Question

Considering Solow Model with technological change. The steady-state level of income per effective labor y^e = 5. The economy reaches the steady state in Year 99. The initial technology T(0) = 1 and T grows constantly at rate $LaTeX: \theta\:=\:0.05$ θ = 0.05. What is the income per capita, y, in Year 100?  (Round to 1 decimal place)

Answer 1

Answer

Steady State occurs when Change in income per effective and change in capital per effective worker = 0 i.e. when state state occurs then income per effective worker will not change any further(and reaches a maximum).

Here It is given that At year 99, Steady state occurs and this means that Change in Income per effective worker( = 0 and i is give that income per effective worker = 5

Income per effective worker = Y/(LT) where L = Population , T = Technology and Y = income and also Note that Y/L = Income per capita(or worker).

Formula :

% change in (A*B) = % change in A + % change in B

% change in (A/B) = % change in A - % change in B

Thus % change in (Y/L) = % change in ((Y/(TL))*E) = % change in Y/(TL) + % change in T

As calculated above that Y/(LT) is constant at steady state implies that % change in Y/(TL) = 0

=> % change in (Y/L) = 0 + % change in E = 0 + θ = 0 + 0.05 = 0.05(i.e. 5%)

So Income per capita will increase by 5% from year 99 to 100.

T = 1 at year = 0 and it is increasing at a rate of 5%.

Formula

F = P(1 + r)ⁿ where, F = final value, P = initial value, r = growth rate and t = time.

So T at Year 99 = 1(1 + 0.05)⁹⁹ = 125.2

So, At year 99, y_e = Y/(LE) = 5 => (Y/L) =  125.24*5 = 626.2

And as discussed above it Y/L increases by 5% from year 99 to year 100.

Hence New Y/L = 626.2 + 0.05*626.2 = 657.5

Hence, Income per capita in year 100 = 657.5