Labor, Resources, and the Production Function

Abstract

This chapter introduces a more-or-less “classical” production function, a widely used tool for analyzing economic behavior in the long run. The version presented here is conventional in its derivation of labor demand. However, it is atypical in that, following the logic of Chaps. 1 and 2, it includes resources along with the usual productive factors of labor, capital, and technology. These are introduced via resource supply curves which Chap. 6 will then feed into the supply side of the labor market. The chapter introduces the Cobb-Douglas specification for the production function and ends with the derivation of the per-worker form.

Appendices

Appendix: Summary of Terminology

Table 1

Table 2

Appendix: Why K? Why N? Why A? Why Z?

It might seem more straightforward to refer to capital as “C” and labor as “L”. Unfortunately, we’ll want those worthy letters for other roles later. C will be taken for “consumption” and L for the demand for money (sometimes also referred to as the demand for liquidity, so it’s not completely bizarre to use L for it). K makes some sense for “capital” because at least it sounds right, and anyway, it was Marx who made the word really famous, and in German it’s “das Kapital.” If you want to make some sense of N for labor, you can imagine it stands for the “number” of workers.

The use of A for technology is purely arbitrary and conventional. Some specifications of the production function, such as in [2], use E to represent the “effectiveness” of labor, which is the logic behind combining that E (or our Z) with N before applying the exponent in the Cobb-Douglas function. But we’re holding E in reserve to stand for “exhaustible” resources when we bring those back in.

The use of Z for input efficiency is, of course, particular to this text. It was chosen because it’s available and because it’s at the opposite end of the alphabet from A.

Appendix: Treatment in Calculus and Algebra

This appendix fleshes out some algebraic details of the Cobb-Douglas function, with a particular focus on diminishing marginal product of capital, diminishing marginal product of labor, and constant returns to scale.

3.1 Diminishing MPK and MPN

Diminishing marginal product of capital (MPK) can be shown algebraically with a little calculus. First, MPK is the first derivative of the production function with respect to K (this means that everything in the expression is treated as a constant except for K):

$$\displaystyle{ \text{MPK} = \partial Y/\partial K =\alpha K^{\alpha -1}(Z \cdot N)^{\delta } \cdot R^{\gamma }. }$$

(The strange symbol ∂ is called “del”, and the expression ∂ Y∕∂ K simply means “the partial derivative of Y with respect to K—in other words, what is the rate at which Y changes as K changes by an infinitessimally small amount.)

“Diminishing MPK” is a claim that, as more and more capital is used with fixed amounts of other inputs, the MPK goes down. In other words, as K gets bigger, MPK gets smaller. In algebraic terms, this is a claim that the second derivative of Y with respect to K is negative. (The second derivative is the derivative of the first derivative.) With the Cobb-Douglas function, this is necessarily true. To start, take the second derivative:

$$\displaystyle{ \partial \text{MPK}/\partial K = \partial ^{2}Y/\partial K^{2} =\alpha (\alpha -1)K^{\alpha -2}(Z \cdot N)^{\delta } \cdot R^{\gamma }. }$$

The notation ∂ ² Y∕∂ K ² denotes the second derivative of Y with respect to K

Now look at the parts. Since Z, K, R, and N are all positive numbers, it must be true that

$$\displaystyle{ K^{\alpha -2}(Z \cdot N)^{\delta } \cdot R^{\gamma }> 0. }$$

And with α between 0 and 1, we know that α is positive while α − 1 is negative. So we know that for any Cobb-Douglas function with α between 0 and 1, MPK gets smaller as K gets bigger.

Diminishing marginal product of labor is exactly analogous to diminishing marginal product of capital: MPN is the first derivative of the production function, only this time with respect to labor rather than with respect to capital. And the second derivative with respect to labor tells you how MPN changes as N increases; as with MPK, a negative second derivative tells you that MPN goes down when N goes up.

Start from the form of the production function in which R has been replaced by ρ N and the N terms have been grouped:

$$\displaystyle{ Y = K^{\alpha }Z^{\delta }\rho ^{\gamma }N^{(\delta +\gamma )}. }$$

The first partial derivative of the production function with respect to N is then

$$\displaystyle\begin{array}{rcl} \partial Y/\partial N& =& (\delta +\gamma )K^{\alpha }Z^{\delta }\rho ^{\gamma }N^{(\delta +\gamma -1)} \\ & =& (1-\alpha )K^{\alpha }Z^{\delta }\rho ^{\delta }N^{-\alpha } = \text{MPN}{}\end{array}$$

(4.8)

and the second derivative of the production function with respect to N is:

$$\displaystyle{ \partial \text{MPN}/\partial N = \partial ^{2}Y/\partial N^{2} = -\alpha (1-\alpha )K^{\alpha }Z^{\delta }\rho ^{\gamma }N^{-\alpha -1}. }$$

By the same reasoning as with MPK, we know that K ^α Z ^δ ρ ^γ N ^−α−1 is positive, and that −α(1 −α) is negative, so the whole thing must be negative.

