$3.(20$ Points$).$ Consider an economy characterized by the following aggregate production function:

Y $=($AN$)1$K

Suppose that the population, N$,$ and productivity, A$,$ are growing at rates gN$,$ and gA$,$ households save a constant fraction of their income, s$,$ and capital depreciates at a constant rate $.$

What is the law of motion for the capital stock in this economy?

Derive the law of motion for capital per effective worker. Show all your

steps $(10$ Points Extra Credit$).$

Solve for the steady$-$state capital stock per effective worker. Show all

steps. Graph the steady$-$state.

What is the impact of an increase in the depreciation rate on steady$-$state capital per effective worker? Graph it$.$

What is the rate of growth of output per worker in the steady$-$state? Given your answer, do you think this model does a good job explaining the causes of economic growth? Why or why not?