Solow model Economics question/ DEADLINE in 8 hours

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2020-Fall-Econ433-Lecture041.pdf

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The Solow Model Math Preliminaries

The value of k if alpha and y are known

Let y = kα

What is k if y = 3.12 and

α = 2.17?

Solution:

First, write down what

you know: 3.12 = (k)2.17

You want to isolate k by getting rid of 2.17

Raise both sides by the

power of (1/2.17).

(3.12)(1/2.17) = (k)(2.17)*(1/2.17)

= (k)*(2.17)*(1/2.17)

Or,

3.120.46083 = k

=1.6894

Check your

calculation:

1.68942.17 ≈ 3.12

Make sure you understand each step.

Economic Growth and its Problems

Some Definitions

K = Capital: tools, machines, and structures used in production

L = Labor: the physical and mental efforts of workers.

Y = GDP of a country.

The 3-D graph on the right shows that both capital and labor increase output.

Math Preliminaries: The Production Function

• An algebraic representation of the graph in the previous slide is given by:

• Y = F(K,L)

• Where Y = output or GDP, K = capital, L = labor and F is a general production function.

• Example: Y = √K √L

• When K = 100, L = 16, we get Y = 40

• Notice that one needs a lot of capital (100) to produce Y (40 units) in this example.

Math Preliminaries: The Production Function

• Production function says that to produce total output or GDP of a country, one needs both inputs: labor (L) and capital (K).

• Note that, in general, production function can take many forms:

• Y = AK • Y = 2K + 15L • Y = 2 √K + 15 √L • Y = KαLβ (where α and β are positive numbers). • How do these production functions look like? • For reasons to be made clear later, economists like smooth

production functions like the one on the next slide.

Have You Seen L?

• Notice that on the previous slide, there is K on the horizontal axis but there is no L visible.

• That’s because we want to see the effect of K on Y - holding L constant.

• What happens to Y when K rises? • When K rises by one unit, MPK (“marginal

product of capital” or the amount by which Y rises for each unit rise of K, becomes smaller and smaller (next slide).

• This is known as the law of Diminishing Marginal Returns (DMR).

Y output

MPK and the production function

K capital

MPK

1 MPK

As more capital is

added, MPK 

Slope of the production

function equals MPK

DMR

• As a factor input is increased, its marginal product falls (other things equal).

• Intuition: Suppose K while holding L fixed

 more machines per worker.

Relative labor shortage.

Output increases at a decreasing rate.

Exercise: Compute & Graph MPK

a. Determine MPK at each value of K.

b. Graph the production function.

c. Graph the MPK curve with MPK on the vertical axis and K on the horizontal axis.

K Y MPK 0 0 1 10 ? 2 19 ? 3 27 ? 4 34 ? 5 40 ? 6 45 ? 7 49 ? 8 52 ? 9 54 ?

10 55 ?

Answers

a. Determine MPK at each value of K.

b. Graph the production function.

c. Graph the MPK curve with MPK on the vertical axis and K on the horizontal axis.

K Y MPK 0 0 n.a. 1 10 10 2 19 9 3 27 8 4 34 7 5 40 6 6 45 5 7 49 4 8 52 3 9 54 2

10 55 1

Answers O

u tp

u t

(Y )

Production function

0 1 2 3 4 5 6 7 8 9 10

Capital (K)

M P

K (u

n it

s o

f o

u tp

u t)

Marginal Product of Capital

0 1 2 3 4 5 6 7 8 9 10

Capital (K)

The Solow Model

2 5

Caution: Speed Bump Ahead!

• The Solow Model is by far the hardest part of this course.

• Much of this material is based on the Required Reading (Mankiw) posted.

• There are many graphs; we also use some algebra and elementary calculus later.

The Solow Model and the Modern Growth Theory

Robert Solow, Nobel Laureate Recipient, National Medal of Science

The Solow Model

• The Model is Used as a Major Paradigm in Economics:

– widely used in policy making

– benchmark against which most recent growth theories are compared

• Looks at the determinants of economic growth and the standard of living in the long run.

