The Solow Growth Model (Part One)

The steady state level of capital and how savings affects output and

economic growth.

Model Background

• Previous models such as the closed economy and small open economy models provide a static view of the economy at a given point in time. The Solow growth model allows us a dynamic view of how savings affects the economy over time.

Building the Model: goods market supply

• We begin with a production function and assume constant returns.

Y=F(K,L) so… zY=F(zK,zL)

• By setting z=1/L we create a per worker function. Y/L=F(K/L,1)

• So, output per worker is a function of capital per worker. We write this as,

y=f(k)

Building the Model: goods market supply

• The slope of this function is the marginal product of capital per worker.MPK = f(k+1)–f(k)

k

y

Change in y

Change in k

y=f(k)kinchange

yinchangeMPK

• It tells us the change in output per worker that results when we increase the capital per worker by one.

Building the Model:goods market demand

• We begin with per worker consumption and investment. (Government purchases and net exports are not included in the Solow model). This gives us the following per worker national income accounting identity.

y = c+I

• Given a savings rate (s) and a consumption rate (1–s) we can generate a consumption function.

c = (1–s)y …which makes our identity,y = (1–s)y + I …rearranging,i = s*y …so investment per worker

equals savings per worker.

Steady State Equilibrium

• The Solow model long run equilibrium occurs at the point where both (y) and (k) are constant. These are the endogenous variables in the model.

• The exogenous variable is (s).

Steady State Equilibrium

• By substituting f(k) for (y), the investment per worker function (i = s*y) becomes a function of capital per worker (i = s*f(k)).

• To augment the model we define a depreciation rate (δ).

• To see the impact of investment and depreciation on capital we develop the following (change in capital) formula,Δk = i – δk …substituting for (i) gives us,Δk = s*f(k) – δk

Steady State Equilibrium

• At the point where both (k) and (y) are constant it must be the case that, Δk = s*f(k) – δk = 0 …or, s*f(k) = δk…this occurs at our equilibrium point k*.

khighklow

• If our initial allocation of (k) were too high, (k) would decrease because depreciation exceeds investment.

• At k* depreciation equals investment.

k

s*f(k),δk

k*

s*f(k*)=δk* s*f(k)

δk• If our initial allocation were too low, k would increase because investment exceeds depreciation.

Steady State Equilibrium (getting there)

• Suppose our initial allocation of (k1) were too low.

k1

k

s*f(k),δk

k*

s*f(k*)=δk* s*f(k)

δkk2=k1+Δk

k3=k2+Δk

k4=k3+Δk

k2 k3 k4

k5=k4+Δk

k5

…

This process continues until we converge to k*

K2 is still too low so…

K3 is still too low so…K4 is still too

low so…K5 is still too

low so…

A Numerical Example

• Starting with the Cobb-Douglas production function we can arrive at our per worker production as follows,

Y=K1/2L1/2 …dividing by L,Y/L=(K/L)1/2 …or,y=k1/2

• recall that (k) changes until, Δk=s*f(k)–δk=0 ...i.e. until, s*f(k)=δk

Changing the exogenous variable - savings

• We know that steady state is at the point where s*f(k)=δk

k**k

s*f(k),δk

k*

s*f(k*)=δk*

δk

s*f(k)

s*f(k)s*f(k*)=δk*

• What happens if we increase savings?

• This would increase the slope of our investment function and cause the function to shift up.

• This would lead to a higher steady state level of capital.

• Similarly a lower savings rate leads to a lower steady state level of capital.

Conclusion

• The Solow Growth model is a dynamic model that allows us to see how our endogenous variables capital per worker and output per worker are affected by the exogenous variable savings. We also see how parameters such as depreciation enter the model, and finally the effects that initial capital allocations have on the time paths toward equilibrium.

• In the next section we augment this model to include changes in other exogenous variables; population and technological growth.

The Solow Growth Model (Part Two)

The golden rule level of capital, maximizing consumption per

worker.

Model Background

• As mentioned in part I, the Solow growth model allows us a dynamic view of how savings affects the economy over time. We also learned about the steady state level of capital.

• Now, we assume policy makers can set the savings rate to determine a steady state level of capital that maximizes consumption per worker. This is known as the golden rule level of capital (k*gold)

Building the Model:• We begin by finding the steady state

consumption per worker.From the national income accounts identity, y = c + iwe get c = y – i

• We want steady state “c” so we substitute steady state values for both output (f(k*)) and investment which equals depreciation in steady state (δk*) giving us c*=f(k*) – δk*

k*gold

k*

f(k*),δk* δk*

f(k*)

c*gold

Below k*gold, increasing k* increases c*

Above k*gold, increasing k* reduces

c*

• Because, consumption per worker is the difference between output and investment per worker we want to choose k* so that this distance is maximized.

• This is the golden rule level of capital k*gold

• A condition that characterizes the golden rule level of capital is

MPK = δ

Building the Model:

• While the economy moves toward a steady state it is not necessarily the golden rule steady state.

• Any increase or decrease in savings would shift the sf(k) curve and would result in a steady state with a lower level of consumption.

k*gold

k*

f(k*),δk* δk*

f(k*)

To reach the golden rule

steady state…

The economy needs the right savings rate.

sgoldf(k*)

sgoldf(k*)

The Transition to the Golden Rule Steady State

• Suppose an economy starts with more capital than in the golden rule steady state.

