Only love money OLG mining

1. Over lapping generation models (OLG or og models) was first created by Samuelson. Aiming at the defect that the theory of monetary economy lacks micro foundation, it studies why people hold money (without intrinsic value) from the perspective of the value storage function of money Under what conditions money has positive value (price), and on this basis (the so-called "micro basis") to establish a theoretical model of monetary analysis. After the establishment of the model, many people responded to it and made a lot of amendments on the basis of the original model. It has become one of the most commonly used models in macroeconomics, especially in monetary economics

in the simplest og model, each person only survives two periods (people born at t time become old people at t + 1 time). There is no proction in the economy, and personal consumption is entirely supported by natural endowments. Young people have the endowment of one unit, while old people have no endowment. If there is a person born at time t and the population growth rate is n, then through proper centralization, there will be

first assume that all endowments are perishable goods, and then relax this restriction. In any period of time, if the commodities of the whole society are distributed to young people, each young person can consume one unit; If the whole part is allocated to the elderly, each elderly person can consume 10 N units. The possibility of social consumption in t period is represented by ab segment in Figure 1, where: represents the consumption of the elderly (x = 2) or young people (x = 1) at Y (y = t, t + 1). Accordingly, the consumption possibility of an indivial in his whole life is represented by the AB segment in Figure 2< According to the nature of indifference curve, the choice of indivial utility maximization should be the point between ab (consume a little when young and old), as shown in point C in Figure 2. However, because endowment cannot be stored, point C cannot be achieved by saving when young and dissaving when old. If there is no money in the economy, indivials can not reach point C by trading. The reason is very simple: young people can only trade with old people, but when they get old, the old people they used to trade with no longer exist. Therefore, in the barter economy, it is impossible to maximize the utility through intergenerational exchange. The only choice for young people is point A. they consume 1 unit of goods when they are young, but not when they are old

now we introce government notes into the og model. Suppose that at time 0, the government gives the elderly h banknotes. If every generation after that believes in the purchasing power of banknotes, then T & gt; The problem of indivial utility maximization born at 0 is:

(1)

S.T. (2)

. The first order condition of maximization is obtained:

(3)

equation (3) implies the function of money demand (the function of money demand expressed by the rate of return of money expressed by deflation):

(4)


thus, the problem of indivial utility maximization is solved, The condition to reach the monetary equilibrium at time t is that the young people absorb all the money of the old people, that is:

(5)

record the deflation rate as, consider time t and T + 1, and use equation (4) and equation (5), If there is:

(6)

then there must be:

G = n (7)

that is, when the amount of money is constant, the deflation rate must be equal to the population growth rate. At this time, the paper money has positive value, and the personal utility reaches the maximum expressed by point C

next, we relax the hypothesis that all commodities are perishable crystals, and study the endowments can be stored, and the storage yield is R & gt- 1 (i.e. storage, but at cost)

if R & lt; n. Then in the barter economy, the consumption possibility of the society at time t and the indivial in the whole life cycle are respectively represented by AB and Ag segments in Figure 3:

it is easy to know that the indivial utility can not exceed the maximum value in the monetary economy represented by point C, so the introction of money can improve the welfare level of all people. But if R & gt; n. Then the barter economy itself has reached Pareto optimality, and money has no existence value (which sets an upper limit on the income of stored money)

Figure 3

we can further relax the assumption that the quantity of money is always h, and investigate the situation that the growth rate of the quantity of nominal money is. Using the same method as before, we can get the following results:

(8)

similarly, the condition of currency equilibrium is G & gt; r. That is:

(9)

equation (9) shows that for the existence of Monetary Equilibrium, the monetary growth rate cannot be too fast< In the beginning of this chapter, the core problem of money growth model is to study the relationship between money growth, inflation and output. In the original overlapping generations model, money does not provide any service that can enter the consumption and utility function except as the medium of intergenerational exchange. Unless there is considerable deflation (it is the rate of return of holding money, which must be higher than other assets), people will not choose to hold money. Therefore, this model is not suitable for analyzing the relationship between inflation and capital accumulation. To study the effect of intergenerational relationship on monetary growth,

it must be improved< The model assumes that: (1) population growth is not taken into account< (2) personal life is divided into two stages< (3) an indivial only works when he is young and gets a salary of W. in addition to current consumption, he saves part of his income m in the form of money and part of his income K in the form of capital. Capital has a rate of return of R. the rate of return of holding money does not mean income, so his real income is expressed by inflation rate< Only young people increase their money and capital holdings. Once they enter the old age stage, they will exchange the money and capital they saved for consumer goods. Holding money only has utility in youth, and the only function of money in old age is as a means of transaction and no longer enters into utility function

