Characteristics of
Production Functions

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 3

Today's Agenda

Part 1: Elasticity of Substitution

Part 2: Scaling

Production functions and their MRTS

Elasticity of substitution

Scaling in the long run (returns to scale)

Scaling in the short run (holding K fixed)

 

What happens as you move
along an isoquant

What happens as you move
between isoquants

Examples of Production Functions

f(L,K) = aL + bK

Linear

f(L,K) = \min\{aL, bK\}

Leontief
(Fixed Proportions)

Cobb-Douglas

f(L,K) = AL^aK^b

Constant Elasticity of Substitution (CES)

f(L,K) = (aL^\rho + bK^\rho)^{1 \over \rho}

Linear Production Function

MP_L =
MP_K =
MRTS =
\displaystyle{a\ {\text{output} \over \text{unit of L}}}
\displaystyle{b\ {\text{output} \over \text{unit of K}}}
\displaystyle{= {a \text{ units of K} \over b \text{ units of L}}}
q=f(L,K)=aL + bK

Leontief (Fixed Proportions) Production Function

f(L,K)=\begin{cases}aL \text{\ \ if }aL \le bK\\bK \text{\ \ if }aL \ge bK\end{cases}
=\min\{aL,bK\}
MP_L =
MP_K =
MRTS =
\begin{cases}a \text{\ \ if }aL < bK\\0 \text{\ \ if }aL > bK\end{cases}
\begin{cases}0 \text{\ \ if }aL < bK\\b \text{\ \ if }aL > bK\end{cases}
= \begin{cases}\infty \text{\ \ if }aL < bK\\0 \text{\ \ if }aL > bK\end{cases}

Cobb-Douglas Production Function

q=f(L,K)=AL^aK^b
MP_L = aAL^{a-1}K^b
MP_K = bAL^aK^{b-1}
MRTS =
\displaystyle{= {a \over b} \times {K \over L}}

CES Production Function

q=f(L,K)=(aL^\rho + bK^\rho)^{1 \over \rho}
\displaystyle{MRTS= {a \over b} \times \left({K \over L}\right)^{1-\rho}}

MRTS for Different Production Functions

aL + bK

Linear

\min\{aL, bK\}

Leontief
(Fixed Proportions)

Cobb-Douglas

AL^aK^b

CES

(aL^\rho + bK^\rho)^{1 \over \rho}
f(L,K)
MRTS
\frac{a}{b}
\begin{cases}\infty \text{\ \ if }aL < bK\\0 \text{\ \ if }aL > bK\end{cases}
\frac{a}{b}\times \frac{K}{L}
\frac{a}{b}\times \left(\frac{K}{L}\right)^{1-\rho}

Interpreting the MRTS
(slope of an isoquant)

"What is the rate at which
one can substitute
one input for another
and keep output the same?"

Another way of thinking about it:

"If we fired one worker and wanted to keep output the same,
how many additional machines would we need to buy?"

Elasticity of Substitution

  • Measures the substitutability of one input for another
  • Key to answering the question: "will my job be automated?"
  • Formal definition: the inverse of the percentage change in the MRTS 
    per percentage change in the ratio of capital to labor, K/L
  • Intuitively: how "curved" are the isoquants for a production function?

Scaling Production

How does a technology respond to increasing production?

Short run: only some resources can be reallocated

Long run: all resources can be reallocated

It depends on the time horizon:

Scaling Production in the Short Run

Suppose \(K\) is fixed at some \(\overline K\) in the short run.
Then the production function becomes \(f(L\ |\ \overline K)\)

pollev.com/chrismakler

When does the production function

f(L) = 10L^a

exhibit diminishing marginal product of labor?

Scaling Production in the Long Run

What happens when we increase all inputs proportionally?

For example, what happens if we double both labor and capital?

Does doubling inputs -- i.e., getting \(f(2L,2K)\) -- double output?

f(2L,2K) > 2f(L,K)
f(2L,2K) = 2f(L,K)
f(2L,2K) < 2f(L,K)

Decreasing Returns to Scale

Constant Returns to Scale

Increasing Returns to Scale

f(L,K) = 4L^{1 \over 2}K

Does this exhibit diminishing, constant or increasing MPL?

Does this exhibit decreasing, constant or increasing returns to scale?

pollev.com/chrismakler

When does the production function

f(L,K) = AL^aK^b

exhibit constant returns to scale?

Next Steps

  • Homework for the last two classes is due on Saturday night
  • Readings (and old videos) for next week have been posted
  • Quizzes for next week will be posted later today