3.2 Constant Returns to Scale

The idea of constant returns to scale is that you could double all three of labor, capital, and resources, or triple them, or multiply them by 1.5, or by any factor, and the output will go up by that same multiple. It doesn’t matter what you multiply K and N and R by, so long as you multiply them all by the same thing. If your production function has constant returns to scale, output will go up by that same multiple.

The way to demonstrate this algebraically with the Cobb-Douglas function (or to test for it in any other function) is to multiply the inputs by a completely general term (call it t), and see whether that same t shows up in output. (Because resources are themselves a function of the labor input, we apply the t factor only to capital and labor.)

We start with arbitrary levels of inputs K ₁ and N ₁, and then we look at a future where those have all increased by some common multiple, t. That means that K ₂ = t ⋅ K ₁ and N ₂ = t ⋅ N ₁. Then Y ₁ = F(K ₁, N ₁) and Y ₂ = F(K ₂, N ₂), and we want to see whether Y ₂ = t ⋅ Y ₁.

With the Cobb-Douglas form we get:

$$\displaystyle\begin{array}{rcl} Y _{2}& =& K_{2}^{\alpha }(ZN_{ 2})^{\delta }(\rho N_{ 2})^{\gamma } \\ & =& (tK_{1})^{\alpha }(ZtN_{1})^{\delta }(\rho tN_{1})^{\gamma } \\ & =& t^{\alpha }K_{1}^{\alpha }Z^{\delta }t^{\delta }N_{ 1}^{\delta }\rho ^{\gamma }t^{\gamma }N_{ 1}^{\gamma } \\ & =& t^{\alpha }t^{\delta }t^{\gamma }K_{1}^{\alpha }Z^{\delta }N_{ 1}^{\delta }\rho ^{\gamma }N_{ 1}^{\gamma } \\ & =& t^{(\alpha +\delta +\gamma )}K_{ 2}^{\alpha }(ZN_{ 1})^{\delta }(\rho N_{ 1})^{\gamma } \\ & =& tY _{1}, {}\end{array}$$

(4.9)

which is what we set out to prove: when you multiply all the inputs by t, the output goes up by a factor of t as well.

If you followed that, you may have noticed the role of the exponents. There were terms of t ^α, t ^δ, and t ^γ, and when you multiply those together, you get t ¹, also known as t. This suggests some slight variants to the Cobb-Douglas form. The exponents on K, ZN, and R could add up to more than 1, in which case there would be increasing returns to scale (doubling the inputs more than doubling the output); or the exponents could add up to less than one (doubling the inputs increases the output by less than double).

Problems

Problem 4.1

The text explained renewables in terms of biologically-based resources. Consider now the supply curve for a non-biological renewable resource.

• (a) What determines the maximum amount you can obtain in a given period (e.g., a year or a quarter)?

• (b) Does obtaining more now affect the amount that will be available in the future?

• (c) What shortcoming(s) might non-biological renewable resources have as substitutes for large amounts of fossil fuel consumption?

Problem 4.2

These questions relate to the per-worker form of the production function discussed in Sect. 4.7. In each case, explain the logic behind your answer.

• (a) What happens to output per worker if there is an increase in capital per worker?

• (b) What happens to output per worker if there is an increase in input efficiency (Z)?

• (c) What happens to output per worker if there is an increase in resource-intensity of labor (ρ)?

• (d) Why does the per-worker form of the production function talk about increased capital per worker, but can speak directly of an increase in input efficiency or an increase in resource-intensity of labor?

Problem 4.3

The calculus-and-algebra appendix explains that the marginal product of labor in a Cobb-Douglas function is:

$$\displaystyle{ MPN = (\delta +\gamma )K^{\alpha }Z^{\delta }\rho ^{\gamma }N^{(\delta +\gamma -1)} }$$

(see Eq. 4.8).

Use the values in Table 4.1.

• (a) What is the real wage?

• (b) Given the choices of 67, 72, 77, 82, 87, 92, which is closest to the equilibrium level of labor? Explain.

• (c) If capital changes from K = 100 to K = 107, which level of N (from the same list of choices as above in part b) is closest to the equilibrium level of labor? Explain.

• (d) If capital stays at its new level of K = 107 and the nominal wage falls from W = 10 to W = 9. 65, which of those same options for N is closest to the equilibrium level?

Table 4.1 Values for Problem 4.3