Consider an economy that lasts for T = ∞ periods. The parameters of the economy are gA = 0.02, gL = 0, s = 0.12, δ = 0.1, α = 0.5. Also, A1 = L1 = 1.

When we refer to steady-states, we always refer to positive ones. That is, we never look at the uninteresting steady-state in which ̃k∗= 0.

1. Compute the steady-state value of capital per unit of effective labor, ̃k∗, where capital per unit of effective labor at any time t is ̃kt = Kt AtLt. You do not need to derive the formula for it, but it could be good practice to
do so.

2. Show that if ̃kt > ̃k∗, then ̃kt+1 < ̃kt, and vice versa that if ̃kt < ̃k∗, then ̃kt+1 > ̃kt. How do we call this property? To answer this question only, do not replace the model parameters with the values provided above. You can choose whether to answer with question analytically (that is, using formulas) or graphically.

3. Assume the economy in period t = 2 is at the steady-state (the same steady state you computed in question 1.). Compute the value of capital, K2. Then, assume that in period t = 2 the growth rate of technology gA increases to 0.04. Keep in mind that when the change happens, K2 has been already determined from savings in period t = 1. Compute ̃k3 and K3. How are they different from ̃k2 and K2? Provide intuition.

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