# Golden Rule savings rate

Golden Rule savings rate

In economics, the Golden Rule savings rate is the rate of savings which maximizes steady state growth consumption in the Solow growth model. It is a term attributedref|PhelpsAttribution to Edmund Phelps who wrote in 1961 that the Golden Rule "do unto others as you would have them do unto you" could be applied inter-generationally inside the model to arrive at some form of "optimum".

In the Solow growth model, a savings rate of 100% implies that all income is going to investment capital for future production, leading to a steady state consumption level of zero. Similarly, a savings rate of 0% implies that no new investment capital is being created, leading to all current capital depreciating without replacement and again a consumption level of zero. Somewhere in between is the "Golden Rule" level of savings, where the savings propensity is set so that per-capita consumption itself is both constant and maximized.

Derivation of the Golden Rule savings rate

If the other parameters of the Solow model are taken as 'given' (beyond the policy-maker's control), but the savings rate can be set exogenously to maximize steady state consumption the solution can be derived from the steady state equations.

Per capita consumption ("c") is the difference between output per capita ("y") and savings per capita:

:$c=y-i$

where "s" is the propensity to save, "c" = per-capita consumption and "k" = the capital/labour ratio (i.e. capital per capita). In the steady state:

:$frac\left\{ partial k\right\}\left\{ partial t\right\} = 0$

Therefore:

$s = frac\left\{ nk \right\}\left\{ y* \right\}$

Where "n" is the constant exogenous population growth rate, and "y*" is the steady state output per worker at the golden rule savings rate.

Measuring the Golden Rule savings rate

Since the Golden Rule represents the optimal savings rate to maximize consumption in the economy, an important question for policy-makers is whether the economy is saving too much or too little. The natural question becomes: "How much capital (k) is needed to achieve the maximum level of consumption per worker in the steady state?"

The explicit solution for the golden rule rate is obtained by differentiating the consumption function: $c = f\left(k\right) - \left(n+d\right)k$with respect to k (capital).

$Savings at golden rule rate = \left(n+d\right)k*/zf\left(k*\right)$

The model implies that if the amount of capital is not at the steady state level, capital levels will adjust to the steady state level. So:

#If k < k* , then sf(k) > (n+d)k and k will rise until it reaches k*
#If k > k*, then sf(k) < (n+d)k and k will fall until it reaches k* [cite book | last = Abel| first = Andrew B.
coauthors= Ben S. Bernake, Gregor W. Smith & Ronald D. Kneebone
title = Macroeconomics, Fourth Canadian Edition
location = Toronto
date = 2005
isbn = 0321306627
]

Policy that can change the savings rate

Various economic policies can have an effect on the savings rate and, given data about whether an economy is saving too much or too little, can in turn be used to approach the Golden Rule level of savings. Consumption taxes, for example, may reduce the level of consumption and increase the savings rate, whereas capital gains taxes may reduce the savings rate. These policies are often known as savings incentives in the west, where it is felt that that the prevailing savings rate is "too low" (below the Golden Rule rate), and consumption incentives in countries like Japan where demand is widely considered to be too weak because the savings rate is "too high" (above the Golden Rule)ref|SovietPolicy.

Private and public saving

Japan's high rate of private saving is offset by its high public debt. A simple approximation of this is that the government has borrowed 100% of GDP from its own citizens backed only with the promise to pay from future taxation. This does not necessarily lead to capital formation through investment (if the revenue from bond sales is spent on present government consumption rather than infrastructure development, say).

Golden rule taxes within economic models

If consumption tax rates are expected to be permanent then it is hard to reconcile the common hypothesis that rising rates discourage consumption with rational expectations (since the ultimate purpose of saving is consumptionref|Frankel1998). However, consumption taxes tend to vary (e.g. with changes in government or movement between countries), and so currently high consumption taxes may be expected to go away at some point in the future, creating an increased incentive for saving. The efficient level of capital income tax in the steady state has been studied in the context of a general equilibrium model and Judd (1985) has shown that the optimal tax rate is zero.ref|Judd1985. However, Chamley (1986) says that in reaching the steady state (in the short run) a high capital income tax is an efficient revenue source.ref|Chamley1986

References

Notes

[http://cepa.newschool.edu/het/essays/growth/optimal/goldengr.htm Origin of the term described at newschool.edu]

Since the golden rule applies only in the steady state an economy not in that state "should not" aspire to the golden rule savings rate, even if the precepts of neo-classical economics growth theory are accepted. For example, the Soviet Union had a famously high savings rate policy in an attempt to "catch up" to the West, the fact that this lowered present consumption below the golden rule rate was justified with the argument that capital accumulation was necessary to reach the world level of industrialization, but that this was a short-term policy of capital deepening.

Frankel, D.M (1998), Transitional Dynamics of Optimal Capital Taxation "Macroeconomic Dynamics", 2, page 493. ( [http://www.econ.iastate.edu/faculty/frankel/tax.pdf David Frankel writes] that a wage tax is the "perfect tool" for influencing the quantity of leisure consumption. Page 495 describes the problem of failing to make government commitment to a tax rate credible).

Chamley, C. (1986), Optimal taxation of capital income in general equilibrium with infinite lives. "Econometrica 54". Chamley writes that before reaching the golden rule steady state capital income taxes are efficient in the sense that they do not promote deadweight loss through intertemporal consumption substitution.

Judd, K.L. (1985), Redistributive taxation in a simple perfect foresight model." Journal of Public Economics" 28, page 59.

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