1. The three-periods intertemporal budget constraint is
$I_1+\frac{I_2}{1+i}+\frac{I_3}{(1+i)^2}=c_1+\frac{c_2}{1+i}+\frac{c_3}{(1+i)^2},$ where the left hand side (LHS) is the present discounted value of life-time wealth. If we set $$c_1=c_2=c_3=c^*$$, then the right hand side (RHS) is $$(1+1/(1+i)+1/(1+i)^2)\times c^*$$.
I1<-100
I2<-200
I3<-50
i<-0.05
pv.life.wealth<-I1+I2/(1+i)+I3/((1+i)^2)

total.discount<-1+1/(1+i)+1/((1+i)^2)

c.star<-pv.life.wealth/total.discount

The present discounted value of life time wealth is 335.8276644. The total discount number is 2.8594104, that is $$(1+1/(1+i)+1/(1+i)^2)$$. Therefore, $$c^*$$ is 117.4464711, which is also this household’s permanent income level.

I1<-120 #100
I2<-200
I3<-50
i<-0.05
pv.life.wealth.c<-I1+I2/(1+i)+I3/((1+i)^2)

total.discount.c<-1+1/(1+i)+1/((1+i)^2)

c.star.c<-pv.life.wealth.c/total.discount.c

The presevent discounted value of life time wealth is 355.8276644. The total discount number is 2.8594104. Therefore, $$c^*$$ is 124.4409199, which is also this household’s permanent income level.

The marginal propensity to consume is defined as $$\Delta C/\Delta I$$ which is the change of C divided by the change of I.

MPC.c<-(c.star.c-c.star)/20

MPC is 0.3497224.

I1<-120 #100
I2<-220 #200
I3<-70 #50
i<-0.05
pv.life.wealth.d<-I1+I2/(1+i)+I3/((1+i)^2)

total.discount.d<-1+1/(1+i)+1/((1+i)^2)

c.star.d<-pv.life.wealth.d/total.discount.d

The presevent discounted value of life time wealth is 393.015873. The total discount number is 2.8594104. Therefore, $$c^*$$ is 137.4464711, which is also this household’s permanent income level.

MPC.d<-(c.star.d-c.star)/20

MPC is 1.

1. continue from a.
I1<-100
I2<-220 #200
I3<-50
i<-0.05
pv.life.wealth.e<-I1+I2/(1+i)+I3/((1+i)^2)

total.discount.e<-1+1/(1+i)+1/((1+i)^2)

c.star.e<-pv.life.wealth.e/total.discount.e

The presevent discounted value of life time wealth is 354.8752834. The total discount number is 2.8594104. Therefore, $$c^*$$ is 124.1078509, which is also this household’s permanent income level.

changeC.e<-(c.star.e-c.star)

$$\Delta C_1$$ is 6.6613799.

1. continue from e. Once we move on to period 2, the household is planning between periods 2 and 3, facing the following intertemporal budget constraint:
$(1+i)A_1+I_2+\frac{I_3}{1+i}=c_2+\frac{c_3}{1+i},$
where $$A_1=I_1-c_1$$.
I1<-100
I2<-200 #200
I3<-50
i<-0.05

c.star.f<-pv.life.wealth.f/total.discount.f
The presevent discounted value of life time wealth is 229.300253. The total discount number is 1.952381. Therefore, $$c^*_2$$ is 117.4464711.
changeC.f<-(c.star.f-c.star.e)
$$\Delta C_2$$ is -6.6613799.