Saving, Capital Accumulation,and Output

Interactions Between Output and Capital

Two important relations in the long run are: The amount of capital determines the

amount of output being produced. The amount of output determines the

amount of saving (=investment), and so the amount of capital being accumulated.

Interactions Between Output and Capital

The Effects of Capital on Output

Under constant returns to scale, we can write the relation between output and capital per worker as follows:

fK

NF

K

N

,1

The Effects of Capital on Output

For now, we make the following assumptions:1. The size of the population, the participation rate, and the

unemployment rate are all constant.

2. There is no technological progress.

Under these assumptions, the first important relation we want to express is between output and capital per worker: Y

Nf

K

Nt t

Which means that, at each period, higher capital per worker leads to higher output per worker.

Effects of Output on Capital Accumulation

Output and Investment: The equations below describe the relation between

private saving and investment:

I sYt t Private saving is equal to investment, and proportional to

income.

S sY 0 1 s

Effects of Output on Capital Accumulation

Investment and Capital Accumulation: The evolution of the capital stock is given by:

11 δ

t t tK ( )K I

+= - +

Effects of Output on Capital Accumulation

Rearranging terms in the equations above, we can articulate the change in capital per worker over time:

In words, the change in the capital stock per worker (left side) is equal to saving per worker minus depreciation (right side).

1 δt t t tK K Y K

sN N N N

+ - = -

Implications of Alternative Saving Rates

Our two main relations are:

First relation:Capital determines

output.

Y

Nf

K

Nt t

K

N

K

Ns

Y

N

K

Nt t t t 1

Second relation:Output determines capital

accumulation

Dynamics of Capital and Output

If investment per worker exceeds depreciation per worker, the change in capital per worker is positive: Capital per worker increases.

If investment per worker is less than depreciation per worker, the change in capital per worker is negative: Capital per worker decreases.

K

N

K

N

K

Nt t t

1 = s f

K

N t

change in capital from change in capital from year year tt to year to year tt+1+1

investment investment during year during year tt

depreciation depreciation during year during year tt

Dynamics of Capital and Output

When capital and output are low, investment exceeds depreciation, and capital increases. When capital and output are high, investment is less than depreciation and capital decreases.

Dynamics of Capital and Output

Dynamics of Capital and Output

Steady-State Capital and Output

The state in which output per worker and capital per worker are no longer changing is called the steady state of the economy. In steady state, the left side of the equation above equals zero.

Y

Nf

K

N

* *

Three observations about the effects of the saving rate:

1. The saving rate has no effect on the long run growth rate of output per worker

which is equal to zero.

2. The saving rate determines the level of output per worker in the long run.

Other things equal, countries with a higher saving rate will achieve higher output per worker.

3. An increase in the saving rate will lead to higher growth of output per worker for some time. (Growth will end once the economy reaches its new – higher - steady state.)

The Saving Rate and Output

A country with a higher saving rate achieves a higher level of output per worker in steady state.

The Saving Rate and Consumption

The Saving Rate and Consumption

The level of capital associated with the value of the saving rate that yields the highest level of consumption in steady state is known as the golden-rule level of capital.

The Saving Rate and Consumption

An increase in the saving rate leads to an increase, then to a decrease, in consumption per worker in steady state.

The Dynamic Effects of an Increase in the Saving Rate

It takes a long time for output to adjust to its new higher level after an increase in the saving rate. Put another way, an increase in the saving rate leads to a long period of higher growth.

Physical Versus Human Capital

The set of skills of workers is called human capital.

An economy with many highly skilled workers is likely to be much more productive than an economy in which most workers cannot read or write.

The conclusions drawn about physical capital accumulation remain valid after the introduction of human capital in the analysis.

Extending the Production Function

When the level of output per workers depends on both the level of physical capital per worker, K/N, and the level of human capital per worker, H/N, the production function may be written as:

Y

Nf

K

N

H

N

,

( , )

Social Capital

The set of institutions, governing laws, norms and cultural habits that might affect the growth potential of an economy. Private property Underground economy Cultural norms

Human Capital, Physical Capital, and Output

An increase in how much society “invests” in human capital—through education and on-the-job-training—increases steady-state human capital per worker, which leads to an increase in output per worker.

In the long run, output per worker depends not only on how much society saves/invests but also on how much it spends on education.

Steady State Growth

In the model we studied until now (with or without human capital), there is no growth in the steady state.

The same model with technological progress will yield growth in the steady state (lecture 16).

Models without exogenous technological progress that have steady state growth are called endogenous growth models (lecture 17).