Q1. Using the Solow growth model with human capital, derive and demonstrate the golden rule for saving. Describe the behaviour of the economy as it moves towards corresponding steady state growth path. What factors are important as it pertains to assessing the desirability and feasibility of obtaining the golden rule growth path?
This paper targets Golden Rule for saving in Solow growth model, and can solve three following problems: 1. using the Solow model with human capital, derive and demonstrate the golden rule for saving. 2. Describe the behaviour of the economy as it moves on the corresponding steady state growth path. 3. What factors are essential when it comes to assessing the desirability and feasibility of achieving the golden rule growth path?
Using the Solow growth model with human capital, derive and demonstrate the golden rule
for saving. :
Let k be the capital/labour ratio (i. e. capital per capita), y be the resulting per capita output ( y = f(k) ), and s be the savings rate. The steady state is thought as a situation in which per capita output is unchanging, which means that k be constant. This involves that the quantity of saved output be precisely what is required to (1) equip any extra personnel and (2) replace any exhausted capital.
In a reliable state, kt+1=kt=k, therefore: sf(k) = (n + )k, where n is the regular exogenous population growth rate, and d is the continuous exogenous rate of depreciation of capital. Since n and d are constant and f(k) satisfies the Inada conditions, this expression may be read as an equation connecting s and k in steady state: any choice of s implies a distinctive value for k (thus also for y) in steady state. Since consumption is proportional to output ( c = (1 ' s)f(k) ), a selection of value for s implies a distinctive degree of steady state per capita consumption. Out of most possible selections for s, one will produce the highest possible steady state value for c and is named the golden rule savings rate.
To uncover the optimal capital/labour ratio, and so the golden rule savings rate, first note that consumption can be seen as the rest of the output that remains after providing for the investment that maintains steady state: c = f(k) ' (n + )k
Differential calculus methods can identify which steady state value for the capital/labour ratio maximises per capita consumption. The golden rule savings rate is then implied by the connection between s and k in steady state (see above).
The equation describing the evolution of the administrative centre sock per unit of effective labour is given by
Substituting in for the intensive form of the Cobb-Douglas, , yields
On the balaced growth path, k is zero, investment per unit of efffective labor is equal to break-even investment per unit of offective labor therefore k remains constant. Denoting the balanced-growth-path value of k as k*, we have, Rearranging to solve for k* yields
To receive the balanced-growth-path value of output per unit of effective labor, substitute equation (2) in to the intensive form of the production function, y = k‹±;
Consumption per unit of effective labor on the balanced growth path is given by. Subsituting euation (3) into this expression yields
After some straightforward algebraic manipulation, this simplifies to
Equation (6) can be easily interpreted. Consumption per unit of effective labour is equal to output per unit of effective labour, k*‹±, less actual investment per unit of effective labour, which on the balanced growth path is the same as break-even investment per unit of effective labour, (n+.
Now use equation (6) to maximize c* with respect to k*. The first-order condition is given by
Note that equation (7) is merely a particular form which is the overall condition that implicitly defines the golden rule degree of capital per unit of effective labour. Equation (7) has a graphical interpretation: it defines the level of k of which the slope of the intensive form of the production function is add up to the slope of the break-even investment line.
Solving equation (7) for the golden rule degree of k yields
, which simplifies to
With a Cobb-Douglas production function, the saving rate required to reach the golden rule is add up to the elasticity of output with respect to capital or capital's share in output (if capital earns its marginal product).
Describe the behaviour of the economy as it moves into the corresponding steady state growth path:
Normally, we assume that the policymaker can merely choose the economy's steady state and jump there immediately. In cases like this, the policymaker would choose the steady state with highest consumption-the Golden Rule steady state. But if we guess that the economy has already reached a reliable state other than the Golden Rule, then we must consider two cases: the economy might start with more capital than in the Golden Rule steady state, or with less. As it happens that the two cases offer very different problems for policymakers.
(Reduce saving rate)
Considering the case where the economy begins at a reliable state with an increase of capital than it could have in the Golden Rule steady state. In cases like this, the policymaker should pursue policies aimed at reducing the pace of saving in order to reduce the administrative centre stock. Suppose that these policies succeed and this at some point-call it time t0-the saving rate falls to the level that will eventually lead to the Golden Rule steady state. Figure 1 shows the behaviour of the economy as it moves towards the golden rule growth path represented by output, consumption, and investment when the saving rate falls. The decrease in the saving rate causes an immediate increase in consumption and a decrease in investment. Because investment and depreciation were equal in the initial steady state, investment will now be significantly less than depreciation, this means the economy is no more in a reliable state. Gradually, the administrative centre stock falls, leading to reductions in output, consumption, and investment. These variables continue to fall before economy reaches the new steady state. Because were let's assume that the new steady state is the Golden Rule steady state, consumption must be higher than it was before the change in the saving rate, even though output and investment are lower. Set alongside the old steady state; consumption is higher not only in the new steady state but also along the entire way to it. When the administrative centre stock exceeds the Golden Rule level, reducing saving is plainly a good policy, for this increases consumption at every time.
