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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

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

(Fixed Proportions)

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

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

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

he can collect 4 fish per net.

L hours of labor, and K nets?

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

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

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

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

\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

your production depends on the number of trucks you have.

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 =

your production depends on the number of trucks you have.

your production depends on the number of workers you have.

L \le 2K

L \ge 2K

\Rightarrow q = 8L

\Rightarrow q = 16K

\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

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\}

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}}

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}

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\}

(Fixed Proportions)

AL^aK^b

(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}

- 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?

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?

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

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

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

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**?

- 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