Solow Model Preliminaries: What is the right production function equation?

• The production function equation that really works in the Solow model looks like this:

•

• Y = KαLβ where α and β are positive numbers and β = 1 ‐ α. In other words, the equation for the preferred Solow‐type production function is:

Y = KαL1‐ α

• What is the big deal about Y = KαL1‐ α ?

• We answer this question in the next slide.

In Praise of Y = KαL1‐ α

• Y = KαL1‐ α produces a well behaved smooth graph such as this:

In Praise of Y = KαL1‐ α

• If both K and L are multiplied by a positive number λ (lambda),

• The new output will be

• = (λ K)α(λ L)1 ‐ α = λ (α+ 1‐ α) (Old Y) = λ.(old Y).

• Thus if β = 1 ‐ α the new Y will be λ times the value of old Y

• This idea is called the property of constant Returns to Scale. Economists like this property because it is consistent with perfect competition.

Math Preliminaries Again: The production Function

• In general aggregate terms:

• Y = F (K, L) or Y is a function of K and L

• Now Define:

• y = Y/L = output per worker (= lowercase y)

• k = K/L = capital per worker (= lowercase k) • Now assuming constant returns to scale just discussed, multiply

each input by : λ where λ is defined as 1/L. λ Y = F (λ K, λ L ) for any λ > 0

• Pick λ = 1/L. Then Y/L = F (K/L, 1)

y = F (k, 1) Now redefine F (k, 1) as

y = f(k) where f(k) = F(k, 1)

How Does y = f(k) Look Like?

DMR

• The law of diminishing returns now shows up again. Marginal product of lowercase y now falls as lowercase k rises.

The production function Output per worker, y

Capital per worker, k

Note: this production function exhibits diminishing MPk as discussed before.

f(k)

MPk = f(k +1) – f(k) 1

Solow Model Preliminaries: National Income or GDP Accounting

• Hold your thought on production function.

• We will now spend a few minutes understanding national income or GDP accounting.

• National Income ≈ GDP = Products produced by domestic firms. Assume first that there is no government.

Look At Everything in Per Capita Terms

• Y = C + I • (income can be either consumed or saved, all

savings are invested; remember, no Government, or G )

• To convert to “per worker” terms: divide both sides by L

• Y/L = C/L + I/L • Which is redefined as: y = c + i

where lowercase c = C/L and lowercase i = I/L

The consumption function

s = the savings/investment rate, the fraction of income that is saved (s is an exogenous parameter; if 20% income is saved, s = 0.20)

Consumption function per worker:

c = (percentage not saved)y = (1–s)y

Example, if s = 0.20, c = 0.80y

Saving and investment

• Just to verify, saving/investment (per worker) is = y – c = y – (1–s)y = s y

• From the national income identity

• y = c + i

Rearrange to get: i = y – c = s y (investment = saving)

• Using the results above,

i = s y = s f(k)

• (because y = f(k))

Output, consumption, and investment

• Look at the next slide carefully.

• If per worker capital stock is k1, per capita GDP will be y1. Out of this y1, c1 is consumed, the rest is (saved and) invested (I1).

Output, consumption, and investment

Output per worker, y

f(k)

sf(k)y1

Capital per worker, k

Output, consumption, and investment

Output per worker, y

f(k)

sf(k)y1

Capital per worker, k

Output, consumption, and investment

• What is wrong with higher and higher capital?

• Higher volume of capital increases GDP, so why not have ever higher volume of capital?

• The problem with high capital is that high capital requires high maintenance cost, i.e., depreciation costs are high.

• We assume that depreciation is a constant proportion of capital stock (say, 5%).

