Reference no: EM133382442

Question 1. Consider a modified version of the Lucas Model where output is a linear function of the aggregate amount of time spent in production, scaled up by labor productivity A: Y = A(1-u)L, where u is the fraction of time spent learning new ideas. Plot the production function with Y on the vertical axis and L on the horizontal axis. Add a dashed line for the production function with a higher u and a dotted line for the production function with a higher A.

Question 2. Let A grow (from an initial value of A₀) exponentially at rate luL, where l governs the productivity of time spent learning new ideas. Solve for the equation that describes the growth rate of output per capita as a function of A₀, l, u, and L.

Question 3.  Consider the following baseline values for the model's parameters: A₀ = 100, l = 1/3000, u = 0.06, and L = 1000. Complete the following Table:

Question 4. Draw the timepath of output per person (y) assuming a constant L until time t*, at which point L jumps to twice its level and then remains at that higher level thereafter (assuming a constant u for the entire time). Also draw the timepath of output per person (y) assuming a constant u until time t*, at which point u jumps to twice its level and then remains at that higher level thereafter(assuming a constant L for the entire time).

Question 5. Suppose that there are suddenly diminishing returns to labor productivity in the production function: Y = A^y(1-u)L, where y=1 would not exhibit diminishing returns but 0<y<1 would. A classmate claims that this would kill off sustained growth because diminishing returns to capital is what makes capital fuel only catch-up growth and not sustained growth. Solve for the growth rate of y and Y to prove to your classmate that diminishing returns in labor productivity does have the effect of killing sustained growth.

Question 6. Suppose that we now include capital in the model so that the production function takes the familiar Cobb-Douglas form: Y = AK^a([1-u]L)^1-a, where A is now Total Factor Productivity (TFP). Taking logs and time derivatives, find theGrowth Accounting Equation: growth rate of output per worker (y) as the sum of the contribution from growth in capital per worker (k), the growth rate of TFP (A), and the contribution from "labor composition" (ie(1-a) times the growth rate of [1-u]).

Question 7. Consider a model where we use the previous production function, the same equation for the growth of the stock of ideas, the capital accumulation equation from the Solow Model, and a constant savings rate (s). Assume that this model exhibits a "balanced growth path" where the endogenous variables, ieoutput (Y), capital (K), and the stock of ideas (A), each grow at constant rates. What does this imply about the ratio of output to capital and hence their growth rates?

Question 8. Given your previous answer; our model's specification for the growth of A; and the assumption of no change in u, solve for the growth rate of output as a function of u, l, L, and a. Compare this growth rate to your answer for the model in question 5(when y=1), where the only difference is the appearance of capital. Given that 1/(1-a) is the sum of a geometric series, interpret capital's role as a multiplier of the direct effect of the growth in knowledge on output.

Question 9. A friend points out that there are measurable characteristics of the workforce that we could add to the model, such as the age of workers (a proxy for experience) and the educational attainment of the workers. Given that we want to keep TFP strictly as a residual of unmeasurable determinants of productivity, which of the remaining two contributors to the Growth Accounting Equation should be where we include age and education?

Question 10. Using our Penn World Tables data on the United States and assuming that the "labor composition" contribution to growth in output per worker is steadily 0.2 over every period, complete the following table (treating A as a residual, ie its value is whatever is required to make the other numbers sum up):