Golden Rule savings rate
The golden rule of accumulation of Edmund S. Phelps says that the consumption per capita or per worker is maximized when the interest rate is equal to the growth rate of gross domestic product. The derivation of this rule is based on a number of simplifying assumptions. The rule has also been developed, such as by time preferences of consumers are taken into account ( Ramsey rule ).
The interest rate obtained with the help of the Golden Rule could then be used as " equal -weight real interest " in the Taylor rule for the determination of the Taylor interest rate.
- 6.1 Profit rate equal to investment ratio
- 6.2 interest rate equal to growth rate
Assumptions
For the derivation of the rule, some assumptions are made:
Under all these assumptions, a mathematical relationship (function) formulate, after which the consumption per worker is C / A is determined by the capital per worker K / A ( denoted k ).
This C / A is to be maximized by derived according to the usual mathematical method according to k, and the result is set equal to zero ( the first derivative is set equal to zero to find the end point ).
In particular:
Steady -state growth rate
The growth of the capital stock is equal to the investment I, which are in turn financed by the savings of S:
Savings rate:
The consumption function:
Capital intensity:
Per capita production:
Linear homogeneous production function
Thus, the production function can also be expressed in per capita variables. The output per worker depends on the capital per worker ( capital intensity ):
The growth rate of population / employment A is given exogenously:
Steady-state growth rate, all sizes are expected to grow at the same rate:
Maximization of per capita consumption
At what steady-state growth rate of consumption per capita is
Maximized?
According to steady state following applies:
So:
Maximization of per capita consumption with respect to k is derived by k and set equal to zero:
Golden rule of accumulation
The marginal productivity of capital must therefore be equal to n the growth rate. Neoclassical is assumed that the marginal productivity of capital is equal to the price for the use of capital, ie equal to the rate of profit and the interest rate.
In addition to accounting for the marginal productivity of capital
Marginal productivity of capital as the partial derivative of F ( K, A ) to K:
Line Rare homogeneity:
Partial invoice ( with application of the chain rule):
So overall:
Empirical verification
Profit share equal to investment ratio
The golden rule of accumulation therefore states that in the optimal case, the interest rate should be equal to the steady-state growth rate, thus in particular:
Multiplying the left and right with K and dividing by Y, then we have:
In which
Is.
Links of the equation is thus the profit ratio ( profit share, the share of profits in GDP) and on the right is the investment rate, the share of investment in GDP. Whether both are actually the same, can be verified empirically.
If the investment ( the same saving rate, including depreciation ) is higher than the profit ratio ( including depreciation, ie equal to the cash flow ratio ), then there is (according to Lutz Arnold S. 54) " dynamic inefficiency " before. According to Arnold studies eligible for the OECD countries to the conclusion that it was the other way around, rather the profit share was higher than the rate of investment.
The profit ratio was present at about one-third, the investment rate at about 10% to 20 %. The profit share is inclusive of Taxes. An IMF study concludes that the global savings and investment rate, measured by the gross domestic product has declined by about 24 % in 1970 to about 22 % in 2004.
While this result is so " dynamic inefficiency " refuted according to Arnold, is on the other hand, the question whether the rule " The profits of today are the investments of tomorrow ( and the jobs of tomorrow) ," the so-called GIB formula that is true if the investment rate is actually significantly lower than the profit share, or show how other investigations, behind the profit rate ever remains ( plant emergency ). In the figure, the gross fixed capital formation as a percentage of overall economic "Cash Flow", shown here profits before tax plus depreciation. The marked propensity returns to the countries of the Triad USA, Japan and Germany tend to be.
Interest rate equal to growth rate
The golden rule can also be checked directly by comparing the growth rate of GDP at the interest rate. In the figure, the difference is formed between GDP growth rate and long-term interest rate. Accordingly, the long -term interest rates since the 1980s were more likely to be low until the 1970s, rather too high. Since the 1980s, so the income of the production factor capital were measured by the golden rule rather too high.