The Production Function
The production function is a schedule that shows the highest level of output the firm can produce from a given combination of inputs. The highest total amount of output the firm can produce given the amount of its outputs is also called the total product of L and K (labor and capital).
An isoquant is a curve showing all of the input combinations that yield the same level of output. An isoquant map is the collection of all isoquants corresponding to a given production function.
How Many Inputs?
More than two inputs are used to produce output. However, all of the key insights that we gain from the two-factor case hold generally.
What Is Output?
Output can be something tangible – bikes, cars, etc., or something non-tangible – education, health services, etc.
The Decision-Making Horizon
The production function tells us whether a given input combination is capable of producing a particular amount of output, but it does not tell us whether the firm can actually obtain the input combination under consideration. For this, we need to consider the time horizon.
A variable factor is a factor whose level can be varied over the relevant planning horizon. A fixed factor is a factor whose level cannot be varied over the relevant planning horizon.
The short run is defined as a time period over which only one of the firm’s inputs is variable, while all the others are fixed. The long run is defined as a period of time long enough that all of the factors are variable and none is fixed. The actual time periods for the short and long run vary depending on the product.
8.2 Properties of the Production Function
Marginal Physical Product (MPP)
The marginal physical product (MPP) of an input is the extra amount of output that can be produced when the firm uses one additional unit of input. If using an additional ∆L units of labor raises output by ∆X, the marginal physical product of labor: MPPL=∆X/∆L
Increasing Marginal Returns
A technology exhibits increasing marginal returns when the marginal physical product of an input rises as the amount of the input used increases. The slope of the total product curve is the marginal physical product of labor. When the firm’s technology exhibits increasing returns to labor, the total product curve gets steeper as we increase the quantity of labor, and the marginal physical product curve is upward sloping.
Constant Marginal Returns
A technology exhibits constant marginal returns when there is a range of input levels over which the marginal product of a factor remains unchanged as the amount of the factor increases. When the technology exhibits constant marginal returns, the marginal physical product of labor is unaffected by changes in the amount of labor employed, and the marginal physical product curve is flat.
Diminishing Marginal Returns
A technology exhibits diminishing marginal returns when the marginal product of an input falls as the amount of the input used increases. When a technology exhibits diminishing marginal returns, the marginal physical product curve falls as the quantity of the input rises, but the curve remains positive. Since the height of the marginal product curve is the slope of the total product curve, the total product increases as the number of workers rises, but at a decreasing rate (the curve becomes flatter as the amount of labor rises).
A Changing Pattern of Marginal Returns
Perhaps the most common situation is that initially there are increasing marginal returns, then a range of constant marginal returns, followed by a range of diminishing marginal returns. In this case, the MPP curve slopes up, is horizontal, and then slopes down.
Marginal Rate of Technical Substitution
The marginal rate of substitution is the rate at which the available technology allows the substitution of one factor for another. It is -1 times the slope of the isoquant. This is a rate of -∆K/∆L. (This is completely analogous to the rate of substitution for the household.)
Two Polar Cases of Factor Substitution
Case I: Perfect Substitutes: Whenever a technology has a constant marginal rate of technical substitution between two inputs, the resulting isoquants are straight lines. Since gasoline and gasohol can always be substituted for each other at the same rate, they are said to be perfect substitutes in this use. Perfect substitutes are two inputs that have a constant marginal rate of technical substitution of one for the other.
Case II: No Factor Substitution: When two products must be used together in a constant proportion, no factor substitution is possible, and the isoquants are right angles that are placed along a ray from the origin whose slope is equal to the proportion.
The Relationship between MPP and MRTS
If a firm hires ∆L additional workers, then the marginal physical product of labor (MPPL) times ∆L gives the total output rise. The firm can now reduce the number of robots it has until the amount of output lost offsets the gain from the extra labor. If the firm employs ∆K fewer robots, the reduction in the total output is MPPK * ∆K. This gives: MPPL*∆L=MPPK*∆K which can be rewritten as -∆K/∆L=MPPL/MPPK
Since -∆K/∆L is the marginal rate of substitution, then: MRTS=MPPL/MPPK
The marginal rate of technical substitution between two inputs is equal to the ratio of the marginal physical products of the inputs.
A technology exhibits a diminishing marginal rate of technical substitution when the rate at which one factor can be substituted for the other falls as the amount of the first factor rises.
Returns to Scale
The degree of returns to scale is the rate at which the amount of output increases as the firm increases all of its inputs proportionately.
Constant Returns to Scale: A technology such that a proportional increase in all input levels leads to output growth in the same proportion. (e.g. Almond Yummies production states that chocolate and almonds have to be used in the exact proportion of 4 ot 1. If the firm doubles the usage of both inputs, it doubles its output.)
Increasing Returns to Scale: A technology such that a proportional increase in all input levels leads to greater than proportionate output growth. (e.g. Because Boeing workers learn through experience, the amount of labor needed to manufacture an aircraft falls dramatically as the number of planes produced rises.)
Decreasing Returns to Scale: A technology such that a proportional increase in all input levels leads to less than proportional output growth. (Some argue that if the firm can simply replicate itelf, then there would never be a reason to observe decreasing returns to scale.)
Graphing Returns to Scale
The degree of returns to scale can be seen graphically using isoquants. A ray drawn from the origin shows all of those input combinations that involve the use of capital in the same proportion. If doubling the each of the inputs results in more than double the output (shown by the isoquant), then there are increasing returns to scale (isoquants seem closer together). If doubling the inputs results in a point on an isoquant representing double the original output, then the technology is subject to constant returns to scale. If doubling the inputs results in a point on an isoquant representing less than double the output, then the technology is subject to diminishing returns to scale (and isoquants seem farther apart).
Marginal Returns and Returns to Scale
The shape of a marginal physical product (MPP) curve reflects the impact of a change in the amount of a single factor, while the degree of returns to scale concerns the impact of a simultaneous change in all of the factors in proportion.
For a production function that allows no substitution, such as “Almond Yummies,” the recipe requires 4 almonds and 1 ounce of chocolate. If the firm currently has 20 almonds, then the marginal physical product of chocolate is equal to one if C is less than 5. If C is greater than or equal to 5, an additional ounce of chocolate has no effect on what the firm can produce – at this point the marginal physical product of chocolate is zero. Since the MPP of chocolate goes from 1 to zero, we might think it has diminishing marginal returns. However, we know from our earlier discussion that this product exhibits constant returns to scale.