Chapter 6
Appendix
The Golden Rule Level of
the Capital-Labor Ratio
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 Ratios
Maximizing 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 capitallabor ratio leads to the highest level of consumption per worker.
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
1 18/11/13 5:22 PM
2 Chapter 6 appendix
Golden Rule Capital-Labor Ratio
To determine the Golden Rule capital-labor ratio, which we will denote by k*G, 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 + n 2k*(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 + n 2k*(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 production 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
+
n
(5)
Marginal Product of Capital = Depreciation Rate + Population Growth Rate
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 depreciation 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 + n 2 = 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.
Z03_MISH4317_WEB_CH06AppA_pp001-005.indd 2
18/11/13 5:22 PM
The Golden Rule Level of the Capital-Labor Ratio 3 Figure 6A1.1
The Golden Rule
Level of the
Capital-Labor Ratio
(a) Solow Diagram
Output per
Worker, y*
In panel (a), steadystate 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 marginal product of capital,
equals the slope of the
depreciation plus capital dilution line, δ + n.
i* = (d + n)k*
Steady-state level of
consumption per worker.
(
e=
d+
n)
p
s lo
Per Worker Production
Function, y* = Ak*0.3
cG*
Golden Rule level.
k*G
Capital-Labor Ratio, k*
(b) Steady-state Consumption
Consumption
per Worker, c*
Steady-state level of
consumption per worker.
cG*
cG*
k*G
Golden Rule level.
Capital-Labor Ratio, k*
Z03_MISH4317_WEB_CH06AppA_pp001-005.indd 3
18/11/13 5:22 PM
4 Chapter 6 appendix
other hand, if k* 7 k*G, 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* = k*G.
Implications of the Golden Rule Capital-Labor Ratio
What 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, sGAk0.3
t , to intersect with
the depreciation plus capital dilution line, 1δ + n 2kt, 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, consumption per worker is at its highest value.
Figure 6A1.2
The Golden Rule
Level of the
Saving Rate
Investment and
Depreciation
plus Capital
Dilution
The Golden Rule level
of the saving rate, sG ,
leads to an investment
function, sGAk0.3
t ,
that intersects with
the depreciation plus
capital dilution line,
1δ + n 2kt , at the
Golden Rule level of the
capital-labor ratio, kG*.
yt = Akt*0.3
cG*
The Golden Rule level occurs
where the investment curve
intersects the depreciation
line plus capital dilution.
(d + n)kt
i* = dkG*
sGAkt0.3
iG*
kG*
Capital-Labor Ratio, kt
Z03_MISH4317_WEB_CH06AppA_pp001-005.indd 4
18/11/13 5:22 PM
The Golden Rule Level of the Capital-Labor Ratio 5 Summary
1.The Golden Rule level of the capital-labor ratio
is the level of the capital-labor ratio that maximizes 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 depreciation rate plus the population growth rate.
3.The Golden Rule level of the saving rate, sG, is
the one that causes the investment function,
sGAk0.3
t , to intersect with the depreciation plus
capital dilution line, 1δ + n 2kt, at the Golden
Rule level of the capital-labor ratio, k*G.
REVIEW QUESTIONS AND PROBLEMS
1.What distinguishes the Golden Rule capitallabor 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 population growth rate?
4.What would be the effect on the Golden Rule
capital-labor ratio of a decrease in the depreciation rate?
5.The following table contains information about the marginal product of capital
Z03_MISH4317_WEB_CH06AppA_pp001-005.indd 5
c­ orresponding 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
MPK 10%
$5,000
$5,500
$6,000
$6,500
9%
8%
7%
6%
6.One of the goals of the Obama administration 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?
18/11/13 5:22 PM