(Increase saving rate)
When the economy begins with less capital than in the Golden Rule steady state, the policymaker must raise the saving rate to reach the Golden Rule. Figure 2 shows what goes on. The upsurge in the saving rate at time t0 causes an instantaneous fall in consumption and a rise in investment. Over time, higher investment causes the capital stock to go up. As capital accumulates, output, consumption, and investment little by little increase, eventually approaching the new steady-state levels. Because the initial steady state was below the Golden Rule, the upsurge in saving eventually leads to a higher degree of consumption than that which prevailed initially.
The upsurge in saving that leads to the Golden Rule steady state eventually raises financial welfare, because the steady-state level of consumption is higher. But achieving that new steady state requires a short amount of reduced consumption. Note the contrast to the case in which the economy starts above the Golden Rule. When the economy begins above the Golden Rule, reaching the Golden Rule produces higher consumption whatsoever points in time. Once the economy commences below the Golden Rule, achieving the Golden Rule requires initially reducing consumption to increase consumption in the foreseeable future.
What factors are essential as it pertains to assessing the desirability and feasibility of achieving the golden rule growth path?
For this question, we firstly discuss the desirability of achieving the golden rule growth path. When deciding whether to attempt to reach the Golden Rule steady state, policymakers have to take into account that current consumers and future individuals are not necessarily the same people. Achieving the Golden Rule achieves the highest steady-state level of consumption and so benets future generations. But when the economy is at first below the Golden Rule, reaching the Golden Rule requires raising investment and so lowering the intake of current generations. Thus, whenever choosing whether to increase capital accumulation, the policymaker faces a trade-off among the list of welfare of different generations. A policymaker who cares more about current generations than about future generations may decide never to pursue policies to attain the Golden Rule steady state. For example, some poor country like Ethiopia have to care more about current generations, since almost all of their consumption limited to basic maintenance, then they can't increase saving rate. By contrast, a policymaker who cares about all generations equally will choose to attain the Golden Rule. Despite the fact that current generations will consume less, an innite quantity of future generations will benet by moving to the Golden Rule.
Thus, optimal capital accumulation depends crucially on how we weigh the interests of current and future generations. The biblical Golden Rule tells us, "do unto others as you'll keep these things do unto you. '' If we heed this advice, we give all generation's equal weight. In cases like this, it is optimal to attain the Golden Rule degree of capital-which is the reason why it is called the "Golden Rule. '' (Mankiw, 2010)
Second, we discuss the feasibility of achieving the golden rule growth path. The main element of this problem is whether policymaker can transform the saving rate effectively by various policies. Thus, we will discuss some policy in several national conditions in this part.
Various monetary policies can have an effect on the savings rate and, given data about whether an economy is saving too much or inadequate, can subsequently be used to approach the Golden Rule level of savings. Consumption taxes, for example, may reduce the degree of consumption and improve the savings rate, whereas capital gains taxes may decrease the savings rate. These policies tend to be known as savings incentives in the west, where it is felt that the prevailing savings rate is "too low" (below the Golden Rule rate), and consumption incentives in countries like Japan where demand is widely regarded as too weak because the savings rate is "too high" (above the Golden Rule).
Japan's high rate of private saving is offset by its high public debt. A simple approximation of the is that the federal government has borrowed 100% of GDP from its citizens backed only with the promise to pay from future taxation. This will not necessarily lead to capital formation through investment (if the revenue from bond sales is allocated to present government consumption rather than infrastructure development). Compared to China's higher rate of private saving, the behaviour of Japan may are based on traditional view or culture, while Chinese folks have to keep higher rate of private saving since the pressure from deficient social security system. Therefore, instead of fiscal policy, building a perfect social security system will bestially release potential consumption. US give a typical example in this respect.
If consumption tax rates are anticipated to be everlasting then it is hard to reconcile the normal hypothesis that rising rates discourage consumption with rational expectations (since the ultimate reason for saving is consumption (Frankel, 1998)). However, consumption taxes tend to vary (e. g. with changes in government or movement between countries), therefore currently high consumption taxes may be expected to disappear completely sooner or later in the future, creating an elevated incentive for saving. Actually, some countries like UK and New Zealand have even increased their consumption taxes. The efficient level of capital income tax in the steady state has been studied in the context of an over-all equilibrium model and Judd (1985) shows that the optimal tax rate is zero. However, Chamley (1986) says that in achieving the steady state (in the short run) a high capital income tax is an effective revenue source.
In economics, the Golden Rule savings rate is the rate of savings which maximizes steady state level or growth of consumption (Phelps, 1966), for example in the Solow growth model. Although the idea can be found earlier in John von Neumann and Maurice Allais's works, the word is generally attributed to Edmund Phelps who wrote in 1961 that the Golden Rule "do unto others as you'll have them do unto you" could be employed inter-generationally inside the model to reach at some form of "optimum".
In the Solow growth model, a reliable state savings rate of 100% implies that all income is going to investment finance for future production, implying a reliable state consumption level of zero. A savings rate of 0% implies that no new investment capital has been created, so that the capital stock depreciates without replacement. This makes a steady state unsustainable except at zero output, which again implies a consumption level of zero. Somewhere among is the "Golden Rule" degree of savings, where in fact the savings propensity is such that per-capita consumption reaches its maximum possible continuous value.
When assessing the desirability and feasibility of obtaining the golden rule growth path, initial monetary situation (whether below or over Golden rule), national culture, and some relational financial model must be looked at.
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