Depreciation Note: some textbooks use “d” as depreciation

Depreciation per

worker, k

Capital per worker, k

k

 = the rate of depreciation = d =

= the fraction of the capital stock that wears out each period

 = the rate of depreciation = d =

= the fraction of the capital stock that wears out each period

1 

I will use δ and “d” interchangeably to refer to the depreciation rate.

Pros and Cons of more k

• More k means more y.

• But more k means more depreciation, which must be funded by savings, making less resources available for new investment.

• Rich economies must spend large amounts of money just to maintain their stock capital (roads buildings, machines etc).

Capital accumulation

Change in capital stock depreciation

k

= investment –

= i – k

Since i = s f(k) , this becomes:

k = s f(k) – k

The basic idea: Investment increases the capital stock, depreciation reduces it.

Remember the old leaky water tank problem?

• A “break even” is achieved when the amount of water coming in is equal to the amount of water going out.

• s.k = δk is the breakeven condition in Solow.

• or, when Δk = 0

Every year s.k = i is added

δk leaks out

The equation of motion for k

income per person:

consumption per person:

y = f(k)

c = (1–s) f(k)

k = s f(k) – k

• This is Solow model’s central equation. Memorize it!

• Determines behavior of capital over time…

• …which, in turn, determines behavior of all of the other endogenous variables because they all depend on k; e.g.,

The steady state

k = s f(k) – k

If investment is just enough to cover depreciation [i.e., if investment equals s f(k) = k ] which must happen when an economy becomes very large, then capital per worker will not grow, it will remain constant forever:

k = 0.

This occurs at only one value of k, denoted k*. It is called the steady state capital stock.

The steady state

Investment and

depreciation

Capital per

worker, k

s f(k)

k

Moving toward the steady state

Investment and

depreciation

s f(k)

k

k = s f(k) − k

depreciation

k investment

k1 k* Capital per worker, k

Moving toward the steady state

Investment and

depreciation k

k = s f(k) − k

s f(k)

k

k1 k* Capital per worker, k

Moving toward the steady state

Investment and

depreciation

s f(k)

k

k = s f(k) − k

k

k1 k2 k* Capital per worker, k

Moving toward the steady state

Investment and

depreciation

s f(k)

k

k = s f(k) − k

investment

depreciation

k

k2 k* Capital per worker, k

Moving toward the steady state

Investment and

depreciation k

k = s f(k) − k

s f(k)

k

k2 k* Capital per worker, k

Moving toward the steady state

Investment and

depreciation

Capital per

worker, k

s f(k)

k

k = s f(k) − k

k

k2 k3 k*

Moving toward the steady state

Investment and

depreciation

Capital per

worker, k

sf(k)

k

k = s f(k) − k

k3 k*

Summary: As long as k < k*,

investment will

exceed

depreciation,

and k will

continue to grow

toward k*.

Summary: As long as k < k*,

investment will

exceed

depreciation,

and k will

continue to grow

toward k*.

Approaching the steady state in the Solow Model: Numerical Example

y c i k ∆kYear k

3 4

… 10 … 25 …



4.000 2.000 1.400 0.600 0.400 0.200

4.200 2.049 1.435 0.615 0.420 0.195

4.395 2.096 1.467 0.629 0.440 0.189 4.584 2.141 1.499 0.642 0.458 0.184

5.602 2.367 1.657 0.710 0.560 0.150

7.351 2.706 1.894 0.812 0.732 0.080

9.000 3.000 2.100 0.900 0.900 0.000

Assume: y = √k; s = 0.3;  = 0.1; Initial k = 4.0

Steady State

Values

Work out this

table

How Did We Get the Steady State Numbers (last row)?

Use

k = sf(k) – k

= 0 in steady state

Exercise: Solve for the steady state based on the numbers on the previous Table

We have:

s = 0.3,  = 0.1, and y = √k = k 1/2

Use the equation of motion

k = s f(k) − k

to solve for the steady‐state values of k, s, and c.

