Solow’s ModelSolow’s Model

(Modeling economic growth)(Modeling economic growth)

Solow model I: Constant Solow model I: Constant productivityproductivity

Assumptions of the modelAssumptions of the model

Population grows at rate Population grows at rate nn

LL’’ = = ((1 + n1 + n))LL

Population equals labor forcePopulation equals labor force

No productivity growthNo productivity growth

Capital depreciates at rate Capital depreciates at rate

1. Per-capital Income1. Per-capital Income

Production function: Production function: Y = FY = F((K, LK, L) )

In “per worker” terms: In “per worker” terms: y = fy = f((kk))

Relationship between variables:Relationship between variables:

L

Kk

L

Yy

From the above we can get:From the above we can get:

Per-person or per-capita income Per-person or per-capita income level (y) depends on each worker’s level (y) depends on each worker’s capital equipment(k).capital equipment(k).

y=f(k) shows DMR.y=f(k) shows DMR.

Can you draw the graph with y and k?Can you draw the graph with y and k?

– Growth rate is measured by the slope of the Growth rate is measured by the slope of the tangent line of the y or f(k) curve.tangent line of the y or f(k) curve.

– Growth rate decreases as the per-capita capital Growth rate decreases as the per-capita capital stock rises. It is true for all countries- stock rises. It is true for all countries- “Convergence”“Convergence”

– Countries that start further away from the Countries that start further away from the steady state grow fastersteady state grow faster

2. Actual Supply of Capital2. Actual Supply of Capital

Assume Assume FIXED SAVINGS RATE or APSFIXED SAVINGS RATE or APS: s : s =S/N/Y/N = savings /income=S/N/Y/N = savings /income

Given an income of Given an income of yy– Actual savingsActual savings= s · y = s f(k)= s · y = s f(k)

EXAMPLEEXAMPLE

Savings rate of 40%Savings rate of 40%– s = s = .4 (you save a fraction of your .4 (you save a fraction of your

income)income)

Can you draw the actual savings Can you draw the actual savings curve in the previous graph you curve in the previous graph you have drawn?have drawn?

Minimum Capital Requirement to just keep up Minimum Capital Requirement to just keep up for each work is proportional to population for each work is proportional to population

growth rate(n) and capital depreciation rate(growth rate(n) and capital depreciation rate(

knkMinimum )(

*if you do not replenish the economy with the minimum requirement of capital, then the level of capital and thus the level of production or income fall.

3. Required Capital for Just Keep-Up

Example)Example)

Y = 100; L = 20; K = 10Y = 100; L = 20; K = 10

y = Y/L = 5y = Y/L = 5

k = K/L = 10/20 = 0.5k = K/L = 10/20 = 0.5

n = 3% ; n = 3% ; Then you need 8% of capital every year to Then you need 8% of capital every year to

keep constant each worker’s capital keep constant each worker’s capital equipment.equipment.

4. Equilibrium or Not4. Equilibrium or Not

The Change in capital per worker is The Change in capital per worker is the actual supply of capital over the the actual supply of capital over the minimum required capitalminimum required capital

knkfsk

knysk

)()(

)(

We may call this net investment.

Thus:Thus:– If If k k > 0: economy accumulates capital > 0: economy accumulates capital

per workerper worker– If If k k < 0: economy reduces capital per < 0: economy reduces capital per

workerworker– If If k k = 0: constant capital per worker: = 0: constant capital per worker:

steady statesteady state

GraphicallyGraphically

k

(+n)k

s f(k)

k*k0

k > 0

k < 0

f(k)

Steady-state Per-capita Income or y* Steady-state Per-capita Income or y* = Y/N is determined where = Y/N is determined where s f(k*) =s f(k*) = ((++nn))k*.k*.

Implications of the modelImplications of the model

The economy converges, over time, The economy converges, over time, to its steady state.to its steady state.– If the economy starts BELOW the steady If the economy starts BELOW the steady

state, it accumulates capital until it state, it accumulates capital until it reaches the steady state.reaches the steady state.

– If the economy starts ABOVE the steady If the economy starts ABOVE the steady state, it reduces capital until it reaches state, it reduces capital until it reaches the steady state.the steady state.

Growth ratesGrowth rates– Capital per worker grows at rate 0Capital per worker grows at rate 0– Output per worker grows at rate 0Output per worker grows at rate 0

– Total capital: Total capital: K = k · L K = k · L grows at rate grows at rate nn

– Total output: Total output: Y = y · L Y = y · L grows at rate grows at rate nn

Comparative statisticsComparative statisticsParameters of the model: Parameters of the model: s, n, s, n, Predictions of the model:Predictions of the model:

In steady state:In steady state:– Higher savings rateHigher savings rate implies higher income per implies higher income per

workerworker– Higher population growthHigher population growth implies lower income per implies lower income per

worker worker

Savings rate and growthSavings rate and growth

k

(+n)k

s1f(k)

kss

s2f(k)

kss2

Note that an increase in savings rate Note that an increase in savings rate do increase the level of income, but do increase the level of income, but not the rate of growth of income. not the rate of growth of income.

Population growth rate and Population growth rate and growthgrowth

k

(+n1)k

sf(k)

kss

(+n2)k

kss2

Technical InnovationsTechnical InnovationsHow is this different for the y curve from an How is this different for the y curve from an increase in savings rate?increase in savings rate?