Answers

Set k in period 1 equal to 5.00. Why does the capital stock now decrease rather than grow as it did when k = 3 in period 1? _________ (Reset the initial value of k to 3.00.)
The larger capital stock results in an equiproportional increase in depreciation (from 0.300 to 0.500 units. Output does not grow proportionately, however, so saving (which is proportional to output, and which equals investment) grows less than depreciation does. Therefore, the capital stock decreases.

One characteristic of the Cobb-Douglas production function is a diminishing marginal product of labor for all levels of employment. How does the curvature of the graph in this worksheet represent this attribute?

The marginal product is the slope of the production function ("rise over run"). This value decreases over the entire employment range, which is to say that diminishing marginal returns sets in at the outset. (Also, marginal product never becomes zero, so adding to k increases y, even though the increments may become only slightly different from zero.)

Have the workbook evaluate the marginal product over these ranges:

k = 10 to k = 15: MP = (2.253 - 1.995)/5 = 0.258/5 = 0.052
k = 15 to k = 20: MP = (2.456 - 2.253)/5 = 0.041
k = 20 to k = 25: MP = (2.627 - 2.456)/5 = 0.034
k = 25 to k = 30: MP = (2.774 - 2.627)/5 = 0.030
k = 30 to k = 35: MP = (2.905 - 2.774)/5 = 0.026

The marginal product is calculated in terms of a discrete (5-unit) change in k (capital per worker). If the change is small, the marginal product approximately equals the slope of the production function. For the Cobb-Douglas function used here, the marginal product at a point is this: MP = ak^a-1.

The interval marginal product is computed using two values of k, 25 and 30. Confirm that the MP computed using the MP formula above, evaluated at k = 27.5, is approximately the same as the value in the workbook.

The point MP when k is 27.5 is MP = 0.3*27.5^-0.7 = 0.02948, approximately, which is quite close to the 0.030 in (4) above.
Set the lower value of k at 27 and the interval at 1. (Doing this requires entering the values rather than using the scroll bar.) Confirm that the interval marginal product is now closer to the slope of the line at k = 27.5.

Over this smaller interval, the interval MP is now 0.029 rather than 0.030 and is, therefore, closer to the slope MP. If you wish see how much closer, change the number of decimals in the spreadsheet to 4 or 5.

(Reset the lower value at 25 and the interval at 5.)

Select two values of k. The lower value of k is __25__ units, and the higher value of k is __35___ units. Confirm each of the following propositions:
Per-worker output, y, does not rise in the same proportion that k increases.
Per-worker consumption and per-worker investment increase in the same proportion as income.

The percentage increase in k is 10/25 or 40 percent.
Output rises from 2.627 to 2.905, or by about 10.6 percent.
Consumption rises from 1.839 to 2.034, or by about 10.6 percent (the calculated percentages are slightly different due to rounding).
Investment rises from 0.788 to 0.872, or by about 10.7 percent (again, rounding causes a slight difference).

Explain the economic meaning of a depreciation function that is a ray (straight line) through the origin.

With this relationship, depreciation is always proportional to the capital stock: A given fraction of that stock wears out each year, and that fraction is independent of the capital stock's size.

Select a value of k that is above the steady state value. Explain the adjustment that occurs within the current period that pushes the economy back toward the steady-state value of k. Also, choose a set of parameter values (s and δ).

We selected a value of k = 10 , s = 0.35, and δ = 0.1. These values imply a steady-state k or about 5.987. At k = 10, depreciation = 1.000 and investment (= saving) equals 0.698. Thus the capital stock decreases by 0.302 units.

How does an increased value of s affect the steady-state value of k? Of the savings level?

Both values increase. The increased saving leads to more investment, causing the steady-state capital level to increase. This, in turn, increases output. Since saving is proportional to output, the saving level increases.

How does an increased value of δ affect the steady-state value of k? Of the savings level?

Both values decrease. The increased depreciation rate causes the steady-state capital level to decrease. This, in turn, decreases output. Since saving is proportional to output, the saving level decreases.

Describe the behavior of each of these values if the initial value of k is below the steady state value: k, i, dep.
Whenever k is below its steady-state value, investment exceeds depreciation. As a result, k grows. Because of the increasing value of k, levels of output, saving (= investment), and depreciation all increase, approaching the steady-state levels.
Describe the behavior of each of these values if the initial value of k is above the steady state value: k, i, dep.
Whenever k is above its steady-state value, investment is less than depreciation. As a result, k decreases. Because of the decreasing value of k, levels of output, saving (= investment), and depreciation all decrease, approaching the steady-state levels.
In all cases, ∆k approaches what value?
The nature of steady-state is such that i = dep, so ∆k must approach zero--from above when the initial k is less than the steady-state value, and from above otherwise.

