doubling all inputs exactly doubles output.
- Decreasing returns to scale: doubling all inputs less than doubles output.
Relationship between Production Function and the Curvature of Long-Run and Short-Run Costs
- If the production function has diminishing \(MP_L\), the short-run cost curve will get steeper as you produce more output
- If the production function has decreasing returns to scale, the long-run cost curve will get steeper as you produce more output.
- What about for constant returns to scale? Increasing returns to scale?
- Homework question 15.1 walks you through these...
Economies and Diseconomies of Scale
Returns to Scale
Has to do with the production function
Economies of Scale
Has to do with cost curves
Increasing Returns to Scale:
double input => more than double output
Decreasing Returns to Scale:
double input => less than double output
Always deals with the long run
Can occur in both the long run and short run
Economies of Scale:
increasing output lowers average costs
Diseconomies of Scale:
increasing output raises average costs
Next Time
- Think about the demand curve facing a firm
- Analyze the price elasticity of demand
- Analyze the total and marginal revenue from producing more output
- Next week: bring cost and revenue together
Econ 50 | Spring 23 | Lecture 16
By Chris Makler
Econ 50 | Spring 23 | Lecture 16
Characteristics of Cost Functions
