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.
Econ 50 | Lecture 02
By Chris Makler
Econ 50 | Lecture 02
Modeling Production with Multivariate Functions
