博弈论试题

GDE Macroeconomics:Week4problems

Sketch answers2012

vdelipsis@ a.karalis-isaac@

1a:In the short run z↑⇒y>y∗therefore population will grow as n(y)=γ(y−y∗)>0.(γis

a constant>0and y∗is the equilibrium output level).But withfixed land L,population

growth implies land per worker will fall l(t)=L/N(t).As land per worker decreases, output per pwroker falls back to its equilibrium level y∗,so population growth returns to

zero and l(t),N(t)stabilze at new equlibrium values N∗

1and l∗

1

with N∗

1

>N∗and l∗

1

and y=y∗.

To provide some graphical intuition,note that the question makes most sense if n(y) refers to a population growth rate,(N −N)/N rather than Williamson’s growth factor N /N.Then we can put

N −N

N

=n(y(t))=γ(y(t)−y∗)

N

N

−1=γ(y(t)−y∗)

N

N

=γ(y(t)−y∗)+1=γy(t)+(1−γy∗)(1) So we see that the model is really the same as Williamson’s Malthus model,but with population growth a linear function of output per worker,rather than the non-linear function g(y)he draws in the textbook.As(1)shows thatγis the gradient of the population growth line,and(1−γy∗)is the intercept,if the growth line is to pass through the origin−so that there is no growth when there is no population−it must be that

0=1−γy∗

y∗=1γ

So we have solved for the equilibrium level of output in terms of the parameters of the model.See Figures1and2at the end,and check you can interpret them(they should be read together).

1b i:With constant technology level z and constant,positive population growth n,output per worker must decline forever towards zero;as the population grows expononetially and the marginal product of new workers falls to zero.Consider moving down the per-worker

production function as l(t)declines.We can see this algebraically by looking at the time derivative:

y(t)=zl(t)β=z

Lβ

N(t)β

(2)

∂y ∂t =

∂y

∂N

∂N

∂t

=−βzLβN(t)−β−1

∂y/∂N

n×N(t)

∂N/∂t

=−βny(t)

<0

As the change in y(t)is a function of y(t)itself,it is easier to consider the growth rate of y(t).Analagously to the discreet-time case in week2,the growth rate will now be the time derivative of ln(y(t)):

∂ln(y(t))

∂t =

1

y(t)

∂y(t)

∂t

=

1

y(t))

(−βny(t)) =−βn

so we see that the growth rate is constant and less than zero;output per worker will decline exponentially to zero.

1b ii:If technology grows at a constant rate,this will have a positive effecton output per worker, which may offset the decline caused by exogenous population growth.So exogenous technology growth may allow us to reconcile a malthusian model with constant population growth,and non-declining living standards.We could go further(though you are prob not expected to at this stage)and ask how much tech growth do we need to avoid decline?

y(t)=z(t)l(t)β=z(t)

Lβ

N(t)β

(3)

⇒∂y

∂t

=y(t)(g−βn)(4)

by following the same procedure as above.Clearly if g>βn then living standards rise forever,etc.

c:Withγ<0the system would be unstable,so any perturbation from initial equilibrium (eg temporary shcok to z)would lead to explosive growth or decline to zero.