Office Hours: The Solow Model

In last week’s Principles of Macroeconomics video, you learned about the steady state level of capital and the Solow model of economic growth. Here are two of the practice questions from that video:

Country A has K=10,000 and produces GDP according to the following equation: GDP=5√K.
1) If the country devotes 25% of its GDP to making investment goods, how much is the country investing?
2) If 1% of all machines become worthless every year (they depreciate, in other words) in Country A, GDP is...?

These are tricky problems! If you're stumped, don’t worry. Mary Clare Peate from the Marginal Revolution University team is here to help.

Are you struggling with a different practice problem or concept from an MRU video? Let us know! Head on over to our feedback forums to suggest a topic for a future "Office Hours" video: http://bit.ly/1psatWs

Additional practice questions: http://bit.ly/1SvNoP8

The Solow Model and the Steady State: http://bit.ly/1YGYiA3

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[Mary Clare] I've reviewed the data online.
I talked to ton of college students.

Everyone is missing this one question.

It's time to make a video.

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Today, we're gonna solve
the following problem from our video

on the Solow Model's steady state.

Country A produces GDP
according to the following equation:

GDP equals five times the square root of K

and has a capital stock of 10,000.

If the country devotes 25% of its GDP
to making investment goods,

how much is this country investing?

Additionally, if 1% of all capital
depreciates every year,

is the country's GDP increasing,

decreasing, or remaining constant
in that steady state?

As always,
it's best to watch the video first

and try to solve this problem by yourself.

If you have remaining questions,
you can always return,

and we'll work through
the problem together.

Ready? This question has two parts.

First, finding how much
this country is investing,

and second, determining
whether or not its GDP is growing.

Fortunately, that first question

is actually a necessary step
for solving the second one.

First things first.

The relevant information from the problem

is in that top right-hand corner
of the board for reference.

As always, it's best to identify steps
for solving the problem.

The first of the two questions
is fairly straightforward.

Simply derive the investment equation
from the GDP equation

and then solve for I, given the current
capital stock of 10,000.

To solve the second question,

we'll need our answer from question one:

the amount of capital we're accumulating
through investment.

We'll then find out how much capital
we're losing to depreciation,

and finally we'll compare the two,
investment to depreciation

to determine whether
the country's capital stock,

and therefore its GDP,
is increasing, decreasing,

or remaining constant
in the steady state.

Let's look a bit more in depth
at this problem by graphing it.

As you can see,
GDP is measured on the y-axis.

In previous Solow questions,

you may have seen this labeled as
total output or Y instead of GDP.

And K, physical capital,
is measured on the x-axis

We know that this country's GDP is
five times the square root of K,

and we've actually already graphed it.

This equation shows that GDP
is a function of K.

As K increases, GDP also increases,

albeit by a smaller amount because
of the law of diminishing returns.

It's also worth noting
that we're actually holding

other variables that could affect
GDP constant.

Things like education,
or population, and ideas.

So increasing capital is the only way
this country's GDP grows.

In our example, this country
has $10,000 dollars worth of capital.

If we plug that into equation,
GDP is 500.

Now we know that GDP is
five times the square root of K.

And we also know that Investment
is 25% of GDP,

therefore, we can substitute five
times the square root of K in for GDP.

And that's it, for step one.

To take a short cut,

since we know GDP in this instance is 500,
25% of 500 is 125.

This country is investing $125 dollars
into capital accumulation.

And that's the answer to step two.

A few quick things to note here.

Several variables are actually
measured along the y-axis.

Not just GDP,
but we're also measuring investment,

and eventually
we're going to add depreciation.

In general, it looks pretty cluttered

if we were to add all of those labels
up to the top.

So we'll just leave it at GDP.

And one other thing to note:
if we're investing 125,

and total GDP is 500,
what's happened to that remaining GDP?

It's being used for consumption,
you know, buying stuff.

One of the follow-up questions
at the end of this video

actually tests your understanding of this.

So while this country is accumulating
125 worth of capital,

we don't yet know
if the country's capital stock overall

is increasing, decreasing,
or remaining constant,

because we don't know how much
of the capital stock is wearing down,

or depreciating.

In the real world,
machines break, laptops die.

Think of physical capital
in your own life.

How many times have you dropped
your iPhone and had to get a new one?

Or how often have you replaced
an old phone, even though it still worked.

So even though capital is being added
to the stock of 10,000 through investment,

some of this 10,000
is also being lost to depreciation,

to those iPhones dropping.

It helps to graph depreciation.

We know from the initial problem

that 1% of all capital stock
is depreciating.

Graphically, 1% times K
can be represented roughly like this:

If capital stock is 10,000,
1% of 10,000 is 100.

So, 100 dollars worth of capital stock
is wearing down,

or depreciating, each year.

We've now solved for step 3.

We now have investment and depreciation,
and can compare the two.

If the country invests
125 worth of capital,

and loses 100 to depreciation,

then investment
is greater than depreciation,

and therefore, the capital stock
will grow by 25 this year,

as represented by the difference
between these two curves.

We can now answer that final question.

The country's capital stock is increasing,

and therefore, so too is GDP.

And that's our answer.

Because remember,
according to the equation

increases in K, increase GDP.

As long as investment
is greater than depreciation

K and GDP will continue to increase

until the country's capital investment
equals depreciation.

At this point, it reaches steady state
because capital gain through investment

is perfectly offset
to capital lost from depreciation.

And therefore, neither the capital stock
nor GDP changes at this point.

As always, please let us know
what you think.

And if you'd like to have
some additional practice,

we've included some extra questions
on Solow and steady state

at the end of this video.

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