Solow-Swan model without technology

Macroeconomic Analysis Revision Q

Continous KCt)

Discrete Kt

As Kapital is a stockrather than a How KC) refers to the amount


of Kapital available at point in timet


KE -refers to the amount of kapital at the & period
beginning -


For Flow variables like output , UC) refers to the quantity of goods produced
-
time E , While It

refers to the quantity of goods produced doing period
t
-




Measuring the change in output over time -
:




Aut =
Ut-UE-1 discrete


output is rising N r E. .


time - When EO
dt -



and falling when (H) O

de


#
=
AD-U(t-AE) continous time
At
At A
as
gets l the less time
closer to zero we have




Basic maths

14 Xz-* log(n)
= =

log(X) + log(z)
2U =
*
-

log(u) log(X) log(z)
= -




Z


*
3u =
y -
log(u)
=

xlogX
4 exp(aX)-log(4) =
xX



For derivatives we have the following -




U =
log(x)- discrete
aXX


n(t) log(x( -) - (+ ) max(t)
=
=
=




dt d(t)

, How calculate the growth rate of an economic variable : -




e.g output (4)

Take the derivative wrt time of the natural log of output


g(u(t)
at
=

=
Egy

>
-

Example :
-

Capital (K)
Find the growth rate of (


k(+) =
) capital labor ratio

Lt)


① Find natural logs
..



.( (t)) log(k(t))
log log(((t))
= -




② Take derivatives of each log variable Nr
.
.
time


g(k(t))
g(kH-GL
=




It at


This is equal to. .
I




-interpretation the
:- the growth rate of capital-per-worker

labor.
the growth rate of capitan


minus growth rate of


If the Capital-labor ratio is not changing overtime , then capital and


labor must be
graving at the same rate




Example 2 : -

Calculate the growth rate of output from the Cobb Douglas production In

x
*
!
-




4 = k


log(4(+)) xlog(((+)) + (1 x) log(L(t))
= - -




then
-
we take derivatives NrE time


log(u(t))
DEA
=

Og(k(t))
x
de
+ (1 x)
-




g(L(t))
It ]