Closed Economy with Aggregate Production Function and Investment: Yt Calculation for t=10, 20, 30

To compute Yt for t = 10, 20, and 30, we need to use the given production function and the capital stock equation.

First, let's calculate the investment made in each period. We know that 20% of total output is invested, so It = 0.2 * Yt.

Next, let's calculate the capital stock for each period using the given equation Kt+1 = 0.95Kt + It.

For t = 0: K0 = 1

For t = 1: I1 = 0.2 * Y1 = 0.2 * F(K0, L0) = 0.2 * F(1, 1) = 0.2 * 0.2√(1)^2 + (1)^2 = 0.2 * 0.2√1 + 1 = 0.2 * 0.2 * 2 = 0.08 K1 = 0.95 * K0 + I1 = 0.95 * 1 + 0.08 = 0.95 + 0.08 = 1.03

For t = 2: I2 = 0.2 * Y2 = 0.2 * F(K1, L0) = 0.2 * F(1.03, 1) = 0.2 * 0.2√(1.03)^2 + (1)^2 = 0.2 * 0.2√1.0609 + 1 = 0.2 * 0.2 * 2.03 = 0.0812 K2 = 0.95 * K1 + I2 = 0.95 * 1.03 + 0.0812 = 0.9775 + 0.0812 = 1.0587

Continuing this process, we can calculate Kt and It for each period.

For t = 10: K10 = 1.8984 I10 = 0.2 * Y10 = 0.2 * F(K9, L0) = 0.2 * F(1.8984, 1) = 0.2 * 0.2√(1.8984)^2 + (1)^2 = 0.2 * 0.2√3.5987 + 1 = 0.2 * 0.2 * 2.0994 = 0.08397 Y10 = F(K10, L0) = 0.2√(1.8984)^2 + (1)^2 = 0.2 * 0.2√3.5987 + 1 = 0.2 * 0.2 * 2.0994 = 0.8399

For t = 20: K20 = 2.4111 I20 = 0.2 * Y20 = 0.2 * F(K19, L0) = 0.2 * F(2.4111, 1) = 0.2 * 0.2√(2.4111)^2 + (1)^2 = 0.2 * 0.2√6.1089 + 1 = 0.2 * 0.2 * 2.4768 = 0.09907 Y20 = F(K20, L0) = 0.2√(2.4111)^2 + (1)^2 = 0.2 * 0.2√6.1089 + 1 = 0.2 * 0.2 * 2.4768 = 0.9927

For t = 30: K30 = 2.8681 I30 = 0.2 * Y30 = 0.2 * F(K29, L0) = 0.2 * F(2.8681, 1) = 0.2 * 0.2√(2.8681)^2 + (1)^2 = 0.2 * 0.2√8.2344 + 1 = 0.2 * 0.2 * 2.8749 = 0.1149 Y30 = F(K30, L0) = 0.2√(2.8681)^2 + (1)^2 = 0.2 * 0.2√8.2344 + 1 = 0.2 * 0.2 * 2.8749 = 1.149

Therefore, Y10 = 0.8399, Y20 = 0.9927, and Y30 = 1.149.