Consider the OLG model with the following specifications:

• Let the utility of household i of cohort t be

cⁱ(t) + cⁱ(t + 1)

where 0 < < 1.

• Preferences are the same across individuals and across generations.
• Also, let Y (t) = K(t)H(t)¹.
• The depreciation rate is 100%: = 1.

Questions

a Find the function G where

K(t + 1) = G(K(t)).

(Show your work.)

Additional specifications: Let us make the following assumptions:

• There is no population growth: N (t + 1) = N (t) t.
• Each young person in the economy works for h_y = 10 hours, while each old person works for h_o = 5 hours:

hⁱ = hⁱ(t), hⁱ(t + 1) = [h_y, h_o] = [10, 5], (i, t).

• Let = 0.8, = 0.5 and K(0) = 0.25.

1. Find K(1).
2. Calculate Y (1) using the income and expenditure approaches.

d) Suppose that due to the public health measures, the labor endowment of old people h₀ is increased strating from t = T . Using the model, discuss how this change will affect the steady state wage w and consumption of young and old workers c_y and c_o. Explain why these changes occur.