# The Theory and Estimation of Production

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The Theory and Estimation of Production
The Production Function Short-Run Analysis of Total, Average, and Marginal Product Long-Run Production Function Estimation of Production Functions Importance of Production Functions in Managerial Decision Making

Learning Objectives Define production function and explain difference between short-run and long-run production function Explain “law of diminishing returns” Define the Three Stages of Production and how it relates to the “law of diminishing returns” Describe different forms of production functions that are used. Briefly describe the Cobb-Douglas function

Importance of chapter To provide a framework for managerial decisions regarding allocation of firms resources Show how managers can determine which inputs and how much of each input to use to produce output efficiently This chapter serves as the foundation for later chapters, which describe in detail pricing and output techniques for managers interested in profit maximization

Production Function Technology available for producing output
Consider a production process that utilizes only capital and labor to produce output K = quantity of capital L= quantity of labor and Q= Output level produced in the production process. The production function is a relation that defines the maximum output that can be produced with a given set of inputs

Q=f( K, L) Mathematically, the production function can be expressed as
Q: level of output K and L: inputs used in the production process

Key assumptions Some given “state of the art” in the production technology. Whatever input or input combinations are included in a particular function, the output resulting from their utilization is at the maximum level.

Short run vs. long run decisions
In the short run some factors of production are fixed and this limits the choice in making input decisions e.g. Car manufacturing company: Capital is fixed but labor and steel can be adjusted making them variable inputs The short run production function is essentially a function of only labor

In the long run the manager can adjust all factors of production in the long run all inputs are variable. If it takes a company 3 years to acquire additional capital machines, then the long run for that company is 3 years and the short run is less than 3 years

In summary: The short-run production function shows the maximum quantity of good or service that can be produced by a set of inputs, assuming the amount of at least one of the inputs used remains unchanged. The long-run production function shows the maximum quantity of good or service that can be produced by a set of inputs, assuming the firm is free to vary the amount of all the inputs being used.

Short run functions Assume Q = F(K,L) = K.5 L.5
K is fixed at 16 units. Short run production function: Q = (16).5 L.5 = 4 L.5 Production when 100 units of labor are used? Q = 4 (100).5 = 4(10) = 40 units

Measures of productivity
Managers must determine the productivity of inputs used in the production process This is useful for evaluating the effectiveness of the production process and making input decisions that maximize profit 3 most important measures of productivity are Total product, Average product and Marginal product

Short-Run Analysis of Total, Average, and Marginal Product
Alternative terms in reference to inputs Inputs Factors Factors of production Resources Alternative terms in reference to outputs Output Quantity (Q) Total product (TP) Product

Average Product: Manager may wish to know, on average, how much each worker contributes to the total output of the firm. AP for an input is Total product divided by quantity use of input

Average Product of Labor
APL = Q/L. Measures the output of an “average” worker. Example: Q = F(K,L) = K.5 L.5 If the inputs are K = 16 and L = 9, then the average product of labor is APL = [(16) 0.5(9)0.5]/9 = 1.33

Marginal Product: Is the change in total output attributable to the last unit of input MP for an input is Change in Total product divided by change in quantity use of input

Average Product (AP): Marginal product (MP):

If MP > AP then AP is rising.
If MP < AP then AP is falling. MP=AP when AP is maximized.

Phases of Marginal Product:
As the usage of an input increases, marginal product initially increases (increasing marginal returns), then begins to decline (decreasing marginal returns) and eventually becomes negative (negative marginal returns)

Increasing, Diminishing and Negative Marginal Returns
Q L Q=F(K,L) MP AP

Law of Diminishing Returns: As additional units of a variable input are combined with a fixed input, at some point the additional output (i.e., marginal product) starts to diminish. Nothing says when diminishing returns will start to take effect, only that it will happen at some point. All inputs added to the production process are exactly the same in individual productivity

The Three Stages of Production in the Short Run
Stage I: From zero units of the variable input to where AP is maximized (where MP=AP) Stage II: From the maximum AP to where MP=0 Stage III: From where MP=0 on

In the short run, rational firms should only be operating in Stage II.
Why not Stage III? Firm uses more variable inputs to produce less output Why not Stage I? Underutilizing fixed capacity Can increase output per unit by increasing the amount of the variable input

What level of input usage within Stage II is best for the firm?
The answer depends upon how many units of output the firm can sell, the price of the product, and the monetary costs of employing the variable input.

