Applied Economics for Business Management

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Applied Economics for Business Management

• Lecture 9

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• Review
• Homework Set 7
• Continue Production Economic Theory
• product-product case

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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.
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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.
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Principle of Enterprise Choice
To illustrate the principle of enterprise choice
in the case of more than one output, lets 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.
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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.
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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.
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As the amount of available resources increase,
the production possibilities curve or product
transformation curve shifts outward.
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Production Transformation Curve
Given the following production transformation
curve
where x0 represents the level of resources
available for the production of y1 and y2.
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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.

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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.
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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

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Isorevenue Line
Since product prices are held constant, the
isorevenue lines are parallel.
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The slope of the isorevenue line is the ratio of
product prices. In this case,
? slope of isorevenue line
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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.
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Rate of Production Transformation
Returning to the equation of the production
transformation curve
?amount of x
available to the
firm for producing y1 and y2
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Production Transformation Curve


Totally differentiate this equation
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Production Transformation Curve
What does this mean?
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Example

Let the product transformation curve be
represented in implicit form as
(explicit form of the transformation
curve)
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Optimization

• In the case of 2 products and 1 variable input,
  we have two optimization problems
• maximizing revenue subject to a resource
  constraint
• (constrained optimization case) and
• (ii) profit maximization case (unconstrained
  optimization)

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Constrained Optimization
Lets consider the constrained optimization case
The constraint is a resource constraint
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Constrained Optimization
Given product prices, the firm maximizes revenue
by moving to the isorevenue line that is tangent
to the product transformation curve
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Constrained Optimization
Objective Function
1st order conditions
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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.
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Constrained Optimization
What about ??
Likewise,
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Constrained Optimization
So ? 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.

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Constrained Optimization
What would happen if
2nd order conditions
for rel max
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Example
Example maximizing revenue subject to the
resource constraint You are given the following
information
Find the optimal combination of y1 and y2.
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But output (ys) can not be negative ?




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(What does ? represent? VMP of x in the
production of y1 and y2) 2nd order condition
? rel max
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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.
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Profit Maximization
We will now determine the optimal level of output
to produce such that profits are maximized.
1st order conditions
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Likewise with similar derivation, we find
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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.
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2nd order conditions
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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
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2nd order conditions
and
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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.
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Unconstrained Optimization
Lets plot these two solutions
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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, ? 0.625 and ? 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 lt r. In the profit max
solution, VMP of x r.
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Note If r 0.625 instead of r 1 then the new
profit max solution is
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1st order conditions


So the new profit max solution with r 0.625 is
the same as the constrained revenue max solution
with ? 0.625