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 We can show all of these production combinations (of y 1 and y 2 ) on a production transformation curve often called the production possibilities curve. Let y 1 be corn yield (in bushels) and y 2 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 y 1 and y 2 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 x 0 represents the level of resources available for the production of y 1 and y 2.

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 y 1 is substituted for y 2 (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 1 and y 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 The constraint is a resource constraint: Let’s consider the constrained optimization case:

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 y 1 and y 2 when products are produced at the optimum.  equal marginal returns to input x in the production of y 1 and y 2.

Constrained Optimization What would happen if 2 nd 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 y 1 and y 2.

But output (y’s) can not be negative 

(What does λ represent? VMP of x in the production of y 1 and y 2 ) 2 nd 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 y 1 and y 2 production such that: i.e., preserving equal marginal returns to input x.

2 nd 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 Recall that in the constrained revenue max solution, λ = and λ is the VMP of x in y 1 and y 2 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 = instead of r = 1 then the new profit max solution is:

1 st order conditions: So the new profit max solution with r = is the same as the constrained revenue max solution with λ = 0.625