For steady state, 0 = s f(k) − k

Or, s f(k) = k ; or, s/  = k/f(k)

Solve for the steady state

s = 0.3,  = 0.1, and y = √k

Use: s/  = k/f(k) LHS = s/ = 0.3/0.1 = 3 RHS = k/f(k) = k/y = k/√k= √ k = 3 Or, k = 9 = steady state capital stock y = 3 = steady state income Steady state consumption = c = 3*0.7 = 2.1 These numbers match the last row of the previous table.

Prediction of the Solow Model:

• Higher s  higher k*.

• And since y = f(k) , higher k*  higher y* .

• Thus, the Solow model predicts that countries with higher rates of saving and investment will have higher levels of capital and income per worker in the long run.

• This conclusion is not the same as the conclusion of the Harrod Domar model.

• Harrod Domar predicted high savings  high growth rate.

• Solow: high savings  high level of income in the long run.

Solow Growth Rates

• What happens to the growth rates in the Solow model?

• Growth rate becomes zero eventually in the Solow model!

• Go back to the previous graphs; at k*, there is no growth any more!

High Saving = Higher Steady State

• Solow emphasizes “K” not “L”, not mother nature (land).

• Investment is at the center of Solow’s analysis. • But Solow also points out that there is a limit to

growth – even if you have high savings. All countries – rich and poor ‐ will reach steady state eventually.

• Once a country reaches steady state – it stops growing.

• From this perspective, Solow points out to a dismal future! We are doomed eventually 

International evidence on investment rates and income per person: seems to support the Solow model

100

1,000

10,000

100,000

0 5 10 3515 20 25 30

Investment as percentage of output

(average 1960‐2000)

Income per

person in

2000 (log scale)

International evidence on investment rates and income per person: seems to support the Solow model

100

1,000

10,000

100,000

0 5 10 3515 20 25 30

Investment as percentage of output

(average 1960‐2000)

Income per

person in

2000 (log scale)

An increase in the saving rate

An increase in the saving rate raises investment… …causing k to grow toward a new steady state:

Prediction: Countries with higher rates of saving and

investment

The Solow model predicts that countries with higher rates of saving and investment will have higher levels of capital and income per worker in the long run.

Are the data consistent with this prediction?

International evidence on investment rates and income per

person

How much should we save?

The Golden Rule: Introduction

Different values of s lead to different steady states.

How do we know which is the “best” steady state?

The “best” steady state has the highest possible

consumption per person: c* = (1–s) f(k*).

An increase in s

leads to higher k* and y*, which raises c*

reduces consumption’s share of income (1–s), which lowers c*.

So, how do we find the s and the k* that maximize c*?

Golden Rule for Goldilocks Savings Rate

The Golden Rule capital stock

= *

gold k

the Golden Rule level of capital, the steady-state

value of k that maximizes consumption

To find it, first express c* in terms of k*:

The Golden Rule capital stock

Then, graph f(k*)

and δk*, looking for

the point where the

gap between them

is biggest.

The Golden Rule capital stock

c* = f(k*) − δk* is

biggest where the

slope of the

production function

equals the slope of

the depreciation

line:

MPK = δ

Maximize Consumption

If you had a semester of calculus, you can easily see that deriving the condition MPK =  is straightforward: The problem is to find the value of k* that maximizes c* = f(k*) − k*. Just take the first derivative of that expression and set equal to zero: f(k*) −  = 0 where f(k*) = MPK = slope of production function, and  = slope of steady-state investment line.

Profit maximization using total revenue and total cost

TR-TC = Consumption in Solow

The transition to the Golden Rule steady state

The economy does NOT have a tendency to move toward the

Golden Rule steady state.

Achieving the Golden Rule requires that policymakers adjust s.

This adjustment leads to a new steady state with higher

consumption.

But what happens to consumption during the transition to the

Golden Rule?

Starting with too much capital

If > * *

gold k k

then increasing c*

requires a fall in s.

