# Macroeconomics and money

### The concept of the Golden Rule value of capital per worker (k*GOLD)

Golden rule rate of savings refers to that savings rate which maximizes consumption growth or steady state level. Although the concept of Golden rule rate of savings can be found in Maurice Allais and John Neumann’s earlier works, it is mostly attributed to Phelps Edmund’s work in 1961. Increased savings would increase capital per labor ratio. The Solow growth model suggests that, a steady state rate of saving of 100% infers that all income earned within a given period will be invested for future production. This implies that the level of consumption is zero. The golden rule stock capital relates to the maximum level of sustained level of permanent consumption.

The level of steady states is determined by the level of savings. The total amount to offset depreciation and maintain a capital-labor ratio constant is the same as total savings. Increase in savings will increase y* and k*, which will in turn lead to an increase in c* (where y* is the marginal output and k* is the marginal product for capital). Increase in savings will also reduce the amount left for consumption c*. k*gold is the value of the state of capital that maximizes C. This means that increased savings from 10% to 15% for instance would increase capital per labor ratio from 1.0 to 1.5.

It is not the tendency of the economy to move in the direction of the golden rule steady state. This is because people have different motives for saving money. Some people prefer to consume their money instead of saving it. Other people on the other hand prefer to save their money or invest in interest bearing bonds or other profitable projects to get future source of income. The different motives of how people handle their money make it difficult for the economy to achieve a golden rule steady state. In order to achieve a golden rule steady state, policymakers are forced to adjust the level of savings. The new adjustment will result to a new steady state that has increased level of consumption. If the level of consumption increases during the golden rule, the level of savings will decrease. Reduced savings in the economy would mean that the level of investments is reduced and may result to a decreased level of employment, which would lead to reduced consumption. Policy makers have to play their role in ensuring that the economy is at a golden rule steady state.

There are several arguments on how golden rule rate of savings was derived. Normally, the capital labor ratio is represented by letter k and y is the per capita output resulting from the combination of capital and labor. Output (y) is normally given as a function of k, that is, y = f (k). The rate of savings is represented by letter s and it measures the amount of income that a person remains with after consumption. Steady state refers to a situation whereby the per capita output y is not changing, implying that k is also not changing, i.e. k is constant. When the level of capital is constant, the amount of money saved will be the same as the amount needed to introduce new capital, and to buy new equipment for new workers. Therefore, in a steady state, f (k) = (n + d) k where d is the constant depreciation rate of capital, and n is the constant rate of population growth. Since the rate of population growth (n) and the rate of depreciation (d) are constants and function of capital f(k) satisfy the condition of the production function curve in achieving economic growth, it is true to say that the function, f(k), connects the rate of savings and the level of capital in a steady state. Since the level of consumption is the same as the total output produced {c = (1 – s) f(k), then choosing the value of savings implies that there is a single level of steady state in per capita consumption.

The golden savings rate rule is whereby an individual chooses the maximum possible value for consumption out of the all the possible choices he has for savings. Policy makers for a country should be able to determine whether the country’s economy is saving too little or too much. Several factors affect the level of consumption and savings in the economy. For instance, if the government imposes a tax on various goods, the prices of these goods will increase. The rate at which these goods are consumed will decrease as compared to when there are no taxes or there are subsidies on those goods. People will prefer to save or invest more of their money and consume less. The reverse is true when there are subsidies to goods consumed in the country. The government may sometimes impose taxes on goods and people do not expect that the prices will fall in the future because of the state of the economy and the level of growth. This will not affect the level of consumption and the rate of savings, making it difficult to compare the relationship between the two (consumptions and savings) after government introduces the tax. That is to say, if the government introduces a tax 15% on basic goods, the level of savings will not increase by 15% because these goods are essential to people, who cannot do without the goods.

### i) Produce a table and a chart that indicates the Steady State value of capital per worker (k*) for this country.

a) Y=Y (K, L)

Output is the function of capital (K) and labor (L).

Y = (KaLb) where a + b = 1 where a and b are constants

a = 0.397 therefore, b = 1 - 0.397 = 0.603

Output Y = K0.397, L0.603

Depreciation = 8% = 0.08 and output saved = 18% = 0.18

Steady state value of capital per worker for the country will be:

1-output saved = output consumed

Output consumed = 1 - 0.18 = 0.82

Depreciation for capital = 0.08 and after depreciation K = 1 - 0.08 = 0.92

0.82 = 0.920.397 L0.603

0.82 = 0.9674L0.603

(0.82/0.9674) = L0.603

0.848 = L0.603

L =

L = 0.761

Capital per labor (K/L) for the first year = 0.92/0.761 = 1.21

Capital per labor after depreciation for the second year = 92%*1.21 = 1.1132

Capital per labor after depreciation for the third year = 92%*1.1132 = 1.024

Capital per labor after depreciation for the fourth year = 92%*1.024 = 0.942

The table and chat below shows the relationship between capital per worker and the value after depreciation of preceding year.

### ii) Produce a table and a chart that indicates the value of k*GOLD for this country

Output Y is a function of both capital (K) and labor (L), that is, Y = KaLb

Savings = investments =18%, therefore, consumption will be 100% - 18% = 82%.

c* = (1-s) f (k*)

Marginal product for capital (MPK) – depreciation = population growth (n)

c* = y*-i*

c* = f(k*) – (depreciation + population)k*

Since there was no population growth, (change in labor)/labor will be one.

Marginal product for capital = (total output/capital)

Output consumed c* = 1 - 0.18 = 0.82

0.82 = f (k*) - (0.08+1)k*

0.09 + 0.82 = k*

K* = 0.91for the first year

K* for the second year is given by 0.91*0.92 = 0.84

K* for the third year is given by 0.84*0.92 = 0.77

K* for the fourth year is given by 0.77*0.92 = 0.71

### iii) Illustrate and comment on the implications for the country of choosing to increase its saving rate from 18% to 25%. If one unit of labor = 1.21 units of capital.

1st depreciation = 8%*1.21 = 0.0968 units of capital

If one unit of labor = 1.21 units of capital

Therefore, how many units of labor will be required for 0.0968 units of capital?

(0.0968*1)/1.21= 0.08 units for labor

Y = (K0.397 L0.603)

Output consumed = 1 - 0.25 = 0.75

0.75 = (K0.397 L0.603), but K = 0.92

0.75 = 0.920.397 L0.603

L0.603 = 0.75/0.9674 = 0.7753

L = = 0.5309

Capital per labor ratio K/L = 0.9674/0.5309 = 1.822

Increased savings from 18% to 25% would increase capital per labor ratio from 1.21 to 1.822. In other words, in the end, the country has increased amount of income and capital per worker. If savings would have reduced, let us say from 25% to 18%, the capital labor ratio would have reduced from 1.822 to 1.21. This means that, in the end, the country would have a decreased level of income and capital per worker. It is true to say that capital labor ratio is mainly affected by the rate of savings.