Question

The idea of permanent income is that consumption depends on a long-run measure of income rather than just on current income. Operationally, we can define permanent income (??) to be the hypothetical constant flowof income that has the same present value as a household’s actual sources of funds.

(a) Use the infinite horizon budget constraint in equation 3.15 (shown below) to obtain a formula for permanent income ??. (if needed, use the result that [1 + 1 + 1 + . . . ] = 1+? )

(3.15)

? + ?2 + ?3 +···+?0(1+?) =? + ?2 + ?3 +···1 (1+?) (1+?)2 ? 1 (1+?) (1+?)2

Answer 1

Suppose current consumption is c

Also say consumption for year 1 is c1

for year 2 is c2, for year 3 is c3 and so on...

Consider a discount rate 'k' to discount these future consumptions...

PV(consumption) = c1/(1+k) + c2/(1+k)^2 + c3/(1+k)^3 + ..... upto infinity

Now lets consider consumptions are increasing every year by growth rate 'g'

So, c1 = c*(1+g)

c2 = c*(1+g)^2

and so on...

Substituting the same, the above equation becomes

PV(consumption) = c*(1+g)/(1+k) + c*(1+g)^2/(1+k)^2 + c*(1+g)^3/(1+k)^3 + ..... upto infinity. (equation 1)

Multiplying the above equation on both sides by (1+g)/(1+k)

PV(consumption)*(1+g)/(1+k) = c*(1+g)^2/(1+k)^2 + c*(1+g)^3/(1+k)^3 + ..... upto infinity. (equation 2)

Subtracting equation 2 from equation 1,

PV(consumption) - PV(consumption)*(1+g)/(1+k) = c*(1+g)/(1+k)

PV(consumption) = c*(1+g)/(k - g)

So,

Yp = c*(1+g)/(k - g)

where

c - current consumption

g - annual growth in consumption

k - annual required return on Yp