Economics Assignment -

Q1. Consider the Malthusian model where Y_t = S^α (A_t L_t )^1-α , A_t+1 = (1 + g)A_t , and L_t+1 = (1+η)L_t . In this case S is the (constant) quantity of land, L is labor, and A is technological progress. Assume that η, g > 0.

a. Derive the growth rate for per capita output.

b. Suppose labor's share is equal to .7. How large must the growth rate of technological progress be relative to the growth rate of population in order for per capita income to grow?

Q2. Consider the Malthusian model where Y_t = S^α (A_t L_t )^1-α , At+1 = (1 + g)A_t , and L_t+1 = (1 + η_t )L_t . In this case S is the (constant) quantity of land, L is labor, and A is technological progress. Assume that (S, L₀ , A₀ ) are all equal to 1, g > 0, and α = 1/3 . In addition, assume that η_t = a + by_t , where is b is greater than zero.

a. Suppose that population in the pre-industrial revolution doubled every 230 years. What is the annual growth rate of population in this case? What would the period population growth rate be if we interpret a period to be 25 years? (Be exact; don't use an approximation here.)

b. Suppose that a period is 25 years. Find the value of g that is consistent with no growth in living standards if the population doubles every 230 years. Use this value of g in the remaining parts of this assignment.

c. Find values for the parameters a and b in the equation determining the population growth rate consistent with the following statements: (i) a period is 25 years; (ii) after the industrial revolution, population doubles every 35 years at the point where per capita income is equal to twice its pre-industrial revolution level.

d. Suppose that the economy is in the Malthusian steady state (per capita income and population growth are both constant). Using Excel, simulate the response of the economy to a plague that destroys half of the population in one 25-year period (say, period 6 of your simulation). Produce a graph showing the evolution of per capita income for 20 periods following the plague. Do the same for the population growth rate. How many years does it take until living standards have returned to within one percent of the steady state? Describe in words how this economy responds to this shock.

e. Again, suppose that the economy is initially in the Malthusian steady state. Simulate the response to a permanent technological improvement that doubles the amount of output that can be produced from a given quantity of inputs in period 6 (a one-time improvement; no change in the growth rate of technological progress). Plot the evolution of per capita income and the
population growth rate for 20 periods following the innovation. Provide intuition for what you have found.

f. Finally, suppose that the economy is initially in the Malthusian steady state and, due to government support of research and development efforts, the growth rate of technological progress increases permanently to 5% per period in period 6. Does this change the steady state? Explain. Produce graphs of the response of per capita income and the population growth rate for 30 periods following the increase the growth rate of technological progress.