EXERCISES:  COMPUTING THE STEADY STATE CAPITAL STOCK
AND THE GOLDEN RULE CAPITAL STOCK.

1.  Given the following Solow model:
Y= AK^.7L^.3                              where Y is output, A is technology, K is capital stock and L is labor.
y = c + s                                  where y is per worker output, c and s are consumption and savings per worker
k_{year 2} = (1-d)k _{year 1} + i _{year 1}   where k is capital stock per worker, d the depreciation rate and i is investment per worker.

a) Assume that the labor force growth rate and the rate of technological progress are both zero in this country.  What is the steady state capital stock per worker, steady state output per worker and steady state consumption per worker if A=3, the rate of depreciation is 20% (0.20) and the savings rate is 25% (0.25)?

b)  Suppose this country starts with a capital stock per worker of 100 (year 1).  Compute output per worker and consumption per worker for years 1 and 2 and the capital stock per worker in year 2.

c)  Compute the golden rule (GR) capital stock.  What savings rate makes the k^ss equal to k^GR?  What is consumption at the golden rule point?

2.  Redo 1(a), but now assume the savings rate is 30%.

3.  Redo 1 (a) and 1 (c), but now assume population (work force) growth of 5% (n=0.05).
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ANSWERS:
1.  a) Steady states:  k = 81.93   y= 65.54   c= 49.16

     b)  year 1:  y = 75.36    c= 56.52
          year 2: k = (1-d)k_{year 1} + savings rate * income _{year 1}
              or     k year 2 = (1-0.2)*100 +  .25 * 75.36    = 80 + 18.84= 98.84.  Note that savings (18.84) is LESS than depreciation (20) which means the capital stock will fall over time (until the steady state is reached).
                    => in year 2: y=74.74    c=56.06.

    c)  The golden rule capital stock is where the marginal product of capital is equal to the cost of maintaining the capital stock per worker.  Thus, we need to find the capital stock where the MPk is equal to rate of depreciation.
    k^GR=2534.92.  Savings Rate = 0.7 (70%).  consumption = 217.28.

2.  Steady states:   k = 150.44   y= 100.3   c= 70.2

3.  a) Steady states:  k = 38.94   y= 38.94   c= 29.2

     c)  Now we need to set MPk = (d+n), since (d+n) is the cost of maintaining the capital stock per worker.
   k^GR=1204.84.  Savings Rate = 0.7 (70%).  consumption = 129.09.