Chapter 7-Solow Growth Model

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Solow Growth Model

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shows how saving, population growth, and technological progress affect the level of an economy's output and its growth over time.

designed to show how growth in the capital stock, growth in the labor force, and advances in technology interact in an economy and how the affect a nations total output of goods and services

Assumes that the production function has constant returns to scale

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y = f(k)

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pg. 189

the amount of capital per worker (little k) determines the amount output per worker (little y)

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Marginal Product of Capital (MPK)

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MPK = f(k+1) - f(k)

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Production functions with constant returns to scale allow us to analyze all quantitities in the economy relative to the size of the labor force

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true


Y/L = F(K/L)

Output per worker is a function of the amount of capital per worker

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Review Pg. 189

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Now!

The slope of the production function IS the marginal product of capital

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The demand for goods in the solow model comes from consumption and investment. Or output per worker (little y) is made up of consumption per worker (little c) and investment per worker (little i)

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y = c + i

This equation is the per worker version of the national income accounts identity for the economy.

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Solow model assumes that each year people save a fraction (s) of their income and consume a fraction (1-s)

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c = (1-s)y

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To see what this consumption function implies for investment substitute (1-s)y for c in the national income accounts identity:

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y = (1-s)y + i

or

i = sy


This equation shows that investment equals saving.

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2 main ingredients of the Solow Model

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Production function:
y = f(k)

Consumption Function:
c = (1-s)y

y = (1-s)y + i

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two forces influence the capital stock

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investment

Depreciation

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by substituting the production function for y, we can express intestment per worker as a function fo the capital stock per worker

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i = s y

or

i = s f(k)

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For any level of capital k, ouput is f(k), investment is sf(k) and consumption = f(k) - sf(k)

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see figure 7-2

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Change in capital stock =

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change in k = i - dk

d = little delta

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Steady state level of capital

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represents the long-run equilibrium of the economy

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y = Y/L

k = K/L

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true

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to calculate the long run steady state rate

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k*/f(k*) = s/d

d = little delta

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The Solow model shows that the saving rate is a key determinant of the steady state capital stock.

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If the saving rate is high, the economy will have a large capital stock and a high level of output in the steady state.

If the saving rate is low, the economy will have a small capital stock and a low level of output in the steady state.

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The association between investment rates and income per pserson is strong

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true

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Steady State

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A condition in which key variables are NOT changing

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Golden Rule Level of Capital

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k*,gold

The steady state value of k that maximizes consumption. The optimal amount of capital accumulation from the standpoint of economic well being

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Steady state consumption per worker

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c* = f(k*) - dk*

According to this equation, steady state consumption is what's left of steady state output after paying for steady state depreciation.

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steady state sconsumption is the gap between output and depreciation

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see Figure 7-7

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Study Figure 7-7,5,4,3,2

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NOW!

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Steady State Consumption is the difference between steady state output and steady state depreciation.

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True

Steady state consumption is MAXIMIZED at the Golden Rule steady state.

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The Golden Rule is described by:

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Marginal Product of Capital (MPK) = delta

MPK = d

Recall that the slope of the production function is the marginal product of capital MPK. The slope of the steady state depreciation (and investment) line (dk*) line IS (d). Bec these two slopes are equal at k*,gold The golden rule is:

MPK = d

At the Golden rule level of capital, the marginal product of capital equals the depreciation rate.

also

MPK - d = 0

At the Golden Rule level of capital, the marginal product of capital net of depreciation is 0.

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Review Bottom half of pg. 200

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yes!

If MPK - d > 0 then increases in capital increases consumption so k* must be BELOW the Golden Rule level.

If MPK - d < 0 then increases in capital decrease consumption, so k* must be ABOVE the Golden Rule level

thus:

Golden Rule Level is (MPK - d = 0)

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Starting with To much Capital

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in this case, the policy maker should pursue policies aimed at reducing the rate of saving in order to reduce the capital stock.

Fig. 7-9:

The reduction in the saving rate causes an immediate increase in consumption and a decrease in investment. bc investment and depreciation were equal in the initial steady state, investment will now be less than depreciation.

Bec we are assuming that the new steady state is the Golden Rule steady state, consumption must be HIGHER than it was before the change in the savings rate.

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Starting with to Little Capital

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the policy maker must RAISE the saving rate to reach the Golden rule.

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When the economy begins ABOVE the Golden Rule, reachig the Golden Rule produces higher consumption at ALL points in time.


When the economy begins BELOW the Golden Rule, reaching the Golden Rule requires initially reducing consumption to increase consumption in the future

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true

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Review & Compare fig 7-9 to 7-10

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yes

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Review Fig. 7-11

change in capital per worker=

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sf(k) - (d+n)k


(d+n)k = break even investment, or the amount of investment necessary to keep constant the capital stock per worker (k)

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fig 7-11 & review top part of pg. 208

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if k is LESS THAN k*, investment is greater than break-even investment, so (k) rises.


If (k) is greater than k*, investmen is LESS THAN break even investment so (k) falls.

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Once the Economy is in the steady state, investmen has 2 purposes

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1. some of it replaces the depreciated capital (dk*)

2.the rest provides the new workers with the steady state amount of capital (nk*)

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Population growth alters teh basic Solow model in 3 ways

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it brings us closer to explaining sustained economic growth


population growth gives us another explanation for why some countries are rich and others are poor.


population growth affects our criterion for determining the Golden Rule (consumption maximizing) level of capital

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MPK - d = n

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in the Godlen Rule steady state, the MPK net of depreciation equals the rate of population growth

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Key Points

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The Solow grwoth model shows that in the long run, an economy's rate of saving determines the size of its capital stock and thus its level of production. The HIGHER the rate of saving, the HIGHER the stock of capital and teh higher the levle of output.



In the Solow model, an increase in the rate of saving has a level effect on income per person: it causes a period of rapid growth, but eventually that growth slows as the new steady state is reached. Thus, although a high saving rate yields a high steayd state level of output, saving by itself cannot generate PERSISTENT economic growth.


Teh level of capital that maximizes steady-state consumption is called the Golden Rule level. If an economy has MORE capital than in the Golden Rule steady state, then reducing saving will increase consumption at ALL points in time. By contrast, if the economy has less captial than in the Godlen Rule steady state, then reaching the Godlen Rule requries increased investment an thuse lower consumption for current generati