Chapter 7: Economic Growth: Malthus and Solow

Problem with Malthusian Model

did no predict the Industrial Revolution. Economic group was in part driven by growth in the stock of capital over time and was no limited by fixed factors of production (such as land)

how to construct consumption per worker in the steady state, c∗, as a function of capital per worker in the steady state, k∗

There is a quantity of capital per worker for which consumption per worker is max- imized, which we denote by k∗gr in the figure. If the steady state quantity of capital is k∗gr, then maximum consumption per worker is c∗. Here, k∗gr is called the golden rule quantity of capital per worker.

Key Property of Malthusian Model

improvements in the production technology of increases in the quantity of land have no effect on the long-run standard of living.

What kind of model is the Solow Model?

exogenous growth model: growth is caused in the model by forces that are not explained by the model itself.

Per-Worker Production Function

f(k) which is defined by f(k)=F(k,1) Slope is the marginal product of capital, MPk. This is because adding one unit to k, the quantity of capital per worker increases y, output per worker, by the marginal product of capital because f(k)=F(k,1) As the slope is MPk and because MPk is diminishing with K, the production function is concave (its slope decreases as k increases)

Problems with Malthusian Model

1. he did not allow for the effect of increases in the capital stock on production 2. he did not account for all of the effects of economic forces on pop- ulation growth

Malthusian Model Property

an improvement in the technology for producing goods leads to increased population growth, so that in the long run there is no improvement in the standard of living. There is no increase in per capita consumption and per capita output. The only means for improving the standard of living is population control

The economy is in a steady state before time T

there is an increase in total factor productivity. Initially, the effect of this is to increase output, consumption, and consumption per worker, as there is no effect on the current population at time T. However, because consumption per worker has increased, there is an increase in population growth.

Consumers: N' = (1 + n)N C = (1 - s)Y

- In each period, a consumer has one unit of time available and we assume consumers do not value leisure, so they supply their one unit of time as labor in each period. - population is identical to labor force because we assume that all members of the population work. - N is the number of workers or labor force - n is the growth rate rate in labor force - Y output as income to consumers - no government sector or taxes - C is current consumption - s is aggregate savings rate

The Steady State Effects of an Increase in Labor Force Growth

- With aggregate output growing at a higher rate, there is a larger and larger "income pie" to split up, but with more and more workers to share this pie. As we show, the Solow growth model predicts that capital per worker and output per worker will decrease in the steady state when the labor force growth rate increases, but aggregate output will grow at a higher rate, which is the new rate of labor force growth. - the steady state effects of an increase in the labor force growth rate, from n1 to n2. Initially, the quantity of capital per worker is k∗1, determined by the intersection of the curves szf(k∗) and (n1 + d)k∗. When the population growth rate increases, this results in a decrease in the quantity of capital per worker from k∗1 to k∗2. Because capital per worker falls, output per worker also falls, from the per-worker production function. That is, output per worker falls from zf(k∗1) to zf(k∗2). The reason for this result is that when the labor force grows at a higher rate, the current labor force faces a tougher task in building capital for next period's consumers, who are a proportionately larger group. Thus, output per worker and capital per worker are ultimately lower in the steady state.

The Steady State Effects of an Increase in Total Factor Productivity

- an increase in the savings rate or a decrease in the labor force growth rate can increase the standard of living in the long run. However, increases in the savings rate and reductions in the labor force growth rate cannot bring about an ever-increasing standard of living in a country. This is because the savings rate must always be below 1 - The Solow model predicts that a country's standard of living can con- tinue to increase in the long run only if there are continuing increases in total factor productivity - The source of continual long-run betterment in a country's standard of living, therefore, can only be the process of devising better methods for putting factor inputs together to produce output, thus generating increases in total factor productivity.

Steady State Effects of an Increase in the Savings Rate

- change is s can occur due to a change in the preferences of the consumers ex. if the consumers care more about the future they save more and s increases - can occur due to a government policy ex. if the government subsidizes savings Result: The increase in s shifts the curve szf(k∗) up, and k∗ increases from k∗1 to k∗2. Therefore, in the new steady state, the quan- tity of capital per worker is higher, which implies that output per worker is also higher, given the per-worker production function y = zf(k∗). Though the levels of capital per worker and output per worker are higher in the new steady state, the increase in the savings rate has no effect on the growth rates of aggregate variables. Before and after the increase in the savings rate, the aggregate capital stock K, aggregate output Y, aggre- gate investment I, and aggregate consumption C grow at the rate of growth in the labor force, n.

