CLASS #8: PRODUCTION: THE PRODUCTION FUNCTION

Ch. 3

• Move away from "endowment economy" where output was treated as given (fruits from tree)
• Understand the determinants of output: capital, labor, and productivity

Introduce a production function that depends on the inputs: capital and labor

Y=F(A,K,L)

where Y is output, K is capital, N is labor, and A is "total factor productivity" total factor productivity: how K and N are "mixed" together in order to produce Y (example: cooking) we will often focus on a Cobb-Douglas production function:

$$Y=AK^\alpha N^{1-\alpha}$$

nice properties of Cobb-Douglas

• increasing in both capital and labor; need both in order to produce
• if you double the inputs, the output doubles
• decreasing marginal returns to capital and labor
• product exhaustion
• constant factor shares

Increasing in both capital and labor

need both in order to produce

Homogeneity of degree one

with twice as much capital and twice as much labor one obtains twice as much output, for given productivity (like recipes)

$$F(A,qK,qL) = A(qK)^\alpha (qN)^{1-\alpha} = qAK^\alpha N^{1-\alpha}$$

Decreasing marginal returns to labor and capital

Adding one more worker, for a fixed amount of capital, will increase production by less and less the larger the number of workers that are already operating the machines Define MPN = marginal productivity of labor, as

$$MPN = \frac{dF}{dN} = (1-\alpha) AK^\alpha N^{-\alpha} = (1-\alpha) A(\frac{K}{N})^\alpha$$

The marginal productivity of labor is an increasing function of the capital/labor ratio Likewise, adding one more unit of capital, for a fixed amount of labor, will increase production by less and less the more capital you already have in place. Define MPK = marginal productivity of capital, as

$$MPK = \frac{dF}{dK} = \alpha AK^{\alpha-1} N^{1-\alpha} = \alpha A(\frac{K}{N})^{\alpha-1}$$

The marginal productivity of capital is a decreasing function of the capital/labor ratio. Aside: How are the remunerations (wages and return to capital) of the two inputs (labor and capital) determined in a competitive economy? Let us study the problem of a firm that hires labor at the market wage w and rents capital at the market interest rate r. As we are in a competitive environment, the firm takes the cost of both labor and capital as given, and chooses the amount of labor to hire and capital to rent with the goal of maximizing profits:

$\begin{array}{l} max_{K,N} Revenues-Cost =\\ max_{K,N} F(A,K,N)-wN-rK \end{array}$

As a result:

$$r = MPK = \frac{dF}{dK} = \alpha AK^{\alpha-1} N^{1-\alpha}$$

$$w = MPN = \frac{dF}{dN} = (1-\alpha) AK^\alpha N^{-\alpha}$$

Product Exhaustion

The remuneration of capital and labor exhausts all production Y - when prices are competitively determined

$\begin{array}{l} wN + rK =\\ =(1-\alpha) AK^\alpha N^{-\alpha} N + \alpha AK^{\alpha-1} N^{1-\alpha} K=\\ = AK^\alpha N^{1-\alpha} = Y \end{array}$

(all production functions that are homogeneous of degree one) Profits are zero

Constant factor shares

What is the fraction of GDP which is to pay labor? and capital? (when prices are competitively determined)

$$\frac{wN}{Y} =\frac{(1-\alpha) AK^\alpha N^{1-\alpha}}{Y} = (1-\alpha)$$

$$\frac{rK}{Y} =\frac{\alpha A K^\alpha N^{1-\alpha}}{Y} =\alpha$$

so factor shares are constant over time, which is what has been observed from the (US) data Since the factor share for labor and capital are respectively 2/3 and 1/3, $\alpha =.33$ seems to be a good guess

What is capital and what is labor?
HUMAN CAPITAL

• Education
• Learning how to operate a machine

This is a very important issue for policy: how much should a country invest in physical capital, and in education?
Including human capital, $\alpha =.6$ may be a better guess Do "real world" data satisfy this equation with $\alpha =.33$?
Notice that per capita output (y) is equal to:

$$y = \frac{Y}{N} = A(\frac{K}{N})^\alpha$$

and the capital to output ratio (k):

$$k = \frac{K}{Y} = A^{-1}(\frac{K}{N})^{1-\alpha}$$

or

$$\frac{K}{N} = (Ak)^{\frac{1}{1-\alpha}}$$

from which we obtain:

$$y = A^{\frac{1}{1-\alpha}} k ^{\frac{\alpha}{1-\alpha}}$$

taking logarithms we get:

$$ln(y) = \frac{1}{1-\alpha} ln(A) + {\frac{\alpha}{1-\alpha}} ln(k)$$

if this relationship holds for each country, it should hold for the world as a whole:

$$ln(\bar{y}) = \frac{1}{1-\alpha} ln(\bar{A}) + {\frac{\alpha}{1-\alpha}} ln(\bar{k})$$

taking differences (assuming A is the same for all countries):

$$ln(y)- ln(\bar{y}) = {\frac{\alpha}{1-\alpha}} (ln(k)-ln(\bar{k}))$$

if $\alpha =.33$, then $\frac{\alpha}{1-\alpha}=.43$. For Switzerland, $ln(k)-ln(\bar{k})=.81$, and therefore we should have $ln(y)- ln(\bar{y}) = .35$ ( if the capital/output ratio is 81 % larger then the world average, per capita output should be 35 % larger ).
For Ethiopia, $ln(k)-ln(\bar{k})=-1.08$, and therefore we should have $ln(y)- ln(\bar{y}) = -.46$ ( if the world average capital/output ratio is 108 % larger than Ethiopia's, the world average per capita output should be 46 % larger than Ethiopia's).
In fact, for Switzerland per capita output is 153 % larger, and for Ethiopia it is 221% smaller.
With $\alpha=.67$ we are closer to reality. (evidence from Chari, Kehoe, McGrattan, 1997, and George Hall)

Total Factor Productivity (TFP or "Solow Residual")

The part of growth in output that is not explained by growth in the two inputs, labor and capital. If we take the logs of the production function:

$$ln(Y_t) = ln(A_t) + \alpha ln(K_t) + (1-\alpha) ln(N_t)$$

at time t and t-1 and subtract

$\begin{array}{c} ln(Y_t) -ln(Y_{t-1}) = ln(A_t)-ln(A_{t-1}) \\ \\ + \alpha (ln(K_t)-ln(K_{t-1}) ) + (1-\alpha) (ln(N_t)-ln(N_{t-1})) \end{array}$

$\begin{array}{ll} ln(A_t)-ln(A_{t-1})= & \mbox{growth in TFP}=\\ \\ ln(Y_t)... ...pha) (ln(N_t)-ln(N_{t-1})) & - (1-\alpha) \mbox{(growth in labor)} \end{array}$