Question

Suppose an economy has the following per-worker production function:
       y = f(k) = 6 k1/2
The saving rate is s = .25 = 25%;
The depreciation rate is d = .15 = 15%.
The initial per worker capital stock (capital-labor ratio) is k = 16.
There is no population growth rate.
What is the steady state
per worker capital stock or capital-labor ratio (k*),    k* = ______
per worker output (y*),                    y* = ______
per worker consumption (c*)?               c* = ______
  

Show the steady state on the graph (next page). Completely label both axes, the production, saving, and label the steady state k*, y*, sy*, and c*.

What is the golden rule saving rate for this economy? sg = _________

y,sy,nk,(n+d)k

                                   k

Now the population growth rate is n = .05 = 5%.What is the steady state
per worker capital stock or capital-labor ratio (k*),    k* =______
per worker output (y*),                    y* =______
per worker consumption (c*)?               c* = ______

     

Show the steady state on the same graph as part a (above). Completely label the steady state k*, y*, sy*, and c*.
What is the golden rule saving rate for this economy? sg = _________

Answer 1

Steady state level of per capital labour stock k* is calculated using the given formula for steady state level of capital in relation to the savings and depreciation rate. k* = 100.

Steady state output per worker is easily calculated by substituting the value of k* in the production function. y* = 60.

Using savings rate to calculate consumption rate gives us the steady state level of consumption c* = 45.

The graph for steady state capital and output is given below.

Using the condition for golden rule of capital, calculate the golden rule level of capitla per worker. Then, using the previous steady state formula, calculate the golden savings rate. s*g=0.5.

Population growth reduces capital per worker. Adding population growth n=0.05 to the formula and calculating k* again gives k* = 56.25, y* = 45, c* = 33.75.

Similar to previous graph, but with population growth reducing capital per worker and subsequently output per worker as well.

EDIT: For the final part, calculate the golden rule capital per worker using the modified formula that includes population growth. Then, use the golden rule capital to calculate s*g.