Answer the following question based on the standard models of analysis developed
ID: 1093630 • Letter: A
Question
Answer the following question based on the standard models of analysis developed in class. The information in the various parts of the question is sequential and cumulative. Be sure to show and discuss each individual effect and not just the net effect of the exogenous shocks.
1. The Solow Growth Model. Suppose that a small open economy with perfect capital mobility and a current account deficit can be described by the Solow Growth Model and is initially at its steady state.
a. Based only on this information use a Solow Growth Model diagram to clearly and accurately show the economy's initial levels of (1) economic output-per-worker, (2) investment-per-worker, (3) balanced investment-per-worker, and (4) the capital-to-labor ratio. This diagram should be drawn in BLACK.
b. Provide an economic explanation of what you have shown in your diagram above
c. Now suppose that a highly contagious disease quickly kills one-quarter of the labor force; then the disease completely disappears. Incorporating only this additional information, clearly and accurately show in your diagram above what effects this would have on (1) economic output-per-worker, (2) investment-per-worker, (3) balanced investment-per-worker, and (4) the capital-to-labor ratio. These effects should be drawn in RED.
d. Provide an economic explanation of what you have shown in your diagram above. Discuss what happens to (1) economic output-per-worker, (2) investment-per-worker, (3) balanced investment-per-worker, and (4) the capital-to-labor ratio. Be sure to explain why these changes take place.
e. In response to these deaths the government undertakes a significant fiscal expansion to provide better housing for the surviving population. Incorporating only this additional information, clearly and accurately show in your diagram above what effects this would have on steady state (1) economic output-per-worker, (2) investment-per-worker, (3) balanced investment-per-worker, and (4) the capital-to-labor ratio. These effects should be drawn in BLUE.
f. Provide an economic explanation of what you have shown in your diagram above. Discuss what happens to steady state (1) economic output-per-worker, (2) investment-per-worker, (3) balanced investment-per-worker, and (4) the capital-to-labor ratio. Be sure to explain why these changes take place.
g. Discuss the adjustment process that occurs during the transition period from the economy
Explanation / Answer
Solow began with a production function of the Cobb-Douglas type:
Q = A Ka L b
where A is multifactor productivity , a and b are less than one, indicating diminishing returns to a single factor, and a + b = 1 , indicating constant returns to scale.
Solow noted that any increase in Q could come from one of three sources:
To concentrate attention on what happens to Q / L or output per worker (and hence, unless the employment ratio changes, output per capita), Solow rewrote the Cobb-Douglas production function in what we shall refer to as per capita form:
Q / L = A K a L b - 1 = A K a / L 1 - b
Q = A K a / L a = A ( K / L ) a
Defining q = Q / L and k = K / L, that is, letting small letters equal per capita variables , we have
which is the key formula we will work with. We will examine how the model works when growth comes through capital accumulation, and how it works when growth is due to innovation.
s = 0.25 q
s = k
Note that if depreciation were only 10 percent of capital stock, the equilibrium condition would be s = 0.10 k . Although this is a more realistic figure for yearly depreciation, we assume 100 percent depreciation for simplicity -- and if you are troubled by the lack of realism, you may think of our time periods as decades rather than years.
s = k
Substitute for s the savings function to obtain:
0.25 q = k
Substitute for output the production function to obtain:
0.25 ( 100 k 0.5 ) = k
Finally, divide through by k 0.5 to obtain:
k 0.5 = 25
and square both sides to get the equilibrium capital stock
k = 625
If the equilibrium capital stock is 625, equilibrium output (found using the production function q = 100 k 0.5 ) will be:
q = 2500
Note that if savings is 1 / 4 of output, this means that equilibrium savings is 625 -- just enough to replace the capital stock next period, and to keep the economy in a steady-state with output at 2500 and capital stock of 625 ever after.
Predictions of the model
If the Solow model is correct, and if growth is due to capital accumulation , we should expect to find
Let A = 100 and a = 0.5 in the Solow per capita production function. Note that a = 0.5 means "take the square root of k" and A = 100 means "then multiply it by 100" to get the ouput per worker.
That is, let our production function be:
q = 100 k 0.5
Consider what happens if we begin with 100 units of capital per worker. We can use the production function to calculate that q = 1000.
The next step is to use the savings function to calculate how much of this output is saved. If s = 0.25 q then 250 units per capita of output are saved -- and the savings of one period become the capital of the next period.
Note that this means in the next period the capital stock will have increased from 100 to 250 .
Since the production function is unchanged, the output next period will be q = 100 (250) 0.5 = 1581
We again note that savings is 0.25 of output; and .25 x 1581 = 395.3, so that savings next period will be 395.3.
Therefore capital in the third period will be 395.3, and output in the third period will be:
q = 100 (395.3) 0.5 = 1988
This procedure can be continued as long as you can punch a calculator; the results for the first 7 periods are:
q = A k a