Unformatted text preview: now how the supply and the demand of GDP in our model work. Now, we need to impose a market clearing
condition, which simply means that we are not going to waste any amount of GDP. Or,
Supply = Demand
Therefore,
Y = F (A, K, L)
And we can write equation 2.4 as,
∆K = sF (A, K, L) − δ K
However, there is a problem with this. As you should remember from last week, we care about GDP per capita, and we
have everything in this equation in levels!! Therefore, I need to divide by the population level, Lt
∆K
F (A, K, L)
K
=s
−δ
L
L
L
Then, let’s move step by step:
• First, we are going to use capital letters for levels, and small letters for variables in per capita terms. For example,
k = K/L y = Y /L c = C/L • So, rewrite our last equation as,
∆K
F (A, K, L)
=s
− δk
L
L
• You can consider 1/Lt as a number. Therefore, taking into account that our production function is a neoclassical
one (constant returns to scale), we can rewrite,
F (A, K, L)
KL
= F A, ,
= f (A, k )
L
LL
We will see an example with a CobbDouglas production function later...
∆K
= sf (A, k ) − δ k
(2.5)
L
• BE CAREFUL: ∆K is not equal to ∆k , because ∆k = ∆ K . That is, the increment of the whole ratio!! But we
L
L
have an increment only in the numerator!
• Therefore, we need to ﬁnd ∆K
L. Take other deep breath, drink a Red Bull, eat a protein bar...and...let’s start!!!! – Step 1: start with ∆k which is the increase in capital per capita over time.
– Step 2: realize that the change of a variable over time, is the derivative of that variable with respect to time.
– Step 3: remember that k = K/L
– Step 4: therefore, it is the derivative of a ratio,
∂K
∂k
L
∆k =
=
∂t
∂t
6 – Step 5: use your Calculus I notes, to get ∆k t =
= ∆k t =
∂K
∂k
L
=
=
∂t
∂t
∆ K × L − ∆L × K
L2
∆K
∆L K
∆K
−
=
− nk
L
LL
L Finally,
∆K
= ∆k + nk
L
Plug this result in equation 2.5
∆k + nk = sf (A, k ) − δ k
∆k = sf (A, k ) − (n + δ )k
or rearranging terms (that is, divide by kt )
∆k
k = sf (A,k)
k − (n + δ ) This equation is called the Fundamental Equation of SolowSwan. And you should memorize it!!! (As soon as possible!!) 3 The fundamental equation of SolowSwan
γk = 3.1 ∆k
sf (A, k )
=
− (n + δ )
k
k Interpretation (We will discuss this in more detail next week, but you need this to solve the problem set...)
The fundamental equation allows us to know the rate of growth of capital per capita once we know the level of capital
per capita today. Why?
• A, s, n, δ are constant numbers and we assume that we know them.
• Hence, if we know k , we can compute γk
We can observe that there are two forces to compute γk . One is positive and the other one is a negative force.
• The positive one i...
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This note was uploaded on 11/18/2012 for the course ECON W3213 ECON W3213 taught by Professor Xaviersalaimartin during the Spring '10 term at Columbia.
 Spring '10
 Xaviersalaimartin

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