Closed Economy Production Function and Output Growth: A Step-by-Step Analysis

To compute Yt for t = 10, 20, and 30, we can use the given equations and information.

First, let's calculate the investment made in each period. We know that 20% of total output is invested in every period, so investment in period t, It, is given by:

It = 0.2 * Yt

Next, we can use the equation Kt+1 = 0.95Kt + It to calculate the capital stock in each period.

For t = 0: K0 = 1

For t = 1: I1 = 0.2 * Y1 K1 = 0.95 * K0 + I1

For t = 2: I2 = 0.2 * Y2 K2 = 0.95 * K1 + I2

And so on, until t = 30.

Now, we can use the aggregate production function Yt = F(Kt, Lt) = 0.2√(Kt)^2 + (Lt)^2 to calculate Yt for each period.

For t = 0: Y0 = F(K0, L0) = 0.2√(1)^2 + (1)^2 = 0.2 * √1 + 1 = 0.2 + 1 = 1.2

For t = 1: Y1 = F(K1, L1) = 0.2√(K1)^2 + (L1)^2 = 0.2 * √(K1)^2 + (L1)^2

And so on, until t = 30.

To calculate Yt for t = 10, 20, and 30, we need to calculate Kt and Lt for each period using the given equations, and then use the aggregate production function to calculate Yt.