k' = szf(k)/(1+n) + (1 - d)k/(1+n) Competitive Equilibrium

- decreasing slope because of the decreasing slope of the per-worker production function f(k) - the 45 line is the line along which k'=k. - once the economy is in the steady state where k=k*, then future capital per worker k'=k* and the capital stock per worker increases from the current period to the future period. - current investment is large relative to depreciation and labor force growth that the per-worker quantity of capital increases but if k>k* then k'<k and the capital stock per worker decreases from the current period to the future period. Investment is small and cannot keep up with depreciation and labor force growth and the quantity of capital delicious, therefor the quantity of capital declines from the current period to the next. If the quantity of capital per worker is smaller than its steady state value, it increases until the quantity of capital per worker is smaller than its steady state and if the quantity of capital per worker is larger than its steady state value it decreases until it reaches the steady state.

How can the golden rule be achieved in the steady state?

- if the savings rate is sgr, then the curve sgrzf(k∗) intersects the line (n + d)k∗, where k∗ = k∗gr. Thus, sgr is the golden rule savings rate. If savings takes place at the golden rule savings rate, then in the steady state the current population consumes and saves the appropriate amount so that, in each succeeding period, the population can continue to consume this maximum amount per person - if the steady state capital stock per worker is less than k∗gr, then an increase in the savings rate s increases the steady state capital stock per worker and increases consumption per worker. However, if k∗ 7 k∗gr, then an increase in the savings rate increases k∗ and causes a decrease in consumption per worker.

Solow Model

- sustained increases in the standard of living can occur in the model, but sustained technological advances are necessary. We suppose that the population grows exogenously (there is a growing population of customers, with N denoting the population in the current period. - Because the Solow growth model predicts that the quantity of capital per worker converges to a constant, k∗, in the long run, it also predicts that the quantity of output per worker converges to a constant, which is y∗ = zf(k∗) from the per-worker produc- tion function.

Economic Growth Facts

1. Before the industrial revolution, standards of living differed little over time and across countries 2. Since the industrial revolution, per capita income growth has been sustained in the richest countries. In the US, average annual growth in per capita income has been about 2%. 3. There is a positive correlation between the rate of investment and output per workers across countries. 4. There is a negative correlation between the population growth rate and output per worker 5. Differences in per capita incomes increased dramatically between 1800 and 1950 6. There is no correlation across countries between the level of output per capital and the average rate of growth in output per capita 7. Richer countries are much more alike in terms of rates of growth in real per capita income than are poor countries

Solow Growth Model Property

A country's standard of living cannot continue to improve in the long run in the absence of continuing increases in total factor productivity. In the short run, the standard of living can improve if a country's residents save and invest more, thus accumulating more capital.

Growth Accounting

Anapproachthatusestheproductionfunction and measurements of aggregate inputs and outputs to attribute economic growth to: (i) growth in factor inputs; (ii) total factor productivity growth.

Effect of an increase in the Savings Rate at Time T

Before time T, aggregate output is growing at the constant rate n (recall that if the growth rate is constant, then the time path of the natu ral logarithm is a straight line), and then the savings rate increases at time T. Aggregate output then adjusts to its higher growth path after period T, but in the transition to the new growth path, the rate of growth in Y is higher than n. The temporarily high growth rate in transition results from a higher rate of capital accumulation when the savings rate increases, which translates into a higher growth rate in aggregate output. As capital is accumulated at a higher rate, however, the marginal product of capital diminishes, and growth slows down, ultimately converging to the steady state growth rate n.

Why does population converge to a steady state?

Ex. the population is currently below steady state, then there will be a large quantity of consumption per working and the population growth rate is large and positive, thus the population will increase Ex. the population is currently above steady state, then there will be a small quantity of consumption per working and the population growth rate will be low and negative, thus the population will decrease.

Labor Supply

N'= N + Births - Deaths Where the birth rate is the ration of births to population, and death rate is the ratio of deaths to population. Hints: assume that each person in the economy is willing to work at any wage and has one unit of labor to supply.

Solow Model vs. Malthusian Model

Solow model implies more optimistic prospects for long-run improvement in the standard of living than the malthusian model, but only to a certain point.

What does the Solow Model Tell us?

That building more productive capacity will not improve long run living standards unless the production technology becomes more efficient. Improvement s in knowledge and technical ability are necessary to sustain growth.

Why does this happen?

