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)

MarginalProductof 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

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