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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As I reviewed the data online

I talked to ton of college students

everyone is missing with this one question

it's time to make a video.

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Today, we are going to solve
the following problems

from our video on the Solow Models steady state.

country A produces GDP according to

the following equation:
GDP=5√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 's investing,

and second, to determine weather or not
its GDP is growing.

Fortunately,
that the first question

is actually a necessary step
for solving the second one.

First thing first.
The relevant information from that problem

is on the top of left hand corner
of the board for reference

As always, it's best way for identifying the steps
for solving the problem.

The first of two questions is very straight forward

simply derived investment (I) 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 were accumulating
through investment

Then find out how much the capital
were losing to the appreciation.

and finally we compare the two,
investment to the depreciation.

to determine weather
the country's capital stock,

and therefore it's GDP is increasing,
decreasing,

or remaining constant
in the steady state?

Let's look a bit more in depth of
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 label does.
Total output Y instead of GDP.

And K, Physical Capital
is measured on the X-axis

We know that this country's GDP is 5√K

and we actually already graphed it.

This equation shows that GDP is function of K

As K increases? GDP also increases.

Although it by a small amount.
because a lot diminishing returns

It's also worth noting
that we are actually holding

other variables could affect GDP constantly.

Things like education, or population,
and ideas

So increase 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 5√K

and we also know the Investment is
25% of GDP,

therefore, we can substitute 5√K in for GDP.

And that's it, for step one.
To take a short cut,

since we know GDP's
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 2

A few quick things to know here

Several variables actually
measured along Y-axis.

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

and eventually,
we are going to add depreciation.

In general, it looked pretty cluttered,
if we added all these labels up to the top.

so we just leave it as GDP

And one other thing to know,
if we investing 125,

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

It's been used for consumption.
You know, buying staff.

One of the following questions
at the end of this video

actually testing understanding this.

So while this country is accumulating 125
worth of capital

We don't yet know
if the country's stock overall

is increasing, decreasing,
or remaining constant.

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

or depreciating.

In a real world,
machine's break, laptops die.

Think physical capital in your own life,

how many times you've dropped iPhone?
and have to get a new one.

or how often you displeased your old phone,
even though it's still worked.

So even though Capital is been added
to stock of 10,000 through investment,

some of these 10,000
is also been lost to depreciation,

to those iPhone's dropping.

It helps to graph the depreciation.

We know from the initial problem,

that 1% of all Capital stock is depreciating.

Graphically, 1%K can be represented
roughly like this.

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

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

or depreciating each year.
We now solved for step 3.

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

If the country invest 125 worth of capital,

and loses 100 to depreciation,

and 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 the equation

increase in K, increases GDP.

As long as investment
is greater than depreciation

K and GDP will continue to increase
until 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 everything.

If you like to have additional practice,

We include some extra questions on
Solow and steady state in this video.

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