Start with the following values: s = 0.3 and δ = 0.1. Leave δ = 0.1 for these exercises.
Reduce s to a value below 0.3. What happens to the equilibrium value of k? of y? of c? Does any of this surprise you?

We selected s = 0.2 The steady state value of k, k*, falls from 4.80 units of capital per worker to 2.69 units. Not surprisingly, y falls, from 1.60 units per worker to 1.35 units. The smaller capital value implies less depreciation. This means that consumption does not fall as much as output. This can be rationalized in either of two ways. Mathematically, c is now 0.8 times y so while y is smaller c is a larger fraction of that smaller number. From another view, y must be allocated either to consumption or to depreciation at a steady state (the capital stock must be maintained), and depreciation has fallen, so c need not fall by as much as y does.

Increase s to a value above 0.3. What happens to the equilibrium value of k? of y? of c? Does any of this surprise you?

We selected s = 0.4. Now k's steady state rises from 4.80 to 7.25. This causes y to increase from 1.60 units to 1.81 units per capita. This is not enough, however, to increase c, which is now only 0.6 times y. As before, depreciation must equal investment (which equals savings). The larger capital stock implies a large enough increase in depreciation that the output gain is not enough to maintain both the consumption level that was possible when s = 0.3 and the capital stock.

Start with the following values: s = 0.3 and δ = 0.1.
Change the depreciation rate to a value other than 0.1. Confirm that changing δ causes k, y, and c to change in the opposite direction. Confirm, too, that s = 0.3 remains the golden rule saving rate.

We increase δ from 0.10 to 0.15. As a result the steady-state value of k falls from 4.80 to 2.69; the value of depreciation falls from 0.48 to 0.40, the output level falls from 1.60 to 1.35 units, and consumption falls from 1.12 to 0.94 units.
Setting the value of s at values above or below 0.30 causes consumption to fall even farther. At s = 0.25, for example, c fall to 0.93 units, as it does at s = 0.35.

Reset δ = 0.1. Click on the "View Table" button, and change the value in cell Y14 to 0.35. This corresponds to an increased productivity of capital.

Given that s = 0.30, determine how this change in the production function affects k, y, and c.
The steady-state values given s = 0.3 change as follows: k increases from 4.80 to 5.42, y increases from 1.60 to 1.81, and consumption increases from 1.12 to 1.26 units.
Confirm that the Golden Rule saving rate is now 0.35.
Increasing s to 35 percent of income results in k rising from 5.42 to 6.87. The resulting higher income causes the sustained consumption level to rise from 1.26 to 1.28 units per year. Any further increase in s will cause c to fall below this level.

The graph shows directly why MPK = depreciation yields a "Golden Rule" outcome. Discuss.
Visual inspection shows that the per-capita consumption level rises until MPK = depreciation, after which it falls.
In the accompanying table, observe that MPK - δ = 0 at the "Golden Rule" steady state. Explain why this is so.
Suppose that MPK > δ. In that case, adding a unit of capital increases output by more than it increases depreciation. That is, it adds to sustainable consumption. Thus, undertaking the investment (increasing k) increases the level of consumption that can be enjoyed perpetually. In contrast if MPK > δ, adding more capital adds more to depreciation than to output, so consumption must fall if the value of k is to be sustained. Indeed, reducing k causes depreciation to fall by more than output does, so sustainable consumption increases.

Select a value of k below the level consistent with the the Golden Rule steady state. The selected value of k is ______. At this value of k, y = _____ and consumption = _____, so saving = _____. This implies that the fraction of income being saving and invested is s = _____.
We selected the following values: a = 0.3, δ = 0.05, and k = 5. Setting a = 0.3 and δ = 0.05 implies that the Golden Rule steady state value of k is k = 12.931. We set k = 5. This value implies y = 1.621 and c = 1.371 (from the Year 0 data in the table). Thus (1.621 - 1.371)/1.621, or about 15%, is the fraction of income being saved. Consumption can rise to about 1.51 units, but this requires raising the saving level (lowering the consumption level) for a few years while the capital stock is being built up.
Discuss the adjustment process if s is raised to the Golden Rule level. If this adjustment involves government policy changes, what barriers to implementation do you see?
For the values that we selected, c does not recover its initial value until Year 56.(View the full table.) Thus, voters are being asked to decrease consumption for over a half-century in return for the promise that consumption will be sustained at a higher level thereafter. Given that people who are currently alive and over 17 years of age are the voting public (and that older people vote in proportionately larger numbers), the political resistance to such a change is likely to be significant.

