Lecture #9. Review Homework Set #7 Continue Production Economic Theory: product-product case.

Lecture #9

Review

Homework Set #7

Continue Production Economic Theory: product-product case

Product-Product CaseThis 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 CaseAlso, 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 ChoiceTo 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

We can show all of these production combinations (of y1 and y2) on a production transformation curve often called the production possibilities curve.

Let y1 be corn yield (in bushels) and y2 be soybean yield (in bushels).

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 CurveGiven 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 LineHow 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 TransformationThe 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 y1 and y2

Production Transformation Curve

Totally differentiate this equation:

Production Transformation Curve

What does this mean?

ExampleLet the product transformation curve be represented in implicit form as:

(explicit form of the transformation curve)

OptimizationIn 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 OptimizationThe constraint is a resource constraint: Let’s consider the constrained optimization case:

Constrained OptimizationGiven product prices, the firm maximizes revenue by moving to the isorevenue line that is tangent to the product transformation curve:

Constrained OptimizationObjective 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 OptimizationWhat about λ?

Likewise,

Constrained OptimizationSo 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 OptimizationWhat would happen if

2nd order conditions:for rel max

ExampleExample: maximizing revenue subject to the resource constraintYou 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:

Profit MaximizationThe second case in the joint products problem is to relax the assumption of a fixed level of input usageor constraint on available resources.We will now determine the optimal level of output to produce such that profits are maximized.

Profit MaximizationWe 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 MaximizationInterpretation 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:

ExampleConsider 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 OptimizationLet’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. 92.5Recall 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