In the transition to the

Golden Rule,

consumption is higher at

all points in time.

Starting with too little capital

If < * *

gold k k

then increasing c*

requires an increase in

Future generations

enjoy higher

consumption, but the

current one

experiences an initial

drop in consumption.

Saving: Golden Rule for Goldilocks Savings Rate

• What matters is long run consumption

• High savings will increase capital stock and income, but may not increase consumption.

• What is the right or the golden savings rate?

Golden Rule for Savings: MPK = δ.

Exercise: Explore golden rule steady state based on the previous numbers

We have:

s = 0.3,  = 0.1, and y = √k = k ½

MPK = 1/(2√k)

As before: for steady state, 0 = s f(k) − k

Or, s f(k) = k ; or, s/  = k/f(k)

Or, sg /MPK = k/f(k)

Or, sg = MPK. k/f(k) = [1/(2√k)]*(k/√k) = 1/2

Savings should be 50% to maximize consumption

Happy Transition to Golden Steady State

Sad Transition to Golden Steady State

Politically Impossible?

Solow Model with population growth

One‐Time change in L or in K

• As the previous slides show, There may be too many people per machine as L rises

• If L rises and if K does not change, then K/L will fall.

• If a large amount of K is destroyed for some reason , K/L will also fall.

Effect of One‐Time Migration to a Country: L rises so k falls

Output per worker, y

f(k)

sf(k)

Capital per worker, k

k1k2

Effect of War that destroys capital stock: K falls, so k falls.

Output per worker, y

f(k)

sf(k)

Capital per worker, k

k1k2

What if L keeps rising?

• It is more difficult to conceptualize a rising L in the Solow model.

• But it can be done quite easily without doing more math.

• Assume that the population (= labor force) grows at rate n. (n is exogenous.).Example: Suppose L = 1,000 in year 1 and the population is growing at 2% per year (n = 0.02).

• Then L = nL = 0.021,000 = 20, so L = 1,020 in year 2.

Rise in L Erodes k

• Since a rise in L erodes k = K/L, in a way, L behaves like the depreciation rate. Just like depreciation erodes K and therefore erodes k= K/L, a higher L also erodes k = K/L.

• If L rises all the time, we must save enough and invest enough to keep K high enough so that K/L does not fall.

• This brings us to the concept of breakeven investment.

Consider the Concept of Breakeven Investment

• ( + n)k = break‐even investment, the amount of investment necessary to keep k constant.

• Break‐even investment includes:

–  k to replace capital as it wears out

– nk to equip new workers with capital

(Otherwise, k would fall as the existing capital stock

would be spread more thinly over a larger population

of workers.)

The Leaky Water Tank Problem Again

• Once again, a “break even” is achieved when the amount of water coming in is equal to the amount of water going out.

• s.k = (δ+n) k : breakeven

• or, when Δk = 0

Every year s.k = i is added

(δ+n) k leaks out

The new equation of motion for k

break‐even investment

actual investment

• With population growth, the equation of motion for k is

k = s f(k) − ( + n) k

The Solow model diagram

Investment, break‐even investment

Capital perk*

k = s f(k) − ( +n)k

( + n )k

s f(k)

worker, k 99

The impact of population growth

Investment, break‐even investment

Capital per 1k *

(+n2)k

(+n1)k

s f(k)

2k *

An increase in n causes an increase in break‐even investment,

worker, k 10 0

An increase in n causes an increase in break‐even investment, leading to a lower steady‐state level of k.

Prediction:

• Higher n  lower k*.

• And since y = f(k) ,

lower k*  lower y*.

• Thus, the Solow model predicts that countries

with higher population growth rates will have

lower levels of capital and income per worker

in the long run.

International evidence on population growth and income per person

100

1,000

10,000

100,000

0 1 2 53 4 Population Growth

(percent per year; average 1960‐2000)

Income per Person

in 2000 (log scale)