Use a calculator or spreadsheet to confirm each of the following (do not just copy the formulas from the workbook; follow the logic of the model to enter the formulas):

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.)

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?

Have the workbook evaluate the marginal product over these ranges:

1. k = 10 to k = 15
2. k = 15 to k = 20
3. k = 20 to k = 25
4. k = 25 to k = 30
5. k = 30 to k = 35

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.  

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.
2. Set the lower value of k at 27 and the interval at 1.  (Dong 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.
   (Reset the lower value at 25 and the interval at 5.)

Open the "MarginalProduct" workbook.  This workbook is the same as the preceding one, except that it shows marginal product as the slope of the tangent.
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Select two values of k. The lower value of k is _______ units, and the higher value of k is _______ 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.

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

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).

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

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

Describe the behavior of each of these values if the initial value of k is below the steady state value: k, i, dep.
Describe the behavior of each of these values if the initial value of k is above the steady state value: k, i, dep.
In all cases, ∆k approaches what value?

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?
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?

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.
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.

Return to Cell Y14 and reset a = 0.3.

The graph shows directly why s = δ  yields a "Golden Rule" outcome. Discuss.
In the accompanying table, observe that MPK - δ = 0 at the "Golden Rule" steady state. Explain why this is so.

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 = _____.

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?

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.
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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.

Mankiw says (p. 218, 5th 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.

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.

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?