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Lecture #7
Lecture Outline
• Review
• Go over Exam #1
• Continue production economic
theory
Maximizing Profit
Using the previous example, let’s now solve for the input
levels which maximize profits.
Maximizing profit 
Maximizing Profit
1st order conditions:
Maximizing Profit
For the first partial derivative:
For the second partial derivative:
Maximizing Profit
2nd order conditions:
and
 Profits are maximized when
Maximizing Profit
Summary:
Least cost combinations and profit maximization.
Maximizing Profit
Summary:
Least cost combinations and profit maximization.
Maximizing output subject to a cost constraint:
Minimizing Cost
Minimizing cost subject to an output constraint:
Profit maximization:
Expansion Path
An expansion path is a locus of points which represent the
tangencies between isoquant and isocost lines assuming
that input prices are constant. Note that these tangencies
are least cost combinations.
Since the tangencies represent points where slopes of
isoquant and isocost lines are equal, we can solve for the
equation of the expansion path:
Slope of isoquant:
Expansion Path
Slope of isocost line:
Using our previous production example:
Expansion Path
 equation of the expansion path
Points of the expansion path are least
cost combinations of inputs.
Expansion Path
(so the least cost solutions lie on the expansion path)
Expansion Path
What about the profit max solution?
So the profit max solution is a least cost solution which
lies on the firm’s expansion path.
Duality
The two constrained optimization methods, constrained
output maximization and constrained cost minimization,
yield a unique relationship called duality. It implies the
possibility of deriving cost functions from production
functions and vice versa.
Given the same cost and production constraints, we see that
both problems (constrained output maximization and
constrained cost minimization) can yield the same optimal
combinations of inputs
Lagrange Multiplier
However, the other bit of information – the
Lagrange multiplier is different yet related. For
the constrained output maximization problem,
In the constrained minimization problem,
Problem
The following production function, based on actual Iowa
experimental data, has been altered slightly to simplify
computations:
Problem
Given the above production function and input prices, what
is the least cost combination to produce a 240 pound hog?
Problem
This is a problem of constrained cost minimization.
Why? Because costs are in cwt and 240 lbs = 2.4cwt
1st order conditions:
Problem
Solve for λ:
From the first equation:
Problem
From the second equation:
Problem
Plug this relationship between
into the third equation.
Problem
or
Problem
 Least cost combination: Feed 448 lbs of corn and 80 lbs
of soybean meal to produce a 240 lb hog (2.4 cwt)
Homogeneous Functions
The function
is said to be homogeneous of
degree k if each independent variable
(the x’s for this function) is multiplied by a scalar λ,
and the dependent variable (in this case y) changes by λk.
So the function is homogeneous of degree k if:
Example
Find the degree of homogeneity for this function.
If the function is homogeneous then we can write:
 the function is homogeneous
of degree 2 (HD2)
Example
What does this mean?
If inputs x1 and x2 are increased by a factor
λ, then output (y) will increase by λ2.
Cobb-Douglas Form
Another example: case of Cobb- Douglas form
Find the degree of homogeneity for this function.
Cobb-Douglas Form
Linearly homogeneous production functions are HD1 or
HD = 1
General Cobb-Douglas production function:
is homogeneous of degree
Another example:
Cobb-Douglas Form
Is this function homogeneous and of what degree?
However, we can not get back to λky.
So this function is not homogeneous.
Factor (Input) Demand Functions
The demand for final products is derived from constrained
utility maximization (objective function of consumers).
The demand for inputs is derived from profit maximization
(objective function of producers).
Take the case of production with one variable input:
Factor (Input) Demand Functions
Recall first order conditions for profit maximization:
We can write the factor demand for input x1 as:
Factor (Input) Demand Functions
Changes in r1 can be represented as movements along the
input demand function.
Changes in p can be represented as shifts of the input
demand function.
Generally, output price is a positive shifter of the input
demand function.
Factor (Input) Demand Functions
We can graph the input demand function for x1 by assuming p
is held constant. We call the input demand function the
derived demand function since it is a result of (or derived
from) profit maximization.
So the input demand function is often called the derived
demand function since it is derived from profit maximization.
For the one variable input case (factor-product case),
economic theory states that the factor demand function is that
portion of the VMP or MVP curve in Stage II.
What is the law of demand? Does it hold for inputs?
Yes. The law of demand underlies the negative relationship
between own price and quantity demanded for final products
as well as inputs.
What about input demands for the factor-factor case?
Given the following production function:
Assume also that p is output price, r1 is the per unit cost of
x1 and r2 is the per unit cost of x2.
Derive the factor demand function for x1.
Input or factor demands are derived from profit
maximization.
1st order conditions:
Solving for x2:
In a similar manner, one can derive the factor demand for x2.
Is the factor demand function for x1 downward sloping?
Is output price, p, a positive shifter of the input demand
function?
So if output price increases  demand for x1 increases
What effect does changes in r2 have on the input
demand for x1?
Own price elasticity of demand: