Lecture #9
 Review
 Homework Set #7
 Continue Production Economic Theory:
product-product case
Product-Product Case
This production relationship involves the production of 2
or more products with a given set of resources or inputs.
In agriculture, farmers seldom are so specialized that they
ignore the profit potential from other crops. For example,
grain producers in the mid-west often grow two or more
crops – e.g., corn and soybeans.
Product-Product Case
Also, grain farmers may often add a livestock enterprise such
as cattle or hogs.
In Hawaii, although farms are not as large, farmers grow a
variety of vegetable crops or combine tropical fruit
production with vegetable production.
So decision-making often involves two or more enterprises
with a goal to maximize profits from a given set of available
resources.
Principle of Enterprise Choice
To illustrate the principle of enterprise choice in the case of
more than one output, let’s take the following case:
(i) the firm has a given amount of each resource, e.g., land,
capital, labor and management.
(ii) the firm can produce two commodities.
Production Transformation Curve
Let y1 be corn yield
(in bushels) and y2 be
soybean yield (in bushels).
We can show all of these production combinations (of y1 and y2)
on a production transformation curve often called the
production possibilities curve.
Suppose we have production of 2 products with a given level
of resources x.
We can combine these two production functions in implicit
form as:
This equation states that 2 products y1 and y2 are
produced with input x such that all units of x are used up
in the production process.
As the amount of available resources increase, the production
possibilities curve or product transformation curve shifts
outward.
Production Transformation Curve
Given the following production transformation curve:
where x0 represents the level of resources available for the
production of y1 and y2.
Production Transformation Curve
• At point S, the firm is not using all available resources.
• Point R lies on the production possibilities curve, so the firm
uses all available resources.
• At point T, the firm cannot produce this output combination
since the firm does not have enough resources.
Isorevenue Line
How much to produce of these two products?
Optimal allocation for these 2 products occur where the
isorevenue line is tangent to the product transformation
curve.
Isorevenue Line
• What is the isorevenue line?
• The term “iso” means equal. So the isorevenue line
shows all possible combinations of the 2 commodities
sold that yield the same revenue.
• Similar to other “iso” concept:
isoquant  equal quantity
isocost  equal cost
Isorevenue Line
Since product prices are held constant, the isorevenue lines are
parallel.
The slope of the
isorevenue line is
the ratio of product
prices.
In this case,
 slope of isorevenue line
Rate of Product Transformation
The rate at which y1 is substituted for y2 (and vice-versa)
without varying the amount of resource x used is called the
rate of product transformation (RPT) (often called the
marginal rate of product substitution (MRPS).
RPT is the slope of the transformation curve at a given point
on the production possibilities curve.
Rate of Production Transformation
Returning to the equation
of the production
transformation curve:
↑amount of x available to the
firm for producing y and y
1
2
Production Transformation Curve
Totally differentiate this equation:
Production Transformation Curve
What does this mean?
Example
Let the product transformation curve be represented in
implicit form as:
(explicit form
of the transformation
curve)
Optimization
In the case of 2 products and 1 variable input, we have two
optimization problems:
(i)
maximizing revenue subject to a resource constraint
(constrained optimization case) and
(ii) profit maximization case (unconstrained optimization)
Constrained Optimization
Let’s consider the constrained optimization case:
The constraint is a resource constraint:
Constrained Optimization
Given product prices, the firm maximizes revenue by
moving to the isorevenue line that is tangent to the product
transformation curve:
Constrained Optimization
Objective Function:
1st order conditions:
Constrained Optimization
So the first order conditions state that for constrained
revenue maximization, the slope of the production
possibilities curve = slope of the isorevenue line.
Constrained Optimization
What about λ?
Likewise,
Constrained Optimization
So λ corresponds to the value of the marginal product of x in
production of y1 and y2 when products are produced at the
optimum.
 equal marginal returns to input x in the production of
y1 and y2.
Constrained Optimization
What would happen if
2nd order conditions:
for rel max
Example
Example: maximizing revenue subject to the resource constraint
You are given the following information:
Find the optimal combination of y1 and y2.
But output (y’s) can not be negative 
(What does λ represent? VMP of x in the production of y1 and y2)
2nd order condition:
 rel max
Profit Maximization
The second case in the joint products problem is to relax
the assumption of a fixed level of input usage
or constraint on available resources.
We will now determine the optimal level of output to
produce such that profits are maximized.
Profit Maximization
We will now determine the optimal level of output to
produce such that profits are maximized.
1st order conditions:
Likewise with similar derivation, we find:
Profit Maximization
Interpretation of first order conditions:
The firm will allocate x to y1 and y2 production such that:
i.e., preserving equal marginal returns to input x.
2nd order conditions:
Example
Consider the previous example:
But instead of constraining
let the product transformation curve have the following
relationship:
Then we can write the profit function as:
2nd order conditions:
and
So far we have these solutions:
Revenue max solution:
Profit max solution:
Why this different?
In the profit max solution, there is no resource constraint.
Unconstrained Optimization
Let’s plot these two solutions:
What is the resource use in the profit max case?
Why does the profit max solution use less resources than
the constrained revenue max case? 37 vs. 92.5
Recall that in the constrained revenue max solution,
λ = 0.625 and λ is the VMP of x in y1 and y2 production.
The market price of x, r, = $1/unit.
So in the constrained revenue max solution VMP of x < r.
In the profit max solution, VMP of x = r.
Note: If r = 0.625 instead of r = 1 then the new profit max
solution is:
1st order conditions:
So the new profit max solution with
r = 0.625 is the same as the constrained revenue max solution
with λ = 0.625