# 0.10 Integration of role of k: production function approach

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At base economic growth is all about sustained growth in productivity of labor and capital.

Productivity gains arise from:

Increase in savings and investment in new physical capital

Increases in savings and investments in new human capital

Discovery of new techniques and technologies that enhance productivity

Institutional changes that foster growth productivity

Later in this book we discuss how societies can obtain more savings to finance such investments.

Much of our discussion in the past few chapters can be brought together by a simple aggregate production function.

## A production function approach to growth

Production function characterizes how inputs (such as capital and labor) are combined to yield various output levels (see Text p. 66, 90-91 and 109-110).

Production functions ordinarily feature diminishing returns to either capital or labor (i.e. a diminishing marginal product of capital or labor).

But some production functions feature constant returns to scale. Some production functions feature fixed coefficients – i.e. that factors such as capital and labor must be utilized in fixed proportions to each other (see Text pp. 108-109).

## A) the cobb-douglas two input production function

The most basic production function is written: Y = L b K c

Where

b = power associated with input L

c = power associated with input K

L = Quantity of Labor inputs

K = Quantity of Physical Capital inputs

This framework first appeared in an article published in 1939 by James Cobb and Paul Douglas (see ____ _____).

A special, useful form of the basic COBB-Douglas function (Y = L b K c ) Where (b + c) = 1: A Linear homogenous production function

When this obtains, we get constant returns to scale (doubling both inputs will give us an output (Y) that is exactly double the initial output). This is called a linear homogenous production function.

In this formulation, the growth rate of an economy is nothing more than the sum of: Productivity growth of capital plus growth in the labor force.

## B) other production function approaches to growth

B) Other Production Function Approaches to Growth

(1) Rudimentary Function: By the mid 1900’s, economists and others thought that output (income) in an economy was best understood as an interaction in which existing technology transformed 3 inputs into output: Land, Labor, and Capital). You could write that relationship as:

1. Y = f (L, P, K) + Z
Y = output,
f = the production function, or technology, that converts inputs into outputs
P = Land
K = Physical Capital (Human Capital not included)
Z = unexplained residuals, or “white noise” from missing factors not included in the analysis.
L = Labor

In this economy, the rate of growth of output would be

dy

d t t = time

(given by the total derivative)

dF (o)

d t of the protection function

In this formulation, Output Changes (∆Y) are entirely due to change in the quantity of factor inputs L,W and K; Changes in Technology play no role at all.

In particular, this model tell us that the more land or workers or Capital a nation has, then the greater will be it growth rate and economic power. This theory attempts to explain what would be expected when a nation puts a larger percentage of its population to work, then its economy will become larger. When the number of “people” increase, as for example there is an expected increase in the female labor force, then the model would yield jump in output. The same holds for land, such one-time jumps can be a very important source of increased economic might. This is one reason why Germany and Japan sought conquest of other nations, it (Hitler stressed “Lebensraum” - more room for life, for Germany).

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