Confirm the following proposition: In the Solow growth model, zero population growth and a 5% depreciation rate yields the same per-capita consumption pattern as an economy with a 1% growth rate and a 4% depreciation rate.
In both cases, the results are as follows (copied from the workbook):
The steady-state k =        12.931    KPW. At this k*, depreciation &
population growth require i =        0.647    KPW per year. Thus,
the steady-state output is y =        2.155    units per year. The steady-
state consumption level is c =        1.509    units per year.

Select a pair of growth rates, given a saving rate other than the Golden Rule saving rate. The two growth rates considered are n₁ = _____ and n₂ = _____. The saving rate is s = _____. Compare the following for the two growth rates:

Steady-state k,
Steady-state y,
Steady-state c,
Golden-rule k,
Golden-rule y, and
Golden-rule c.

We set n₁ = 2% and n₂ = 4%. Our selected saving rate is s = 0.40.Also, we left δ = 5%.

Steady-state k is 12.061 with 2% population growth versus 8.423 with 4% population growth.
Steady-state y is 2.111 with 2% population growth versus 1.895 with 4% population growth.
Steady-state c is 1.266 with 2% population growth versus 1.137 with 4% population growth.
Golden-rule k is 7.996 with 2% population growth versus 5.584 with 4% population growth.
Golden-rule y is 1.866 with 2% population growth versus 1.675 with 4% population growth.
Golden-rule c. is 1.306 with 2% population growth versus 1.173 with 4% population growth.

Mankiw says (p. 218, 6th edition), "Now, however, because k = K/(L x E), breakeven investment includes three terms: to keep k constant δk is needed to replace depreciating capital, nk is needed to provide capital for new workers, and gk is needed to provide capital for the new "effective workers" created by technological change."

Select a set of parameters, and confirm that the numbers add up the way Mankiw says they must.

We use the following values: δ = 0.05, s = 0.30, n = 0.015, and g = 0.025. The implied steady-state k is about 5.58 units. This value of k implies an output level of 5.58^0.3 = 1.6749, approximately, so investment (= saving) is 0.3 times output, or about 0.5025 units. Depreciation is 0.05k = 0.279 units, the added k needed to accommodate the added number of workers is 0.015k = 0.0837 units, and the added k needed to provide for the new "effective workers" is 0.025k = 0.1395 units. Thus 0.279 + 0.0837 + 0.1395, or 0.5022 units are required (difference at 4^th decimal place due to rounding).

Select a couple of adjacent years in the table and confirm that per-capita consumption grows at a rate equal to the rate of technological change.
We selected these values: δ = 0.03, n = 0.015, g = 0.025, and a = 0.3. Thus, we selected the Golden Rule saving rate. This is not necessary to the analysis. Consider years 4 and 5. C = 0.476 in Year 4 and 0.482 in Year 5. Thus the growth is 0.06/0.482 = 0.12448, which differs from g = 0.025 only due to rounding.

As the text mentions, the efficiency units are selected so that L = 0 and Y = 0 in the initial year. This normalization is useful for showing the rates of growth of Yand C, but it obscures the fact that, to be on this normalized path, that is to achieve a given level of steady-state per-capita consumption at each point on the path, technology must be higher if population grows faster. Select two population growth rates and determine the initial value of E for each. If the technology (E) required to sustain the faster-growing population were available to the slower-growing one, by how much would the initial value of Y increase in the economy with the lower population growth rate?

Given the values above, k must be 8.00. For Y = 1 when K = 1 in the initial year K must equal 4.286 and E must equal 0.536.
Increasing n from 0.015 to 0.025 changes k to 6.61, K to 3.750, and E to 0.536. The faster-growing workforce implies a smaller steady-state value of k. In order to have an inital output of 1 when L = 1, therefore, these workers would have to be more productive.
Suppose that the workers in the first economy were as efficient as those in the second economy must be in order to have Y = 1 when L = 1. Then, their initial year Y would be:
Y = 1*4.286^0.3*0.568^0.7 = 1.0415.
[Y = L^(1-a)E^(1-a)K^a. In the initial year L = 0, so Y = E^(1-a)K^a = E^0.7K^0.3.]

Period	k = K/L	y = Y/L	c = C/L	i = (Y-C)/L	dep	change_k
1	3.000	1.316	0.921	0.395	0.300	0.095
2	3.095	1.326	0.928	0.398	0.309	0.088