Applied Economics for Business Management Lecture 6 Lecture

Applied Economics for Business Management Lecture #6

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

Continue Factor – Product Case: Output elasticity: Output elasticity or the elasticity of production can be written as:

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

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

Example

Stages of Production • 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).

Stages of Production Boundary between Stages I & II:

Stages of Production Boundary between Stages II & III:

Stages of Production What’s happening in Stage I?

Stages of Production What’s happening in Stage II?

Stages of Production What’s happening in Stage III?

Stages of Production 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.

Stages of Production 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).

Profit Maximization First order condition(s):

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

Profit Maximization First order condition(s): What is the economic meaning of this first order condition? So the critical value for x 1 occurs where the value of the marginal product for this input is equal to its price.

Profit Maximization 2 nd 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.

Example You are also given that Find the profit maximizing level of input usage for x 1.

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

Example What is the level of profit?

Example 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.

Two variable factors and one product case: 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

Isoquant 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.

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).

Isoquant 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.

Isoquant 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.

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

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

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

Optimal combination of inputs in production: 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)?

Optimal combination of inputs in production: Assume a given production function: Specify a given level of costs (a constraint):

Optimal combination of inputs in production: Assume a given production function: Specify a given level of costs (a constraint): Objective function: maximize output subject to a cost constraint

Optimal combination of inputs in production: First order conditions: Solving for λ in the first two equations:

Isocost ┌slope of cost line └slope of isoquant or The 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 combination of inputs.

Second Order Condition The second order condition is a bordered Hessian: for maximum

Constrained Output Maximization 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.

Optimal Combination of Inputs in Production 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:

Optimal Combination of Inputs in Production 1 st order conditions: The least combination occurs where: ┌slope of isoquant └slope of isocost

Optimal Combination of Inputs in Production The least combination occurs where: ┌slope of isoquant └slope of isocost The least combination occurs at the tangency between isoquant and isocost. This is the same conclusion as the case of constrained output maximization.

Optimal Combination of Inputs in Production 2 nd order conditions: for a minimum

Profit Maximization 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:

Profit Maximization 1 st 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.

Profit Maximization 2 nd order conditions:

Example 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.

Example One way to find optimal combinations of inputs is constrained output maximization. Set up the objective function as: 1 st order conditions:

Example From the 1 st order conditions, Substitute into 3 rd equation:

Example Substitute into 3 rd equation:

Example 2 nd order conditions: output is maximized subject to the given cost constraint when

Constrained Cost Minimization 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.

Constrained Cost Minimization For this method, the objective function is written as: In this example, y 0 is assumed to be 8. 1 st order conditions:

Constrained Cost Minimization 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.

Constrained Cost Minimization Second order conditions: critical values will minimize costs subject to the output constraint