Output, y

Consumption, cInvestment, i

t0

At t0, the savings rate is reduced.

Time

• This causes an immediate increase in consumption and an equal decrease in investment.

• Over time, as the capital stock falls, output, consumption, and investment fall.

• The new steady state has a higher level of consumption than the initial steady state.

The Transition to the Golden Rule Steady State

• Suppose an economy starts with less capital than in the golden rule steady state.

Output, y

Consumption, c

Investment, i

t0

At t0, the savings rate is increased.

Time

• This causes an immediate decrease in consumption and an equal increase in investment.

• Over time, as the capital stock grows, output, consumption, and investment increase.

• The new steady state has a higher level of consumption than the initial steady state.

Conclusion

• In this section we used our knowledge that savings affects the steady state and chose the savings rate to maximize consumption per worker. This is known as the golden rule level of capital (k*gold)

• In the next section we augment this model to include changes in other exogenous variables; population and technological growth.

The Solow Growth Model (Part Three)

The augmented model that includes population growth and technological

progress.

Model Background

• As mentioned in parts I and II, the Solow growth model allows us a dynamic view of how savings affects the economy over time. We learned about the steady state level of capital and how a golden rule steady state level of capital can be achieved by setting the savings rate to maximize consumption per worker. We now augment the model to see the effects of population growth and technological progress.

Steady State Equilibrium

• By expanding our model to include population growth our model more closely resembles the sustained economic growth observable in much of the real world.

• To see how population growth affects the steady state we need to know how it affects the accumulation of capital per worker. When we add population growth (n) to our model the change in capital stock per worker becomes…Δk = i – (δ+n)k

• As we can see population growth will have a negative effect on capital stock accumulation. We can think of (δ+n)k as break-even investment or the amount of investment necessary to keep capital stock per worker constant.

• Our analysis proceeds as in the previous presentations. To see the impact of investment, depreciation, and population growth on capital we use the (change in capital) formula from above,Δk = i – (δ+n)k …substituting for (i) gives us,Δk = s*f(k) – (δ+n)k

Steady State Equilibrium with population growth

• At the point where both (k) and (y) are constant it must be the case that,Δk = s*f(k) – (δ+n)k = 0

…or,s*f(k) = (δ+n)k…this occurs at our equilibrium point k*.

k*k

InvestmentBreak-even Investment

s*f(k*)=(δ+n)k* s*f(k)Investment

Break-even investment (δ+n)k

At k* break-even investment equals

investment.

Like depreciation, population growth is one reason why the

capital stock per worker shrinks.

The impact of population growth

• Suppose population growth changes from n1 to n2.

• This shifts the line representing population growth and depreciation upward.

k2*k

InvestmentBreak-even Investment

s*f(k)

(δ+n2)k (δ+n1)k

k1*

An increase in “n”

…reduces k*

• At the new steady state k2* capital per worker and output per worker are lower

• The model predicts that economies with higher rates of population growth will have lower levels of capital per worker and lower levels of income.

The efficiency of labour

• We rewrite our production function as…Y=F(K,L*E)where “E” is the efficiency of labour. “L*E” is a measure of the number of effective workers. The growth of labour efficiency is “g”.

• Our production function y=f(k) becomes output per effective worker since…y=Y/(L*E) and k=K/(L*E)

• With this augmentation “δk” is needed to replace depreciating capital, “nk” is needed to provide capital to new workers, and “gk” is needed to provide capital for the new effective workers created by technological progress.

Steady State Equilibrium with population growth and technological progress

• At the point where both (k) and (y) are constant it must be the case that,Δk = s*f(k) – (δ+n+g)k = 0

…or,s*f(k) = (δ+n)k…this occurs at our equilibrium point k*.

k*k

InvestmentBreak-even Investment

s*f(k)Investment

Break-even investment (δ+n+g)k

Like depreciation and population growth, the labour augmenting

technological progress rate causes the capital stock per worker to shrink.

s*f(k*)=(δ+n)k*

At k* break-even investment equals

investment.

The impact of technological progress

• Suppose the worker efficiency growth rate changes from g1 to g2.

• This shifts the line representing population growth, depreciation, and worker efficiency growth upward.

k2*k

InvestmentBreak-even Investment

s*f(k)

(δ+n+g2)k (δ+n+g1)k

k1*

An increase in “g”

…reduces k*

• At the new steady state k2* capital per worker and output per worker are lower.

• The model predicts that economies with higher rates of worker efficiency growth will have lower levels of capital per worker and lower levels of income.

Effects of technological progress on the golden rule

• With technological progress the golden rule level of capital is defined as the steady state that maximizes consumption per effective worker. Following our previous analysis steady state consumption per worker is…c* = f(k*) – (δ + n + g)k*

• To maximize this…MPK = δ + n + gorMPK – δ = n + g

• That is, at the Golden Rule level of capital, the net marginal product of capital MPK – δ, equals the rate of growth of total output, n+g.

Steady State Growth Rates in the Solow Model with Technological Progress

Variable Symbol Steady-State Growth Rate

Capital per effective worker

k=K/(E*L) 0

Output per effective worker

y=Y/(E*L)=f(k) 0

Output per worker

Y/L=y*E g

Total output Y=y(E*L) n+g

Conclusion

• In this section we added changes in two exogenous variables (population and technological growth) to the Solow growth model. We saw that in steady state output per effective worker remains constant, output per worker depends only on technological growth, and that Total output depends on population and technological growth.