Lecture #6

• Lecture outline• Review• Go over Homework Set #5• Continue production economic theory

Output elasticity:Output elasticity or the elasticity of production can be written as:

This is a special type of production function called the Cobb-Douglas form.

For this type of production function, the output elasticity is equal to the exponent.

• The production function can be divided into three stages. • The definition of each stage is based solely on a technological basis without any reference to price (either product price or factor price).

Boundary between Stages I & II:

Boundary between Stages II & III:

What’s happening in Stage I?

What’s happening in Stage II?

What’s happening in Stage III?

Which stage does the producer operate in?Stages I and III are technically infeasible. For stage III,of the input is negative. So an additional unit of input will decrease output.

The producer will not choose Stage I because increases throughout Stage I. Increasing implies that technical efficiency is increasing throughout Stage I. Consequently, it is best to produce beyond Stage I.

The rational or feasible stage of production is Stage II where diminishing returns occur (as denoted by a positive but declining MPP).

First order condition(s):

First order condition(s):What is the economic meaning of this first order condition?

First order condition(s):What is the economic meaning of this first order condition?

So the critical value for x1 occurs where the value of the marginal product for this input is equal to its price.

2nd order condition:

So profit maximization is occurring on the downward sloping portion of the MPP curve or the area of diminishing returns or diminishing MPP.The negative second derivative verifies that the critical value is a rel max.

You are also given thatFind the profit maximizing level of input usage for x1.

Use 2nd derivative to verify max: critical value represents a rel max

What is the level of profit?

Why does this firm produce in the short run?For this production period, the loss is less than fixed costs (recall FC = $100). In the longer term, if the firm anticipates continued losses, it will decide to shut down.

In this case, we deal with 2 variable factors, a set of fixed factors, and a single output.The production function looks like this:

or simply write the equation as

Technically, we are dealing with three dimensional space called a production surface. However, we can plot this surface in two dimensions by assuming one of the three variables is constant. Most frequently held constant is output and the curve that is derived is called the isoquant.

The isoquant is a curve which shows combinations of inputs, which yield a specific and constant level of output (y).

The isoquant is very similar to the indifference curve (from demand theory).

Like the indifference curve, the isoquant is downward sloping which illustrates the substitution of one input for the other while producing a specific amount of output.

We can illustrate this substitution by examining the slope of the isoquant. Also, we can specifically measure the degree of substitution by calculating the slope at a specific point on the isoquant.

The rate of technical substitution (RTS) is a similar concept to the rate of commodity substitution (RCS) in demand theory.

or the slope of the production isoquant is equal to the negative ratio of marginal products.

or the slope of the production isoquant is equal to the negative ratio of marginal products.

There are two ways to illustrate the optimal combination of inputs in production for the factor-factor case:(i) maximize output subject to a cost constraint and (ii) minimize cost subject to an output constraint

How do we illustrate method (i)?

Assume a given production function:

Specify a given level of costs (a constraint):

Assume a given production function:

Specify a given level of costs (a constraint):Objective function: maximize output subject to a cost constraint

First order conditions:

Solving for λ in the first two equations:

┌slope of cost line

└slope of isoquant or 21,xxRTSThe cost line or budget line for production is called the isocost line.

The first order conditions state that the variable factors are combined in an optimal manner when the ratio of marginal products is equal to the ratio of factor prices. This optimal combination is called the least cost combination of inputs.

The second order condition is a bordered Hessian:

So for the case of constrained output maximization (where the constraint is costs), the optimal occurs where the ratio of marginal products is equal to the ratio of factor prices. This occurs where the isoquant is tangent to the isocost line.

The optimal combination of inputs can also be determined in the factor-factor case by constrained cost minimization. For this case the objective function can be written as follows:

1st order conditions:

The least cost combination occurs where:┌slope of isoquant

└slope of isocost

The least cost combination occurs where:┌slope of isoquant

└slope of isocost

The least cost combination occurs at the tangency between isoquant and isocost.

This is the same conclusion as the case of constrained output maximization.

2nd order conditions:for a minimum

Consistent with finding the optimal combination of inputs, is the question of determining the optimal level of input use.To answer this question, we assume that the firm is a profit maximizer and has the following objective function:

1st order conditions:

The first order conditions state that for profit maximization inputs should be utilized such that the value of their marginal product is equal to the factor price.

2nd order conditions:

Assume perfect competition with output and input prices:

Also, assume also the firm wishes to spend $80 in production costs with fixed cost (FC) = $20.

One way to find optimal combinations of inputs is constrained output maximization.

Set up the objective function as:

1st order conditions:

From the 1st order conditions,

Substitute into 3rd equation:

Substitute into 3rd equation:

2nd order conditions:

output is maximized subject to the given cost constraint when

λ can be interpreted as the change in output (y) given a $1 change in C (costs).So λ can be interpreted as or the reciprocal of marginal cost. The alternative formulation to solve for the optimal combination of inputs is constrained cost minimization.

For this method, the objective function is written as:

In this example, y0 is assumed to be 8.1st order conditions:

First 2 derivatives Solving these equations simultaneously yields λ = 10 reflects the change in cost given a one unit increase in output (the constraint). So for this formulation, λ represents the MC.

Second order conditions:

critical values will minimize costs subject to the output constraint