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♪ (music) ♪

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

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Everyone is missing this one question.

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It's time to make a video.

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♪ (music) ♪

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

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on the Solow Model's steady state.

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Country A produces GDP
according to the following equation:

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GDP equals five times the square root of K

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and has a capital stock of 10,000.

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

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how much is this country investing?

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

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is the country's GDP increasing,

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decreasing, or remaining constant
in that steady state?

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

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and try to solve this problem by yourself.

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If you have remaining questions,
you can always return,

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and we'll work through
the problem together.

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Ready? This question has two parts.

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First, finding how much
this country is investing,

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

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Fortunately, that first question

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is actually a necessary step
for solving the second one.

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First things first.

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The relevant information from the problem

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

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

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The first of the two questions
is fairly straightforward.

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Simply derive the investment equation
from the GDP equation

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

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To solve the second question,

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we'll need our answer from question one:

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the amount of capital we're accumulating 
through investment.

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

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

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to determine whether 
the country's capital stock,

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and therefore its GDP,
is increasing, decreasing,

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or remaining constant 
in the steady state.

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

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

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In previous Solow questions,

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

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

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

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and we've actually already graphed it.

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This equation shows that GDP
is a function of K.

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As K increases, GDP also increases,

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

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

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other variables that could affect
GDP constant.

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Things like education,
or population, and ideas.

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

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

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If we plug that into equation,
GDP is 500.

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

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

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

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And that's it, for step one.

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To take a short cut,

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

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This country is investing $125 dollars
into capital accumulation.

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And that's the answer to step two.

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A few quick things to note here.

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Several variables are actually 
measured along the y-axis.

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

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and eventually
we're going to add depreciation.

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In general, it looks pretty cluttered

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

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So we'll just leave it at GDP.

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

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

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

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

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actually tests your understanding of this.

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So while this country is accumulating
125 worth of capital,

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

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is increasing, decreasing,
or remaining constant,

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

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or depreciating.

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In the real world, 
machines break, laptops die.

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Think of physical capital
in your own life.

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

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

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

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

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to those iPhones dropping.

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It helps to graph depreciation.

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We know from the initial problem

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that 1% of all capital stock
is depreciating.

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

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

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

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or depreciating, each year.

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We've now solved for step 3.

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

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If the country invests
125 worth of capital,

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and loses 100 to depreciation,

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then investment
is greater than depreciation,

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

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as represented by the difference 
between these two curves.

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We can now answer that final question.

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The country's capital stock is increasing,

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and therefore, so too is GDP.

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And that's our answer.

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Because remember,
according to the equation

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increases in K, increase GDP.

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As long as investment 
is greater than depreciation

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K and GDP will continue to increase

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until the country's capital investment
equals depreciation.

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

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is perfectly offset 
to capital lost from depreciation.

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

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As always, please let us know
what you think.

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

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

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at the end of this video.

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♪ (music) ♪