the golden rule level of the capital-labor ratio · 2014. 8. 15. · 2 chapter 6 appendix golden...

1

We have seen that a higher saving rate s leads to a higher level of output per worker. Does this conclusion mean that more saving is always better? Clearly not, since an s of 100% would leave no income left over for buying food or shelter. To identify the saving rate that produces the highest level of economic well-being, we use the Solow model to compare steady states at different saving rates.

Steady States at Different Capital-Labor RatiosMaximizing consumption per worker is a natural choice for maximizing the economic well-being of workers in an economy in the steady state.1 After all, people care more about the amount of goods and services they consume than the amount of capital or output in the economy. Policy makers may try to influence the national saving rate to maximize consumption per worker in the steady state.

But what saving rate achieves this? To answer this question, the policy maker needs to identify the steady-state level of the capital-labor ratio that maximizes consumption per worker, which is known as the Golden Rule capital-labor ratio. The name is a reference to the Bible, which states that “you should do unto others as you would have them do unto you.” In economic terms, policy makers should weigh the well-being of all future generations equally. In this case, you would want all generations to have the same maximum level of consumption, which is achieved when the steady-state capital-labor ratio leads to the highest level of consumption per worker.

Chapter 6 Appendix

The Golden Rule Level of the Capital-Labor Ratio

1There is clearly more to life than just consumption. People also care, for example, about the amount of leisure they have, and we are ignoring such considerations here.

Z03_MISH4317_WEB_CH06AppA_pp001-005.indd 1 18/11/13 5:22 PM

2 Chapter 6 appendix

Golden Rule Capital-Labor RatioTo determine the Golden Rule capital-labor ratio, which we will denote by kG* , we need to compare steady-state levels of consumption per worker, c*, for different capital-labor ratios, k*. We start by recognizing that, since y = c + i, we can write consumption per worker in the steady state as follows:

c* = y* - i* (1)

where the * indicates values at the steady state. In other words, consumption per worker at the steady state is just income minus investment per worker at the steady state. The value of income per worker at the steady state is straightforward: it comes directly from substituting k* into the production function:

y* = Ak*0.3 (2)

At the steady state, the capital stock is not changing. As we saw in Equation 10 in the chapter, investment must be equal to depreciation plus capital dilution, i.e.,

i* = 1δ + n2k* (3)

We plot the steady-state levels of y* and i* against the steady-state level of k* in panel (a) of Figure 6A1.1. Substituting Equations 2 and 3 into Equation 1,

c* = Ak*0.3 - 1δ + n2k* (4)

The difference between y* and i*, which equals the steady-state level of consumption per worker, c*, is shown by the colored area between the y* and i* curves in panel (a) of Figure 6A1.1, and is also shown in panel (b).

Notice in panel (a) that c* is at its maximum value cG* when the slope of the produc-tion function curve, which is the marginal product of capital, is equal to the slope of the depreciation plus capital dilution line, which is δ + n, the depreciation rate plus the rate of population growth. The Golden Rule level of the capital-ratio, kG*, occurs when2

MPK = δ + nMarginal Product of Capital = Depreciation Rate + Population Growth Rate

(5)

To see why the condition in Equation 5 makes sense, let’s consider what happens when k* 6 kG*. In this case, the marginal product of capital is higher than the deprecia-tion rate plus the population growth rate. Adding another unit of k* adds more output, MPK, than the increase in investment, δ + n, so that consumption, c*, must rise. On the

2We can also derive this condition using calculus. From Equation 1, c* = y* – i*. To maximize c*, differentiate this equation with respect to k* and set it to zero. That is, dc*>dk* = dy*>dk* - 1δ + n2 = 0. This equation thus produces the condition dy*>dk* = δ + n. Since dy*/dk* is the marginal product of capital, this condition is the same as in the text. Note that we need to modify this condition when adding productivity growth to the Solow model.


the Golden rule level of the Capital-labor ratio 3

Figure 6A1.1

The golden rule Level of the Capital-Labor ratioIn panel (a), steady-state consumption per worker, c* (shown in panel (b)), is at its maximum value at kG* when the slope of the production function curve, which is the mar-ginal product of capital, equals the slope of the depreciation plus capi-tal dilution line, δ + n.

k*G

Per Worker ProductionFunction, y* = Ak*0.3

slope

= (d

+ n)

Output perWorker, y*

Steady-state level ofconsumption per worker.

