Production Theory: Inputs, Outputs, and Costs
Theory of Production: Exercises in Production
1. Production with One Input Variable: Total Output, Average Product, and Marginal Product
The production function is an equation, table, or graph that shows the maximum quantity (q) of a product that can be produced per unit of time for each set of alternative inputs, when using the best available production techniques.
For example, consider an agricultural production curve using different alternative quantities of labor per unit of time to cultivate a fixed area of land and record the corresponding quantities of the product. In most cases, where at least one of the factors or inputs of production is fixed, we call this a short-term situation. The average product of labor (APL) is defined as the total product (TP) divided by the number of units of labor employed. The marginal product of labor (MPL) is the change in TP per unit change in the quantity of labor employed.
2. Shape of the Average and Marginal Product Curves
The APL and MPL curves can be determined from the TP curve. The APL at any point on the TP curve is given by the slope of the line from the origin to that point. Generally, the APL curve initially rises, reaches a maximum, and then falls, but remains positive as long as TP is positive.
The MPL between two points on the TP curve equals the slope of the line connecting those two points. The MPL curve also initially rises, reaches a maximum (before the APL reaches its maximum), and then declines. MPL becomes zero when TP is at its maximum, and negative when TP begins to decline. The declining part of the MPL curve illustrates the law of diminishing returns.
3. Stages of Production
Using the relationship between the APL and MPL curves, we can define three stages of production for labor. Stage I goes from the origin to the point where APL is at its maximum. Stage II goes from the maximum point of APL to the point where MPL is zero. Stage III covers the range where MPL is negative. The producer will not operate in Stage III, even with free labor, because total output could be increased by employing less labor per unit of land. The producer will not operate in Stage I because Stage I for labor corresponds to Stage III for land (where the MP of land is negative). Thus, the only relevant stage for the rational producer is Stage II.
4. Production with Two Variable Inputs: Isoquants
Let’s now examine the case where a firm has only two factors of production, labor and capital, and both are variable. Since both factors are variable, we have a long-term situation.
An isoquant shows the different combinations of labor (L) and capital (K) with which a firm can produce a specific quantity of a product. A higher isoquant indicates a greater quantity of product, and a lower isoquant indicates a smaller quantity of product.
5. The Marginal Rate of Technical Substitution
The marginal rate of technical substitution of L for K (MRTSLK) refers to the amount of capital that a firm can give up by increasing the amount of labor employed by one unit while remaining on the same isoquant. MRTSLK is also equal to MPL/MPK. As the firm moves down along an isoquant, the MRTSLK decreases.
6. Characteristics of Isoquants
Isoquants have the same characteristics as indifference curves:
- In the relevant range, they have a negative slope.
- They are convex with respect to the origin.
- They never intersect.
7. Isocosts
An isocost shows all the different combinations of labor and capital that a firm can purchase, given the firm’s total budget and the prices of the factors. The slope of an isocost is determined by the expression PL/PK, where PL is the price of labor and PK is the price of capital.
8. Producer Equilibrium
A producer is in equilibrium when they maximize the output for a given total budget. Another way of saying the same thing is that a producer is in equilibrium when they reach the highest isoquant, given their isocost. This occurs where an isoquant is tangent to an isocost. At the point of tangency, the absolute slope of the isoquant equals the absolute slope of the isocost. That is, in equilibrium, MRTSLK = PL/PK. Since MRTSLK = MPL/MPK, at the equilibrium point:
MPL/PL = MPK/PK or MPL/MPK = PL/PK
This means that, in equilibrium, the MP of the last dollar spent on labor equals the MP of the last dollar spent on capital. The same would be true for other factors if the firm had more than two factors of production.
9. Expansion Path
If the firm’s total budget changes while the prices of labor and capital remain constant, the isocost shifts parallel to itself, upward if the total budget increases and downward if it decreases. These different isocosts will be tangent to different isoquants, thus defining different equilibrium points for the producer. By connecting these equilibrium points, we obtain the firm’s expansion path. This is analogous to the income-consumption curve seen in consumer theory.
10. Factor Substitution
If, starting from a producer equilibrium position, the price of one factor decreases, the equilibrium position will be disturbed. In the process of restoring equilibrium, the producer will substitute the now relatively cheaper factor for the other in production until equilibrium is restored. The extent to which the firm can substitute K for L in production as a result of a change in the relative prices of the factors is called the elasticity of technical substitution, and is measured by:
εsubstitution = %Δ(K/L) / %Δ(MRTSLK)
11. Constant, Increasing, and Decreasing Returns to Scale
We have constant, increasing, or decreasing returns to scale if, when all inputs are increased by a given proportion, output increases in the same, a greater, or a smaller proportion, respectively.