5. All the government's income from seigniorage (inflation tax) is transferred to the elderly, set as X

2. Model construction

the indivial welfare function or utility function can be written as:

W = w () (10)

where: and (I = 1, 2) represent the consumption and the actual monetary balance held in two stages of life<

for young people, the following welfare functions need to be maximized:

(11)

indivial consumption in the first and second stages of their life is:

= w-k-m (12)

= K (1 + R) + m / (1 +) + X (13)

the first order condition for maximizing welfare function is:

(14)


(15)


where: represents indivial time preference. The meaning of condition (15) is that in the first stage, the marginal utility of using income for consumption is equal to the marginal utility of saving in the form of money and using it for consumption in the second stage. It is a modification of the condition in strawski model

3. The inflation effect in the overlapping generation model

next, we further investigate the effect of inflation on capital accumulation. Under the equilibrium condition, the real interest rate (marginal return of capital) and wages (assuming that output is only distributed between wages and capital gains, so output minus capital gains is wages) are determined by:

(16)

(17)


note that f (k) in the above formula is actually the per capita output of young people, assuming that only young people work. For the sake of simplification, it is still assumed that there is no population growth, and the number of young people is equal to the number of old people, then the inflation rate is equal to the money growth rate, and the seigniorage per capita of young people () is equal to the transfer payment per capita to the old people (x). In the equilibrium state, the steady-state consumption path is:

(18)

(19)


the original utility maximization first-order conditional equations (14) and (15) become:

(20)

(21)


by direct comparison with equation (18) - equation (20), the static state shows that as long as the steady-state conditions of the model are satisfied, there is Dk / D & gt; 0 That is to say, in the steady state, the rising inflation rate will make the steady-state capital density rise, and the monetary non neutrality will be established< However, this conclusion is very sensitive to the assumptions of the model. If the above assumptions are modified, the conclusion may change, Now let's examine the impact of the amendment to the fifth assumption (that is, seigniorage is fully transferred to the elderly)

if all seigniorage () is transferred to young people, then equations (18) and (19) become:

(22)

(23)

now the effect of inflation on steady-state capital is determined by equation (20) - equation (23). The comparative static analysis shows that under this condition, the steady-state per capita capital is less than that when seigniorage tax is transferred to the elderly, and the consumption path is smoother. However, the steady-state effect of inflation on Capital Holdings is still positive (Dk / D & gt; 0) < The former models assume that money only provides utility in the first stage of the life cycle and only serves as a medium of transaction in the second stage, which is of course unrealistic. Drazen's model modifies this by assuming that money provides utility at both stages of life, so that both the elderly and the young hold real balances. Under this condition, if seigniorage is transferred to young people, the rising inflation rate will lead to the growth of young people's demand for capital, but the rising inflation rate will also increase the seigniorage collected by the government. If seigniorage is paid to the elderly, the income in the second stage will increase, which will rece the savings in the first stage and the capital demand. However, even in the Drazen model, the seigniorage transfer effect is not strong enough to reverse the relationship between inflation and capital accumulation under steady-state conditions, and the Tobin effect still holds

defects of three-stage and two-stage OLG model

in the framework of utility maximization, the two-stage OLG model amends the strawsky model by introcing the overlapping generations problem. However, its conclusion supports Tobin effect, which greatly encourages the monetary theorists who believe in the non neutrality of money. However, we can also see from a large number of assumptions that this model has many shortcomings, most of which are not in line with people's actual experience. The most obvious thing is that the division of the two stages of the life cycle is too rigid. In real life, even if the young and old stages can be divided, each stage still contains a period of perhaps decades. In these decades, it is simply assumed that "young people" work, "old people" do not work, "young people" accumulate money and capital, "young people" accumulate money and capital, and "old people" do not work It is unreasonable for the "old people" to consume only the accumulated money and capital

there are quite a lot of arguments about whether these defects can be corrected, but experience tells us that the more appropriate the model is