The reason is that the marginal product of capital is diminishing. Output per worker can grow only as long as capital per worker continues to grow. However, the marginal return to investment, which is determined by the marginal product of capital, declines as the per-worker capital stock grows. In other words, as the capital stock per worker grows, it takes more and more investment per worker in the current period to produce one unit of additional capital per worker for the future period. Therefore, as the economy grows, new investment ultimately only just keeps up with depreciation and the growth of the labor force, and growth in per-worker output ceases.

Competitive Equilibrium

Two markets in current period 1. current consumption goods are traded for current labor 2. current consumption goods are trades for capital The labor market and capital market must clear in each period. In the labor market, the quantity of labor is always determined by the inelastic supply of labor, N. Because the supply for labor is N no matter what the real wages, the real wage adjusts in the current period so that the representative firm wishes to hire N worker, letting S denote the aggregate quantity of savings in the current period. In equilibrium S=I however, because S=Y-C

The Representative Firm

Y = zF(K, N): production function - Y is current output - z is current total factor productivity Y/N=zF(K/N,1) - Y/N is output per worker - K/N is capital per worker - this equation tells us that if the production function has constant returns to scale, then output per worker depends only on the quantity of capital per worker. y = zf(k) - y is output per worker - k is capital per worker - f(k) is the per-worker production function

Malthusian Theory Equation

Y = zF(L, N) This equation shows how current aggregate output, Y, is producing using current inputs of land, L, and current labor, N, where z is total factor productivity and F is a function having these same properties. Hints: - think of Y as perishable food that perishable from period to period. - in this economy, there is no investment, therefore no saving and no government spending - L is in a fixed supply.

Per-worker production function

describes the quantity of output per worker y that can be produced for each quantity of land per worker l, with the function f defined by f(l) = F(l,1).

Malthus's Argument

any advances in the technology for producing food would lead to further population growth, with the higher population reducing the average person to the subsistence level of consumption they had before the advance in technology. Population and level of aggregate consumption could grow over time, but in the long run there would be no increase in the standard of living.

tyler banks

best boyfriend

As the population grows after period T

consumption per worker falls (given the fixed quantity of land), until consumption per worker converges to c∗, its initial level, and the population converges to its new higher level N2∗.

pessimistic Malthusian result

improvements in the technol- ogy for producing food do not improve the standard of living in the long run. A better technology generates better nutrition and more population growth, and the extra pop- ulation ultimately consumes all of the extra food produced, so that each person is no better off than before the technological improvement.

What does the Solow Model tell us?

tells us that if the savings rate s, the labor force growth rate n, and total factor productivity z are constant, then real income per worker cannot grow in the long run. Thus, since real income per worker is also real income per capita in the model, we can take y as a measure of the standard of living. The model then concludes that there can be no long-run betterment in living standards under these circumstances.

Consumption per Worker and Golden Rule Capital Accumulation c∗ =zf(k∗)-(n+d)k∗.

that there is a given quantity of capital per worker that maximizes consumption per worker in the steady state. This implies that an increase in the savings rate could cause a decrease in steady state consumption per worker, even though an increase in the savings rate always increases output per worker.

Golden Rule Property

the slope of the per-worker production function where k∗ = k∗gr is equal to the slope of the function (n + d)k∗. That is, because the slope of the per-worker production function is the marginal product of capital, MPK, at the golden rule steady state we have MPk=n+d when capital is accumulated at a rate that maximizes consumption per worker in the steady state, the marginal product of capital equals the population growth rate plus the depreciation rate.

An increase in z on the Steady State

this can be interpreted as an improvement in agricultural techniques. That is, suppose that the economy is initially in a steady state, with a given level of total factor productivity z1, which then increases permanently to z2

What would happen with population control?

this would have the effect of reducing the rate of population growth for each level of consumption per worker. It causes increases output per worker and consumption per worker, and everyone is better off in the long run

What happens in the long run?

when the economy converges to the steady state quantity of capital per worker, k∗, all real aggregate quantities grow at the rate n, which is the growth rate in the labor force. The aggregate quantity of capital in the steady state is K = k∗N, and because k∗ is a constant and N grows at the rate n, K must also grow at the rate n. Similarly, aggregate real output is Y = y∗N = zf(k∗)N, and so Y also grows at the rate n. Further, the quantity of investment is equal to savings, so that investment in the steady state is I = sY = szf(k∗)N, and because szf(k∗) is a constant, I also grows at the rate n in the steady state. As well, aggregate consumption is C = (1 - s)zf(k∗)N, so that consumption also grows at the rate n in the steady state. In the long run, therefore, if the savings rate, the labor force growth rate, and total factor productivity are constant, then growth rates in aggregate quantities are determined by the growth rate in the labor force.