Today's Agenda

Theory of the Firm
Review of Econ 1 Treatment of Production and Cost
Total, Marginal, and Average Costs
Short-Run and Long-Run Costs
Curvature of Cost Functions

The Econ 1 Approach

Greg Mankiw, Principles of Economics

Dollars

Dollars Per Unit

Greg Mankiw, Principles of Economics

The Econ 50 Approach

pollev.com/chrismakler

Generally speaking, if capital is fixed in the short run, then higher levels of capital are associated with _______ fixed costs and _______ variable costs for any particular target output.

\text{Total Cost}: TC(q) = F + VC(q)

\text{Total Cost}: TC(q) = F + VC(q)

\text{Average Cost}: ATC(q) = \frac{TC(q)}{q} = \frac{F}{q} + \frac{VC(q)}{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{Total Cost}: TC(q) = F + VC(q)

\text{Average Cost}: ATC(q) = \frac{F}{q} + \frac{VC(q)}{q}

\text{Average Cost}: ATC(q) = \frac{F}{q} + \frac{VC(q)}{q}

Average Costs

TC(q) = 64 + {1 \over 4}q^2

TC(q) = 64 + {1 \over 4}q^2

ATC(q) = {64 \over q} + {1 \over 4}q

ATC(q) = {64 \over q} + {1 \over 4}q

\text{Total Cost}: TC(q) = F + VC(q)

\text{Total Cost}: TC(q) = F + VC(q)

\text{Marginal Cost}: MC(q) = TC'(q) = 0 + 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{Total Cost}: TC(q) = F + VC(q)

\text{Marginal Cost}: MC(q) = TC'(q)

\text{Marginal Cost}: MC(q) = TC'(q)

Marginal Cost

TC(q) = 64 + {1 \over 4}q^2

TC(q) = 64 + {1 \over 4}q^2

MC(q) = {1 \over 2}q

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{ increasing}

MC = AC \Rightarrow AC \text{ constant}

MC = AC \Rightarrow AC \text{ constant}

MC < AC \Rightarrow AC \text{ decreasing}

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

TC(q) = wL^c(q | \overline K) + r \overline K

{dTC(q) \over dq} = w \times {dL^c(q) \over dq}

{dTC(q) \over dq} = w \times {dL^c(q) \over dq}

= w \times {1 \over MP_L}

= w \times {1 \over MP_L}

Long-Run Costs

Cost Minimization: Lagrange Method

\mathcal{L}(L,K,\lambda)=

\mathcal{L}(L,K,\lambda)=

wL + rK +

wL + rK +

(q - f(L,K))

(q - f(L,K))

\lambda

\lambda

\frac{\partial \mathcal{L}}{\partial x_1} = w - \lambda MP_L

\frac{\partial \mathcal{L}}{\partial x_1} = w - \lambda MP_L

First Order Conditions

\frac{\partial \mathcal{L}}{\partial x_2} = r - \lambda MP_K

\frac{\partial \mathcal{L}}{\partial x_2} = r - \lambda MP_K

\frac{\partial \mathcal{L}}{\partial \lambda} = q - f(L,K) = 0 \Rightarrow q = f(L,K)

\frac{\partial \mathcal{L}}{\partial \lambda} = q - f(L,K) = 0 \Rightarrow q = f(L,K)

= 0 \Rightarrow \lambda = w \times {1 \over MP_L}

= 0 \Rightarrow \lambda = w \times {1 \over MP_L}

= 0 \Rightarrow \lambda = r \times \frac{1}{MP_K}

= 0 \Rightarrow \lambda = r \times \frac{1}{MP_K}

MRTS (slope of isoquant) is equal to the price ratio

\text{Also: }\frac{\partial \mathcal{L}}{\partial q} = \lambda = \text{Marginal cost of producing last unit using either input}

\text{Also: }\frac{\partial \mathcal{L}}{\partial q} = \lambda = \text{Marginal cost of producing last unit using either input}

\text{Set two expressions }

\text{Set two expressions }

\text{for }\lambda \text{ equal:}

\text{for }\lambda \text{ equal:}

\frac{MP_L}{MP_K} = \frac{w}{r}

\frac{MP_L}{MP_K} = \frac{w}{r}

f(L,K) = \sqrt{LK}

f(L,K) = \sqrt{LK}

Long Run (can vary both labor and capital)

L^c(q) = \sqrt{\frac{r}{w}}q

L^c(q) = \sqrt{\frac{r}{w}}q

K^c(q) = \sqrt{\frac{w}{r}}q

K^c(q) = \sqrt{\frac{w}{r}}q

TC^{LR}(q) = wL^c(q) + rK^c(q)

TC^{LR}(q) = wL^c(q) + rK^c(q)

=2\sqrt{wr}q

=2\sqrt{wr}q

Short Run with Capital Fixed at $\overline K$

L^c(q | \overline K) = {q^2 \over \overline K}

L^c(q | \overline K) = {q^2 \over \overline K}

TC(q) = wL^c(q | \overline K) + r \overline K

TC(q) = wL^c(q | \overline K) + r \overline K

={wq^2 \over \overline K} + r \overline K

={wq^2 \over \overline K} + r \overline K

f(L,K) = \sqrt{LK}

f(L,K) = \sqrt{LK}

Long Run (can vary both labor and capital)

TC^{LR}(q)=2\sqrt{wr}q

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

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^{LR}(q)=8q

TC(q)=64 + {1 \over 4}q^2

TC(q)=64 + {1 \over 4}q^2

TC(q)=64 + {1 \over 4}q^2

TC(q)=64 + {1 \over 4}q^2

F = 64

F = 64

VC(q) = {1 \over 4}q^2

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 | \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

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(q,\overline{K}) \ge TC^{LR}(q)

= TC^{LR}(q)

= TC^{LR}(q)

TC^{LR}(q) \text{ is the lower envelope of the family of }TC(q)\text{ curves}

TC^{LR}(q) \text{ is the lower envelope of the family of }TC(q)\text{ 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?

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
Friday: bring cost and revenue together

Production and Cost
for a Firm

Today's Agenda

Unit I: The “Real Economy"

Unit II: Consumers and Prices

Unit III: Theory of the Firm

Unit III: Theory of the Firm

Unit III: Theory of the Firm

Unit III: Theory of the Firm

Unit III: Theory of the Firm

The Econ 1 Approach

The Econ 50 Approach

Short-Run Total Cost of $q$ Units

Total, Fixed and Variable Costs

Average Costs

Average Costs

Marginal Cost

Marginal Cost

Marginal cost tends to "pull" average cost toward it:

Marginal grade = grade on last test, average grade = GPA

Relationship between Average and Marginal Costs

Relationship between Marginal Cost and Marginal Product of Labor

Long-Run Costs

Cost Minimization Subject to a Utility Constraint

Cost Minimization Subject to an Output Constraint

Cost Minimization: Lagrange Method

Expansion Path

Long-Run Total Cost of $q$ Units

Relationship between
Short-Run and Long-Run Costs

Returns to a Single Input

Returns to Scale (Scaling all inputs.)

Relationship between Production Function and the Curvature of Long-Run and Short-Run Costs

Economies and Diseconomies of Scale

Next Time

Econ 50 | Lecture 17

Econ 50 | Lecture 17

Chris Makler PRO

Production and Cost for a Firm

Econ 50 | Lecture 17

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Production and Cost
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