Manager’s role of using the right level of input:
e.g. restaurant manager must hire the “correct” number of servers If product is sold at \$3 on the market and each unit of labor costs \$400, how many units of labor should be hired to maximize profit?

First, determine the benefit of hiring an additional worker
First, determine the benefit of hiring an additional worker. Each worker increases the firm’s total output by her marginal product. This increase can be sold in a market at a price of \$3 Thus the benefit from each unit of labor is \$3 x MP of worker This number is known as the Value marginal product of labor = VMP VMPL = P x MPL.

It is profitable to hire units of labor so long as their additional output value exceeds their cost.
So, employ labor as long as VMP exceeds their wage (w) To maximize profits, a manager should use inputs at levels which their marginal benefits equal the marginal cost. Specifically for labor, VMPL = w For capital: value of marginal product of capital equals the rental rate: VMPK = r,

Alternative Terminology (using Marginal Revenue product)
Total Revenue Product (TRP): market value of the firm’s output, computed by multiplying the total product by the market price. TRP = Q · P Marginal Revenue Product (MRP): change in the firm’s TRP resulting from a unit change in the number of inputs used. MRP = = MP · P

Total Labor Cost (TLC): total cost of using the variable input, labor, computed by multiplying the wage rate by the number of variable inputs employed. TLC = w · X Marginal Labor Cost (MLC): change in total labor cost resulting from a unit change in the number of variable inputs used. Because the wage rate is assumed to be constant regardless of the number of inputs used, MLC is the same as the wage rate (w).

Summary of relationship between demand for output and demand for input
A profit-maximizing firm operating in perfectly competitive output and input markets will be using the optimal amount of an input at the point at which the monetary value of the input’s marginal product is equal to the additional cost of using that input. MRP = MLC

Multiple variable inputs
Consider the relationship between the ratio of the marginal product of one input and its cost to the ratio of the marginal product of the other input(s) and their cost. Other factors may outweigh this relationship Political/Economic risk factors`

The Long-Run Production Function
In the long run, a firm has enough time to change the amount of all its inputs. Effectively, all inputs are variable. The long run production process is described by the concept of returns to scale.

If all inputs into the production process are doubled, three things can happen:
output can more than double increasing returns to scale (IRTS) output can exactly double constant returns to scale (CRTS) output can less than double decreasing returns to scale (DRTS)

One way to measure returns to scale is to use a coefficient of output elasticity:
If EQ > 1 then IRTS If EQ = 1 then CRTS If EQ < 1 then DRTS

Returns to scale can also be described using the following equation
hQ = f(kX, kY) If h > k then IRTS If h = k then CRTS If h < k then DRTS

Graphically, the returns to scale concept can be illustrated using the following graphs.
Q X,Y IRTS Q X,Y CRTS Q X,Y DRTS

Estimation of Production Functions
Forms of Production Functions Cobb-Douglas Production Function: Q = aLbKc Both capital and labor inputs must exist for Q to be a positive number Can be increasing, decreasing, or constant returns to scale b + c > 1, IRTS b + c = 1, CRTS b + c < 1, DRTS Permits us to investigate MP for any factor while holding all others constant Elasticities of factors are equal to their exponents

Estimation of Production Functions
Forms of Production Functions Cobb-Douglas Production Function Can be estimated by linear regression analysis Can accommodate any number of independent variables Does not require that technology be held constant Shortcomings: Cannot show MP going through all three stages in one specification Cannot show a firm or industry passing through increasing, constant, and decreasing returns to scale Specification of data to be used in empirical estimates`

Marginal Rate of Technical Substitution (MRTS)
The rate at which two inputs are substituted while maintaining the same output level.

Cost Minimization Marginal product per dollar spent should be equal for all inputs: But, this is just