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Solow Growth Model

- Real GDP in US is 5X its level 50 years ago; per capita real GDP is 3X.
- In some poor countries, real GDP per person is only 2-5% of US.
- Using the production function with only K, L, and Θ, Robert Solow developed a very abstract theory to capture growth.

Solow Model

- To keep the analysis as simple as possible, we will pretend that G=0, NX=0.
- After we develop the model, we can see how changing these parameters will affect the results.

- First, the contribution of capital to growth and the importance of savings to capital accumulation will be discussed.

Solow Model

- The impact of increasing available labor (dubbed population growth) will be discussed after the basic model is understood.
- The impact of technology change will be the subject of the next chapter.

Accumulation of Capital

- If L and Θ are fixed, the only factor of production that will bring about growth of Y is K.
- We will use the same approach of supply of Y and demand for Y we used before, to determine how much K will increase each period.

Supply of Y

The production function is the familiar one with

constant returns to scale. The little trick of

defining z allows us to show the output as per

worker real GDP and the input as capital per

worker (also called capital-labor ratio).

The lower case depiction of the production

function, therefore, says that per worker output

depends on capital per worker.

Basic Rule of Derivatives

When x changes by one unit, by how many units will Z change?

Negative exponent means reciprocal.

Marginal Product of Capital

Using an arbitrary Cobb-Douglas

function, we can see how the

production function can be

presented in terms of GDP per

worker.

The MPK will be decreasing as

capital increases because as K

goes up the denominator

increases.

Likewise, when capital-labor ratio

(k) increases, the marginal

product of k decreases.

The exercise here shows that

MPK=MPk and it doesn’t matter

If one uses Y or y.

Demand for Y

The total output (GDP) is divided between C, I, G, and NX. For

simplicity, pretend that G=0 and NX=0. Then, Y = C + I

Let’s show this equation as per worker:

Output per worker (y) is determined by capital

per worker (k). Given k, we know what y will be.

The output per worker (y) will be divided between

consumption per worker and investment per

worker according to the size of savings rate (s).

The higher the savings rate, more of the output

will be used for investment and less for

consumption.

Relationship of i and y

y

f(k)

y1

y=f(k) and i=sy

which is the

same as i=sf(k)

Consumption

per worker, c

sf(k)

y=c+i

Invesment

per worker, i

k1

k

What happens if s rises?

Depreciation

- Some capital stock is used up.
- Some capital stock becomes obsolete.
- Some capital stock is broken.
- Collectively, let’s say, in general a certain percentage of the capital will be lost per year: δK.

Capital Accumulation

- Investments add to the capital stock.
- Depreciation subtracts from the capital stock.
- Net capital accumulation, ΔK, then, must be I – δK.
- Per worker: Δk = i – δk
- Alternately, Δk = sf(k) – δk

Capital Accumulation and Steady State

δk, i

δk

δk2>i

Δk*=i

sf(k)

δk1<i

k*

k2

k (Capital per worker)

k1

Calculating k*

- At k*, sf(k)=δk or sf(k*)= δk*
- Likewise, s/δ=k*/f(k*)
- Suppose s=0.4, δ=0.2, f(k)=10k0.3333
- 2 = k/10k0.3333
- 8 = k2/10
- 80 = k2
- k* ≈ 9

Test of Solow Model

- Solow model says, ceteris paribus, higher investment rates bring higher steady-state capital and higher income per worker.
- How does one test this?
- What does Figure 7-6 show?

Golden Rule of Capital

- What steady state level of capital per worker is optimal?
- Define optimal as maximum consumption per worker (well-being = consumption).
- The higher the s, the higher the k.
- But which k is the best one?

Golden Rule k

- Consumption per worker is at maximum when the slope of δk is exactly equal to the slope of f(k).
- Slope of δk is δk/k = δ.
- Slope of f(k) is dy/dk. But dy/dk = dY/dK (see slide #7)
- When δ = MPK, consumption per worker is maximized.

Example

If y = 10k0.25, and δ = 0.15, what is the golden rule k?

MPK = 2.5k-0.75

MPK = δ

2.5k-0.75 = 0.15

k0.75 = 16.6667

k = 16.66671.333

k ≈ 42.6

Compare with slide # 15!

Population Growth

- Population growth rate is given as n.
- If the population growth is equal to labor force growth, next year’s L will be (1+n)L.
- To distinguish this year’s L from next year’s, let’s say Lt+1 = (1+n)Lt.
- For K/L to be constant, the growth rate of K should also be n.
- If y*=f(k*), then at equilibrium Y, K, and L all grow at the rate of n.

Population Growth

- At equilibrium, i had to just match depreciation. Now that K has to also grow at rate n to keep k constant, i has to compensate for both depreciation and required capital growth:

Δk = i – δk – nk

Δk = i – (δ + n)k

It has to be nk because growth rate of L

has to match growth rate of K.

Population Growth

- The impact of n on the model is to make the δk line steeper.
- The steady state will now be
0 = sf(k) - (δ+n)k

i = (δ+n)k

Population Growth

- In the steady state consumption per worker is not increasing but GDP (Y) is increasing at rate n.
- A higher n implies a lower k and a lower y. Do higher n countries have lower per capita incomes? Figure 7-13.
- Golden Rule capital is now MPK = δ+n

Is Population Growth a Curse or a Blessing?

- Malthus: resource constraint
- Kremer: innovation and technology.

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