Macroeconomics

Homework 5

1)

a) the IS curve: ln Yt= ln Y(t+1) – (1/Ɵ)rt

so the slope is: drt/dyt (is) = -Ɵ/Yt. That means that an increase in Ɵ will result in a steeper curve.

LM curve: Mt/Pt = Yt^(Ɵ/v) (1+rt / rt)^(1/v)

Ln(Mt/Pt) = (Ɵ/v) ln Yt +(1/v)ln(1+rt) – (1/v)ln rt.

0 = (Ɵ/v)(1/Yt)dYt + (1/v)(1/(1+rt)) drt – (1/v)(1/rt)drt.

The slope is: drt/dyt (LM) = (Ɵrt(1+rt))/Yt. That means that an increase in Ɵ will result in a steeper curve.

b) the curve IS is not affected by the value of V. while curve LM shifts upwards, since a decrease in v will result in an increase for the demand for real money.

c) IS is not affected byΓ(.)

optimal money holdings: BΓ’(Mt/Pt) = (it/(1+it)) U’(Ct)

B(Mt/Pt)^(-v) = (it/1+it) Yt^-Ɵ

Mt/Pt= B^(1/v) Yt^(Ɵ/v) (1+rt/rt)^(1/v)

So this means that the LM curve will shift downwards.

2)

a) AC= (PC/)+(αYP/2)i

AC/ = -(PC/^2) + (αYP/2)I = 0

C/^2 = αYi/2

So *=(2C/αYi)^(1/2)

b) average real money holdings: M/P= αY/2

M/P = (αY/2) (2C/αYi)^(1/2)

M/P= (αCY/2i)^(1/2)

Ln(m/p) = (1/2)(lnα+lnY+lnC-ln2-lni)

(1/(M/P))((M/P)/i) = -(1/2)(1/i)

Elasticity of real money with respect to i: ((M/P)/)(i/(M/P)) = -1/2

The elasticity with respect to Y : ((M/P)/Y)(Y/(M/P)) = ½

Average real money holdings increase in Y, and decrease in i.

4)

a)when p is at a level that generates maximum output, LS meets LD.

b) when p is above the level that generates maximum output, will cause unemployment.

7)

a)

b)i)

ii)

iii)

13)

a) the asset has an expected rate of return r. capital gain/loss plus dividends per unit time = rvp. There is no dividends per unit time while searching for the palm tree, and there is b probability per unit time of capital gain of (vc-vp)-c. the difference in the price of the asset is(vc-vp) and –c is what the asset pays, so at the end we have rvp=b(vc-vp-c)

b) there is probability aL that a person will find another person with a coconut and trade with that person and gain u̅. the difference in the price of the asset is (vp-vc). So we end up with

rvp=al(vp-vc+u̅).

c) vp=(rvc/aL)+vc-u̅.

r((rvc/aL )+vc-u̅)= b(vc-(rvc/aL)-vc+u̅-c)

vc(r(r+aL+b))/aL = u̅(r+b)-bc

the value of being in state C: vc= (aL(u̅(r+b)-bc)) / r(r+aL+b)

the value of being in state p: vp= ((u̅(r+b)-bc)/(r+aL+b)) + (aL(u̅(r+b)-bc)/r(r+aL+b)) - u̅

so finally

vc-vp = (bc+u̅aL)/(r+aL+b).

e) vc-vp ≥c

vc-vp = (bc+u̅a(b/a))/(r+a(b/a)+b) = (bc+bu̅)/(r+2b)

(bc+bu̅)/(r+2b) ≥ c

That means that

Bc+bu̅≥c and c(r+2b-b) ≤ bu̅

So finally we have

c≤ bu̅ / (r+b).

f) it is a steady-state equilibrium for no one who finds a tree to climb it for any value of c>0.

Yes there are values of c which there is more than one steady-state equilibrium for 0<c< bu̅/(r+b)

Yes, L = b/a has a higher welfare than L=0. When L=0 people don’t gain any utility since they don’t climb a tree and don’t have a chance to trade with other people and gain a coconut.

0 1 2 3 4 5 -3 -2.2000000000000002 -1.8 -1.8 -2.2000000000000002 -3

0 1 2 3 4 5 7 6.5 5.5 3.5 1

0 1 2 3 4 -2 -2.5 -3.5 -5.5 -8

LD 1 2 3 4 5 6 7 8 9 10 10 9 8 7 6 5 4 3 2 1 LS 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

AD 2 3 4 5 4 3 2 1 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9 10 9 8 7.5 7 6.7 6.5 6.3 6.1 5.9

0 1 2 3 4 5 -2.6 -2 -1.4 -1.1000000000000001 -0.95 -0.87 0 1

0 1 2 3 4 7 5 4.3 3.5 2