Question

II) A consumer behavior is described by the equation Y4 = C+ + S where Y, Ct and S stand respectively for the income, consumption and saving of the consumer at the time period t. We assume that Ct = {Y{+1 – 7 and S = 3 Y at any time period t. 1. Write down the difference equation of consumption of the consumer. 2 marks For the subsequent questions, we assume that the consumption of the consumer satisfies the relation: Ct = 11Ct-1 + 1. 2. Find his consumption over time given that Co = 450. 1 mark 3. What is his consumption at period t = 5? 1 mark 4. Determine the steady state level of consumption C. 1 mark 5. Is the difference equation of consumption convergent? If yes, to what value does it converge. 1 mark

Answer 1

1) solving for Difference equation

Yt = Ct + St

Yt = (Yt+1)/5 - 7 + 58Yt/65

So, 7Yt = 13*Yt+1 - 7*65

Yt+1 = (7/13)*Yt + 35

.

2) if standard first order autonomous Difference equation

Xt = aXt-1 + b

then solution of Difference equation :

Xt = (a)^t*(X0 - X*) + X*

X*= b/(1-a)

so, comparing , a= 7/13, b= 49/13, b/(1-a) = 49/6

solution path of C

Ct = (7/13)^t *(450 - 49/6) + 49/6

3)

t= 5,

Then Ct = 28.17

4)

Steady state value C= C*= 49/6

5) as if t tends to infinity,

as |a|< 1, Ct tends to C*= 49/6

so it is convergent solution , converges to 49/6