Efficiency+in+production

Matt Stark MBA 651

Managers must determine which and how many of each input should be used in order to most efficiently produce output. When doing this, the production process is considered to utilize two specific inputs to produce output: **capital** and **labor (Pressman 2005)**. This is generally accepted since most production processes consist of machiner (capital) and people (labor). The process utilized to produce output is expressed as an engineering relation called the **production function**. The production function defines the maximum amount of output that can be produced with a given set of inputs (Baye 2006). Where //K// equals capital, //L// denotes labor, and //Q// is the level of output, the production function is mathematically expressed as:

//Q = F(K,L)//

In the short run, a manager's input decisions are limited due to some factors of production being fixed. **Fixed factors of production** are the inputs that managers are unable to adjust because they are going to be used no matter the amount of output being produced. When using the production function, capital is generally the input that is the fixed factor of production. This leaves only labor as the input which the manager can adjust to produce output effeciently. Therefore labor is a **variable factor of production** because managers can alter the amount of this input used to alter production. It can then be implied that the **short run** is the amount of time fixed factors of production exist and the **long run** is point in time in which a manager can begin to alter all inputs of production.
 * //Production Efficency in the Short Run//**

In order for a manager to effectively determine how much of each input to use in production, the manager must measure the productivity of each input. One of those measures is **total product (TP)**, which is the maximum level of output that can be produced with a given amount of inputs (Baye 2006). It is assumed when calculating total product that the labor input is consisting of workers who are putting forth maximal effort. Another measure of productivity is **average product (AP)**, which is the output produced per unit of output (Baye 2006). Average product is measured in terms of each input, therefore when using the production function, the average product of both labor and capital are calculated. The third measure of productivity is **marginal product (MP),** which is the change in total output attributable to the last unit of an input (Baye 2006). It is calculated by dividing the total output by the change in input, and like average product is expressed in terms of each input. The average and marginal product of capital and labor are defined as respectively:




 * **Table 1** ||  ||   ||   || (ΔQ / ΔL) || (Q / L) ||
 * K || L || ΔL || Q || MPL || APL ||
 * 3 || 0 || 1 || 0 || 0 || 0 ||
 * 3 || 1 || 1 || 75 || 75 || 75 ||
 * 3 || 2 || 1 || 250 || 175 || 125 ||
 * 3 || 3 || 1 || 500 || 250 || 167 ||
 * 3 || 4 || 1 || 800 || 300 || 200 ||
 * 3 || 5 || 1 || 1,100 || 300 || 220 ||
 * 3 || 6 || 1 || 1,400 || 300 || 233 ||
 * 3 || 7 || 1 || 1,700 || 300 || 243 ||
 * 3 || 8 || 1 || 1,950 || 250 || 244 ||
 * 3 || 9 || 1 || 2,125 || 175 || 236 ||
 * 3 || 10 || 1 || 2,050 || -75 || 205 ||

However in the short run the amount of capital used cannot be adjusted, meaning there is no change in capital and marginal product of capital is cannot be calculated. For every production function, the marginal product of labor follows a certain pattern of increasing to a maximum point at a certain amount of labor then decreases for each additional unit of labor, as shown in Table 1 above. This makes sense because if too many laborers are hired, eventually they would be getting in each other's way, prohibiting space and decreasing output. This is best shown through plotting total product, average product, and marginal product on a line graph.



As you can see up to a certain number of workers hired, the slope of the total product curve gets steeper. This portion of the graph represents **increasing marginal returns**, or the range of input usage over which marginal products increases (Baye 2006). At this point output starts to decrease with every additional unit of labor. The range from this point until the point where marginal product reaches zero again is referred to as **decreasing marginal returns**. In this portion of the graph, the total product curve starts to flatten out and eventually reach an apex (Truett and Truet 2006). It is this point where a firm will start to experience **negative marginal returns** to labor, as shown by the part of the total product curve that is downward sloping. You will also see that the marginal product curve dipped below zero output.

In order to maximize profits, a manager must ensure that the firm operates at the right point on the production function (Bayre 2006). In the short run, this means hiring the optimal level of labor (since in the short run capital is a fixed input). To do this, the manager will determine the **value marginal product of labor,** or the value of the output produced by the last unit of input (Baye 2006). This is calculated by multiplying the price of the output by the marginal product of the last laborer hired. If this product is greater than the unit cost of the additional worker, then the manager should hire the worker, and vice versa. The formula for value marginal product of labor is:



The manager should then continue to hire additional workers until the last additional worker's value marginal product of labor no longer covers their cost. This is called the **profit-maximizing input usage rule,** which defines the demand for labor in the short term for a profit-maximizing firm (Baye 2006). When the VMPL is graphed, you see the demand curve for labor is upward sloping until it reaches the range of diminishing marginal returns, at which point the curve then slopes downward because of the law of diminishing returns. Remember when too many workers are hired, production efficiency begins to decrease, which is shown by the demand curve of labor.



