problem 1 write each production function given

Question

Problem 1: Write each production function given below in terms of output per person y a Y/L and capital per person k K /L. Plot these per person versions in a graph with y on the vertical axis and k on the horizontal axis. (You can assume A is a constant positive number).

a)  Y = AK¹PL²/³ and Y = AK³/⁴0⁴ (plot them on the same graph).

b) Y = K.

c)Y = K + AL

d) Y = K - AL

Problem 2: Consider the production model that we studied in class (Chapter 4) and assume that the production function is now given by Y = AK³/⁴4¹¹⁴. Everything else in the model remains unchanged.

a)  Reproduce the analogue of Table 4.1 for this new economy. That is, state carefully what are your endogenous variables, parameters, and what equations you have in order to solve for equilibrium.

b) Fully describe the solution of the model (in other words provide formulas that relate each endogenous variable of the model with the known parameters).

c)  Let A = 1, K = 100, L = 1000. What are the equilibrium values of the wage, the rental rate of capital, and total output?

d) How does the equilibrium wage that you reported in (c) change if L = 1500 and everything else remains the same? Why?

e)  Again consider general values for A, R, L. What is the equilibrium value of output per person?

Problem 3: The (steady state) equilibrium value of output per person in the Solow growth model (Chapter 5) is given by

y* = A^3/2 (s/(d)^1/2    

The equilibrium value of output per person in the production model (Chapter 4) is given by

y* = Ak^1/3

a) In (2), y* depends on equilibrium capital per person (i.e. k*). Is that true in (1) as well?

b) 'Why do you think the technological parameter A enters with a different exponent in the equations? Or equivalently, what is the difference in the two models that leads to this different result?

Problem 4: Suppose a country enacts a tax policy that discourages investment. As a result, the value of the parameters now goes to a smaller value s^-t.

a) Assuming that the economy starts at its initial steady state, use the Solow model to explain what happens to the economy (after the change of i) over time and in the long run.

b) Draw a graph showing how output evolves over time (put Y_r on the vertical axis and time on the horizontal axis). What happens to economic growth over time?

Problem 5: Suppose the level of TIT in an economy rises permanently from A to A'.

a) Assume that the economy starts at the initial steady state. Use the Solow model to expalin what happens to this economy (after the change of TFP) over time and in the long run.

b) Draw a graph showing how output evolves over time, and explain what happens to per capita income.

c) How is the response of the economy to the increase in TFP different from the economy's response to an increase in the investment rate (like the one of Problem 4 above)? 

Problem 6: In class we pointed out that, in the baseline Solow model, the variables K*,Y* remain constant in the long-run (that is, there is no long run growth since the economy settles at the steady-state).

a) Verify that the varibles y*, C*, c* are also constant in the long-run.

b) Now suppose that everything in the Solow model remains the same excpet one assumption: labor supply is not constant over time any more, but assume that it grows at a constant rate per period. Intuitively (no need to write down any equations), will the variables K*,Y* y*,C*,e* still reach a long run steady state?