Golden Rule level.

cG*

cG*

(a) Solow Diagram

Capital-Labor Ratio, k*

k*G

Consumptionper Worker, c*

Steady-state level ofconsumption per worker.

Golden Rule level.

Capital-Labor Ratio, k*

(b) Steady-state Consumption

cG*

i* = (d + n)k*


4 Chapter 6 appendix

other hand, if k* 7 kG* , MPK 6 δ + n, then adding another unit of k* increases output by MPK, which is less than the increase in investment, δ + n, so that consumption, c*, falls. Consumption per worker in the steady state, c*, therefore reaches a maximum at k* = kG* .

Implications of the Golden Rule Capital-Labor RatioWhat saving rate would a policy maker choose if he or she wants the economy to get to the Golden Rule capital-labor ratio? Figure 6A1.2 provides the answer. The Golden Rule level of the saving rate, sG, is the one that causes the investment function, sGAkt0.3, to intersect with the depreciation plus capital dilution line, 1δ + n2kt, at the Golden Rule level of the capital-labor ratio, kG*. If the saving rate were higher than sG, then the intersection of the investment curve and the depreciation line would be at a steady-state level of k* that would be higher than kG*, and so the steady-state level of consumption per worker would fall. Similarly, if the saving rate were lower than sG, then the intersection of the investment curve and the depreciation line would be at k* 6 kG*, and the steady-state level of consumption per worker would also fall. When the saving rate is at sG, con-sumption per worker is at its highest value.

Figure 6A1.2

The golden rule Level of the Saving rateThe Golden Rule level of the saving rate, sG, leads to an investment function, sGAkt0.3, that intersects with the depreciation plus capital dilution line, 1δ + n2kt, at the Golden Rule level of the capital-labor ratio, kG*.

Capital-Labor Ratio, kt

Investment andDepreciationplus Capital

Dilution

(d + n)kt

sGAkt0.3

yt = Akt*0.3

kG*

i* = dkG*

iG*

cG*

The Golden Rule level occurswhere the investment curveintersects the depreciationline plus capital dilution.


3. The Golden Rule level of the saving rate, sG, is the one that causes the investment function, sGAkt

0.3, to intersect with the depreciation plus capital dilution line, 1δ + n2kt, at the Golden Rule level of the capital-labor ratio, kG* .

SummaRy 1. The Golden Rule level of the capital-labor ratio

is the level of the capital-labor ratio that maxi-mizes consumption per worker in the steady state.

2. The Golden Rule level of the capital-labor ratio occurs when MPK = δ + n, that is, when the marginal product of capital equals the depre-ciation rate plus the population growth rate.

corresponding to each capital-labor ratio (measured as the value of the capital stock per capita). Use it to determine the Golden Rule capital-labor ratio if the depreciation rate is 5% and the population growth rate is 2%.

k $4,500 $5,000 $5,500 $6,000 $6,500

MPK 10% 9% 8% 7% 6%

6. One of the goals of the Obama administra-tion is to develop and encourage the use of new technologies, in particular within the energy industry. What would be the effect of an increase in technology on the Golden Rule capital-labor ratio?

1. What distinguishes the Golden Rule capital-labor ratio from other possible capital-labor ratios? What determines whether the economy will operate at the Golden Rule capital-labor ratio?

2. What would be the consequences for future generations of a saving rate that is lower than the Golden Rule level of the saving rate?

3. What would be the effect on the Golden Rule capital-labor ratio of an increase in the popula-tion growth rate?

4. What would be the effect on the Golden Rule capital-labor ratio of a decrease in the depre-ciation rate?

5. The following table contains informa-tion about the marginal product of capital

REVIEW QuESTIONS aND PROBLEmS

the Golden rule level of the Capital-labor ratio 5


the golden rule level of the capital-labor ratio · 2014. 8. 15. · 2 chapter 6 appendix golden...

Documents