There are three different type of algebraic forms of production functions. The most simple form is the **linear production function,** which assumes that a production function has a perfect linear relationship between all inputs and total outputs. The formula for the linear production function is:

//Q F(K,L) aK + bL//

In this formula a and b are constants, making inputs perfect substitutes. Therefore if a 2 and b 1, then this equation assumes that one unit of capital will always be twice as productive as one unit of labor. **The Leontif or fixed proportions** **production function** always implies that inputs are used in fixed proportions. Therefore one unit of capital and one unit of labor are necessary to produce one unit of output. (Baye 2006). The last type of production function is called the **Cobb-Douglas production function,** which assumes there is some degree of substitution exists between inputs, but the inputs cannot be perfectly substituted. This is because when Cobb and Douglass attempted to study the elasticity of supply of labor and capital, they graphed a series of both inputs and output and found that the output curve lay in between the capital and labor curve at approximately on quarter of the distance between the input curves (Adams and Felipe 2005). This production function is expressed as:

//Q F(K,L) Kˆa x Lˆb//

A manager's task to optimize long-term production efficiency is more complex since both inputs can be adjusted in the long run. This means that different combinations of inputs can be used to produce the same amount of output, the manager must then choose the right amount of each input to minimize the cost of producing the desired level of output. The basic tool used for doing this is the isoquant, which defines the combinations of labor and capital that yield the same amount of output (Baye 2006). When depicted graphically, isoquants are convex in shape.
 * //Production Efficienct in the Long Run//**



This is because the inputs are not perfectly substitutable, which if they were the isoquants would be linear-shaped. Therefore it would take increasing amounts of one input to replace each unit of the other input that is not used, which is called the **law of diminishing marginal rate of technical substitution.** The rate at which this is done is called the **marginal rate of technical substitution (MRTS).** This ratio is the absolute value of the slope the isoquant. The MRTS of labor to capital is expressed as:

//MRTS = MPL / MPK//

Different combinations of capital and labor can can result in the same cost of producing output for a firm. When described graphically, these combinations of inputs that cost the firm the same amount fall on the isocost line. The total cost of inputs (C) is found by using the wage rate (cost of labor) and the rental rate (cost of capital) in the following equation:

//wL + rK = C//

Because K is a linear funciton of L along the isocost line, the slope and y-intercept of the isocost line is determined by algebraically manipulating this formula:

//K = C/r - (w/r)L//

By using this linear function one can plot the isocost line on the graph by using C/r as the y-intercept and -w/r as the slope. If the firm wishes to produce more output, the isocost line will move further away from the origin due to the higher costs associated with increasing input levels. However if the cost of inputs change, that will simply increase or decrease the slope of the isocost line.

In order for a firm to produce on the production function in the long run, the manager must minimize the cost of inputs used in production. To do this the manager must determine the combination of labor and capital necessary to produce the desired level of output at he lowest possible cost to the firm. Managers use isoquants and the isocost line to determine the the optimal combination of inputs to minimize cost. The closest point to the origin where the isoquant and isocost line meet is the point on the graph that defines the **cost-minimizing input mix** (Baye 2006)



At this point on the graph is where the slopes of the isoquant and the isocost line are the same. If you recall, the absolute value of the slope of an isoquant is reflects the marginal rate of technical substitution, which means the MRTS is equal to the slope of the isocost line at the cost-minimizing input mix:

//MRTS = w / r//

Therefore in order to minimize the cost of producing the desired level of output, a firm should utilize the combination of inputs such that the MRTS is equal to the ratio of input prices. This is referred to as the **cost-minimizing input rule.**



This can be affected by price changes of inputs. If the price of an input rises, that will cause the slope of the isocost line to become steeper and change the cost-minimizing input mix (remember the slopes of the isoquant and the isocost line are the same at the point of cost minimization). Therefore the manager must adjust the combination of inputs buy substituting the amount of input which experienced a price increase with the input mix whose price stayed the same. **This is called optimal input substitution.** (Baye 2006)


 * REFERENCES**

Adams, F, Gerard and Felipe, Jesus. " 'A Theory of Production' The Estimation of the Cobb-Douglas Functino: A Retrospective View." __Eastern Economic Journal.__ Vol. 31 No. 3 (2005): 427-443

Baye, Michael R. __Managerial Economics and Business Strategy__. Rev. ed 5. New York: McGraw-Hll, 2006

Pressman, Steven. "What is Wrong with the Aggregate Production Function?" Eastern Economic Journal. Vol. 31 No. 3 (2005): 422-425

Truett, Dale B. and Truett, Lila J. "Production Function Geomoetry with "Knightian" Total Product." Journal of Economic Education. Summer 2006: 348-358.

Questions:

1. Which production function assumes that all inputs are used in fixed proportions? a. Cobb - Douglass b. Leontif c. linear d. long run

2. All inputs of production can be adjusted in the _. a. long run b. short run c. cost-minimizing input mix d. value marginal product of labor

3. The _ is the change in output attributable to the last additional input. a. total product b. average product c. marginal product d. labor

4. At what range on the graph does total output begin to decrease and marginal product is zero?

a. increasing marginal returns b. cost-minimizing input mix c. negative marginal returns d. diminishing marginal returns

5. In order to minimize the cost of producing the desired level of output, a manager should choose the combination of inputs such that the MRTS is equal to the ratio of the prices of each input. What is this called?

a. optimal input substitution rule b. cost-minimizing input rule c. profit-maximizing input usage rule d. production function

Answers

1. a 2. a 3. c 4. d 5. b