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# Characteristics of Production Functions

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 3

Goal for Today: Analyze how different functional forms can be used to model different production processes

f(L,K) = aL + bK

# Linear

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

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

# Constant Elasticity of Substitution (CES)

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

f(L,K) = 2L + 4K

f(L,K) = 2L + 4K

# If Chuck uses a net (K), he can collect 4 fish per net.

MP_L = {df \over dL} = 2 {\text{fish} \over \text{hour}}
MP_K = {df \over dK} = 4 {\text{fish} \over \text{net}}
MRTS =
= {\text{1 net} \over \text{2 hours}}

Intuition: no matter how many hours he works, and how many nets he uses,
he can catch the same number of fish with 1 net as he can using 2 hours of labor.

f(L,K) = 2L + 4K

# If Chuck uses a net (K), he can collect 4 fish per net.

MRTS = {\text{1 net} \over \text{2 hours}}

What does an isoquant look like?

What does the isoquant for $$q = 20$$ look like?

L

K

# 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

# Linear Production Function

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

What happens to an isoquant if $$a$$ increases?

L

K

f(2,1) = 16

f(3,1) = 16
f(2,2) = 16
f(4,2) = 32

# If you have less than twice as many workers as trucks, your production depends on the number of workers you have.

L \le 2K
L \ge 2K
\Rightarrow q = 8L
\Rightarrow q = 16K
f(L,K)=\begin{cases}8L & \text{ if }L \le 2K\\16K & \text{ if }L \ge 2K\end{cases}
=\min\{8L,16K\}
MP_L=\begin{cases}8 & \text{ if }L \le 2K\\0 & \text{ if }L \ge 2K\end{cases}
MP_K=\begin{cases}0 & \text{ if }L \le 2K\\16 & \text{ if }L \ge 2K\end{cases}
=\begin{cases}\infty & \text{ if }L \le 2K\\0 & \text{ if }L \ge 2K\end{cases}
MRTS =

# If you have less than twice as many workers as trucks, your production depends on the number of workers you have.

L \le 2K
L \ge 2K
\Rightarrow q = 8L
\Rightarrow q = 16K

# Leontief Production Function

\displaystyle{MRTS = \begin{cases}\infty & \text{ if }L \le 2K\\0 & \text{ if }L \ge 2K\end{cases}}
q=f(L,K)=\min\{8L, 16K\}

What do the isoquants look like?

L

K

# 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}
f(L,K) = \min\{2L, K\}

# 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}}

# Cobb-Douglas Production Function

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

What does the isoquant for $$q = 4$$ look like?

MRTS = {2K \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}}

aL + bK

\min\{aL, bK\}

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}

## 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?

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

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When does the production function

f(L) = 10L^a

exhibit diminishing marginal product of labor?

# 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)

# 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?

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When does the production function

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

exhibit constant returns to scale?

## Next Steps

• Homework for today's class is due on Saturday night
• Readings and quizzes for next week have been posted
• Homework for next week will be posted later today

By Chris Makler

# Econ 50 | Lecture 03

Characteristics of Production Functions

• 440