Office Hours: The Solow Model

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MRUniversity：
认识经济，了解世界

看了我们网站上的资料，也跟很多学生谈过

发现这问题大家都不会

所以觉得该是做个视频解说的时候了

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Office Hours:
新古典经济增长模型

新古典经济增长模型的稳定状态
我们之前已有教学视频

现在来解其中一道题目

A国的GDP产出是以下列方程式表示：
GDP=5√K

资本存量为1万

若该国25%的GDP用于生产投资财

则该国总投资额为多少？

另外，若每年所有资本消耗1%

那经济模型达到稳定状态时
该国GDP是增加、减少还是持平？

一如既往，最好先看过视频
然后试着自己做做看

若还有问题，可以再回来

大家一起解决

准备好了没？
此问题其实有两部分

首先得算出该国的投资总额

然后判断GDP是否成长？

幸运的是，首题是解开次题的必要步骤

按部就班

题目中的相关信息已写在白板左上角供大家参考

同样地，最好先理出解题步骤

第一个问题相当明确

只要从GDP等式中拆解出投资要素

然后以资本存量1万的条件，解出投资额(I)

至于次题则需首题答案

也就是投资过程中所累积的资本

即能算出资本消耗量

最后比较投资额与资本消耗

藉此判断该国的资本存量以及GDP
于稳定状态下是增减或持平

我们来详解这个问题

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

In previous Solow questions,

you may have seen this label does
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 5√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 5√K

and we also know that 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 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 know,
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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