Question

1. Malthusian Model of Growth Notation: Yt Aggregate output; Nt Population size; L¯ Land (fixed); ct Per capita consumption Production: Aggregate production function is Yt = F(Nt , Lt) = zN2/3 t L 1/3 t Population Dynamics: Nt+1 = g(ct)Nt Population growth function: g(ct) = (3ct) 1/3 Parameter Values: Land: L¯ = 1000 for all t. Productivity parameter: z = 1                                                       

   
   

   (a) Solve for the steady state of this economy (Steady state: Nt+1 = Nt). Report steady state values for c and N.

   
   

   (b) Suppose the economy is at its Steady State Equilibrium (SSE). Analyze the effect of the following changes on SSE consumption per capita and population. For each case, discuss in words what we learn from the exercise.

   i. All else the same, z decreases.

   ii. All else the same, L¯ increases. Illustrate this experiment using a diagram with c on the y-axis and N on the x-axis (see lecture for an example).

   
   

2.  Consider the Solow Model with the following features: aggregate production function at time t is Yt = K0.5 t N0.5 t , K is capital and N is labour. The savings rate is s, population growth rate is n and the depreciation rate is δ. Capital accumulation: Kt+1 = Kt(1 − δ) + Xt , where Xt is the amount of investment at time t, which is equal to total savings.

 (a) Derive the capital per worker accumulation function. Using this function argue that, all else equal, the growth rate of output per worker, y, decreases as capital per worker, k, increases.

 (b) Solve for the steady-state levels of capital per worker, k ss, and output per worker, y ss , as a function of s, n and δ. Argue that, all else equal, y ss increases if s increases. 

(c) Suppose the government aims to maximize consumption per worker in the steady state. What level of the savings rate should the government target? Explain.