Modeling Production with Multivariable Functions

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 2

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Today's Agenda

Part 1: Math Review

Multivariable Functions

Level Sets

Partial Derivatives

Implicit Differentiation

 

Part 2: Production Functions

Production Functions

Isoquants

Marginal Products of Labor and Capital

Marginal Rate of Technical Substitution

Multivariable Functions

x
f()
z = f(x,y)
y
z

[INDEPENDENT VARIABLES]

[DEPENDENT VARIABLE]

\text{Level set for }z=\{(x,y)|f(x,y)=z\}
\displaystyle{{df \over dx} = \lim_{\Delta x \rightarrow 0} {f(x + \Delta x) - f(x) \over \Delta x}}
\displaystyle{{\partial f \over \partial x} = \lim_{\Delta x \rightarrow 0} {f(x + \Delta x, y) - f(x, y) \over \Delta x}}
\displaystyle{{\partial f \over \partial y} = \lim_{\Delta y \rightarrow 0} {f(x, y + \Delta y) - f(x, y) \over \Delta y}}

Univariate Chain Rule

\text{Example: }h(x)=(3x + 2)^2
h(x)=f(g(x))
{dh \over dx} = {df \over dg} \times {dg \over dx}
f(g)=g^2
g(x)=3x+2

Multivariable Chain Rule

\text{Example: }h(x,y)=(3x+y)^2
h(x,y)=f(g(x,y))
{\partial h \over \partial x} = {df \over dg} \times {\partial g \over \partial x}
f(g)=g^2
g(x)=3x+y

Total Derivative Along a Path

\text{How does }f(x,y)\text{ change along a path?}
\Delta f \approx
\displaystyle{{\partial f \over \partial x} \times \Delta x}
\displaystyle{{\partial f \over \partial y} \times \Delta y}
+
\text{Suppose the path is defined by some function }y(x):
\Delta f \approx
\displaystyle{{\partial f \over \partial x} \times \Delta x}
\displaystyle{{\partial f \over \partial y}}
+
\displaystyle{\times {dy \over dx}}
\displaystyle{\times \Delta x}
\displaystyle{\Delta f \over \Delta x} \approx
\displaystyle{{\partial f \over \partial x}}
\displaystyle{{\partial f \over \partial y}}
+
\displaystyle{\times {dy \over dx}}

Total Derivative Along a Path

\displaystyle{\Delta f \over \Delta x}\ \ \ \ \ \ \approx
\displaystyle{{\partial f \over \partial x}}
\displaystyle{{\partial f \over \partial y}}
+
\displaystyle{\times\ \ \ \ \ \ {dy \over dx}}

The total change in the height of the function due to a small increase in \(x\)

The amount \(f\) changes due to the increase in \(x\)

[indirect effect through \(y\)]

The amount \(f\) changes due to an increase in \(y\)

The amount \(y\) changes due to an increase in \(x\)

[direct effect from \(x\)]

y(x)=4-0.4x

Derivative Along a Level Set

f(x,y)=z

Take total derivative of both sides with respect to x:

Solve for \(dy/dx\):

\displaystyle{{\partial f \over \partial x}}
\displaystyle{{\partial f \over \partial y}}
+
\displaystyle{\times \ \ {dy \over dx}}
=
0
\displaystyle{{dy \over dx}}
= -
\displaystyle{{\partial f \over \partial x}}
\displaystyle{{\partial f \over \partial y}}

IMPLICIT FUNCTION THEOREM

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Consider the multivariable function

f(x,y) = 4x^{1 \over 2}y

What is the slope of the level set passing through the point (1, 5)?

Consider the multivariable function

f(x,y) = 4x^{1 \over 2}y

What is the slope of the level set passing through the point (1, 5)?

\displaystyle{{\partial f(x,y) \over \partial x}=}
\displaystyle{{\partial f(x,y) \over \partial y}=}

Slope of level set =   —

Production Functions

Production Functions

A mathematical form describing how much output is produced as a function of inputs.

Labor \((L)\)

Capital \((K)\)

Production Function  \(f(L,K)\)

Output (\(q\) or \(x\))

Lecture 2: Production Functions

Labor

Fish

🐟

Capital

Coconuts

🥥

[GOODS]

⏳

⛏

[RESOURCES]

Unit I: Scarcity and Choice

Economics is the study of how
we use scarce resources 
to satisfy our unlimited wants

Resources

Goods

Happiness

🌎

⌚️

🤓

Isoquants

Economic definition: if you want to produce some amount \(q\) of output, what combinations of inputs could you use?

Mathematical definition:
level sets of the production function

\text{Isoquant for }q = \{(L,K)\ |\ f(L,K) = q\}

Isoquant: combinations of inputs that produce a given level of output

Isoquant map: a contour map showing the isoquants for various levels of output

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What happens to isoquants after an improvement in technology?

Marginal Products of Labor and Capital

Economic definition: how much more output is produced if you increase labor or capital?

Mathematical definition:
partial derivatives of the production function

\displaystyle{MP_L = {\partial f(L,K) \over \partial L}}

These are both rates: they are measured in terms of units of ouptut per unit of input.

\displaystyle{= \lim_{\Delta L \rightarrow 0} {f(L + \Delta L, K) - f(L, K) \over \Delta L}}
\displaystyle{MP_K = {\partial f(L,K) \over \partial K}}
\displaystyle{=\lim_{\Delta K \rightarrow 0} {f(L, K + \Delta K) - f(L, K) \over \Delta K}}

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Consider the production function

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

What is the expression for the marginal product of labor?

Marginal Rate of Technical Substitution (MRTS)

Economic definition: the rate at which a producer can substitute one input for another while keeping output at the same level

Mathematical definition: slope of an isoquant

Recall: by implicit function theorem,
the slope of a level set is given by

\displaystyle{MRTS = {MP_L \over MP_K}}
\displaystyle{\left.{dy \over dx} \right|_{f(x,y) = z} = -{\partial f/\partial x \over \partial f/\partial y}}

Therefore the formula for the MRTS is

(absolute value)

Labor (L)

Capital (K)

Intuition behind the formula for the MRTS

Example: 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}}

Summary

So far we've modeled an important economic activity
using multivariable calculus.

We looked at the economic meaning for
various properties of the production function:

Partial Derivatives

Marginal Products

Level Sets

Isoquants

Slope of a Level Set

Marginal Rate of Technical Substitution

In section and on Friday, we'll analyze how
different functional forms can be used to model different kinds of technologies.