12.1 Production and Factor Employment with a Single Variable Input
Production functions generally take the form
\(q \equiv \phi (a, b, c, ...)\)
That states that the output quantity (q) depends in some specified way upon the amounts a, b, and c... of the resource inputs A, B, and C...
Suppose all inputs are held fixed except for the amount of A. Then we can write the function as
\(q \equiv q(a)\)
These production functions are often governed by the Law of Diminishing Returns:
If the amount a of input A increases, with other inputs held fixed, the rate of increase of Total Product (q) - that is, the marginal product - eventually begins to fall. This is the point of diminishing marginal returns. As the input amount increases further, Average Product also begins to fall. This is the point of diminishing average returns. And as use of input A rises further, even Total Product may fall (the case of extreme overabundance hindering production). This would be the point of diminishing total returns.
From Production Function to Cost Function
Cost functions reflect the underlying production function with the prices of inputs, either renting or purchasing. To begin, suppose the firm is a price-taker (no monopsony power). For any given combination of inputs, the Total Cost will be
\(C \equiv h_a a + h_b b + h_c c + ...\)
When some factors are held fixed, perhaps all except for A, the Total Cost can be divided into a fixed component and a variable component
\(C \equiv F + V \equiv F + h_a (q)\)
This states that the Total Cost depends on the fixed cost, the hire-price, and the production function of A. We can measure the marginal cost using this equation
\(MC \equiv \frac{\Delta C}{\Delta q} = \frac{h_a \Delta a}{\Delta q} = \frac{h_a}{\Delta q / \Delta a} = \frac{h_a}{mp_a}\)
Here is is clear that the MC necessarily increases as MP decreases. This explains why the marginal cost curve must eventually rise. We can show the relationship between average variable cost and average product as such
\(AVC \equiv \frac{V}{q} = \frac{h_a a}{q} = \frac{h_a}{q / a} = \frac{h_a}{ap_a}\)
As the average product falls, the average cost increases. The relationship holds even after adding in fixed costs.
The Firm's Demand for a Single Variable
The production function is a technological relation between inputs and outputs. In deciding upon the amounts of factors to employ, firms must decide concerning factor prices as well. The firm faces a horizontal product supply curve at the market price of the input. In deciding whether or not to employ an additional unit of A, the firm must balance the hire-price with the benefit gained - both additional physical output and what that output generates as revenue.
Combining the product price and the marginal product leads to the concept of Value of the Marginal Product:
\(vmp_a \equiv P * mp_a\)
This curve is the firm's demand for the input, taking the same shape as the marginal product curve, just shifted up to account for the price. The factor employment condition for a price-taking firm will thus be
\(vmp_a = h_a\)
Monopolists will use the marginal revenue product -- the marginal revenue times the physical marginal product.
12.2 Production and Factor Employment with Several Variables
With more than one factor to be considered, the graphs of the production function becomes multi-dimensional. Now there are "families" of total product curves, marginal product curves, and the other relevant information. They all follow the same shape (for the Law of Diminishing Returns applies) but now all must be taken into account. The classic example of this is the Cobb-Douglas production function
\(q = \kappa a^\alpha b^\beta\)
If the sum of the exponents exceeds 1, there will be increasing returns to scale since the doubling of both inputs more than doubles output. Exactly equaling 1 is constant returns to scale and decreasing returns to scale would be less than 1.
Factor Balance and Factor Employment
The question of factor balance asks what are the best input proportions at any given level of cost or output. This is a preliminary to asking about factor employment -- the actual amounts of inputs to hire at any given set of factor hire-prices.
Unlike consumers, the firm can decide its budget constraint, or how much cost to incur. These lines will be intersected by the Scale Expansion Path (SEP) that shows the best combination of inputs at each level of cost
Recall that the consumer's optimum condition is expressed as equality between the product price ratio \(\frac{P_x}{P_y}\) (absolute slope of the consumer's budget line) and the marginal rate of substitution in consumption (MRSc, the absolute slope of the indifference curve). MRSc was defined as the ratio at which a person was just willing to substitute a small amount of Y for a small amount of X in the consumption basket, leaving the consumer at the same level of utility or indifference.
Correspondingly here, the marginal rate of substitution in production, MRSQ, is defined as the mount of input B that can be substituted for a small change in input A
\(MRS_Q \equiv - \frac{\Delta b}{\Delta a} |_q \equiv \frac{mp_a}{mp_b}\)
The slope of the isocost line is \(- \frac{h_a}{h_b}\) and the tangency condition is
\(\frac{mp_a}{mp_b} = \frac{h_a}{h_b}\)
Leading us to the Factor Balance Equation:
\(\frac{mp_a}{h_a} = \frac{mp_b}{h_b}\)
Inputs are balanced when the marginal products per dollar are equal for all resources employed. This gives us the Scale Expansion Path.
The Firm's Demand for Inputs
Since some inputs are used jointly, the firm's demand for them will be interrelated. When a firm maximizes profit by equating marginal cost to marginal revenue, it automatically satisfied the Factor Employment Conditions:
\(mrp_a = h_a\)
\(mrp_b = h_b\)
Two inputs are complementary if increased use of one raises the marginal product of the other. They are anticomplementary is vice versa (close substitutes). These differences give rise to differing elasticities for the demand curves, which are always flatter than the marginal revenue product curves.
12.3 The Industry's Demand for Inputs
After a fall in hire-price, industry-wide output increases and so product price falls - thus lessening the firms' incentive to hire more of the cheapened input A. This product-price effect makes the industry demand curve steeper than the simple aggregate of the individual demand curves for the factor. On the other hand, the entry-exit effect cuts in the opposite direction. A fall in hire-price increases firms' profits, inducing new firms to enter and thereby flattening the industry demand curve for input A.
Two contesting models, the competitive model and the monopsonist model, have been used in analyzing minimum-wage laws. If the competitive model is applicable, so that firms are price-takers in the input market for labor, a legally imposed minimum wage higher than the previous equilibrium will raise wages for those workers who remain employed but disemploy others. If the monopsony model applies, the effects are mixed. Over a certain range it is even possible that wages adn employment may both increase. However, there is much evidence accumulated by economists in favor of the competitive model.