Fiscal Policy and Monetary Policy (Part 1)

Fiscal Policy and Monetary Policy (Part 1) – Pan Deng’s Notes on Macroeconomics

China's fiscal situation

Taxes can be roughly divided into two categories

• Direct Taxes: Taxes levied directly on businesses and individuals
  • income tax
  • property tax
• Indirect Taxes: Taxes on Goods and Services
  • VAT
  • tariff

Compared with direct taxes, indirect taxes are easier to collect; therefore, in developing countries, indirect taxes account for a larger proportion of the total tax revenue. But indirect tax is a regressive tax system, which is not conducive to adjusting income distribution. In developed countries, general direct taxes are the bulk;

However, the public finance caliber does not cover all government activities in China. In addition to public finance, the Chinese government also has government funds (including land transfer fees), social insurance funds, and state-owned capital management, which correspond to revenue and expenditure activities.

• Government funds: The Budget Law of the People's Republic of China stipulates: "The budget of government funds refers to the funds collected, received or raised from specific objects within a certain period of time in accordance with the provisions of laws and administrative regulations, which are earmarked for specific public undertakings. Development revenue and expenditure budget.” According to this regulation, government funds are collected from specific objects in accordance with laws and regulations, and are earmarked for specific public utilities. For example, the Three Gorges Reservoir Fund, lottery public welfare funds, and cultural undertaking construction fees are all government funds.

  • The largest item of income in government funds is land transfer income. According to China's current land system, all land used for urban construction is owned by the state, while rural land is owned by village collectives. However, the law stipulates that commercial housing can only be built on urban construction land. If rural land is to be converted into land for urban construction, it must be sold to the government through the process of "land expropriation", and then the government will "modify" the nature of the land.

• Social insurance fund: According to the provisions of the "Social Insurance Law of the People's Republic of China", social insurance funds include basic pension insurance funds, basic medical insurance funds, work-related injury insurance funds, unemployment insurance funds and maternity insurance funds. The "National Social Security Fund Regulations" stipulates: "The National Social Security Fund is the national social security reserve fund", and the income of the social insurance fund mainly comes from the contributions of our employees. In 2016, the total income of social insurance funds accounted for 6.5% of my country's GDP.

• State-owned capital operating income: From its name, we can know that this item is the income created by operating state-owned capital. The income from state-owned capital operation mainly includes the profits handed over by state-owned enterprises, bonuses and dividend income from state-owned shares, and income from the transfer of state-owned capital property rights. In 2016, the government's total income from state-owned capital operations was about 260.9 billion yuan, accounting for 0.4% of China's GDP that year.

China's fiscal surplus

Automatic fiscal stabilizer : When the economy is good, there are more taxes, fiscal surpluses, and less fiscal expenditures; when the economy is down, unemployment subsidies increase, taxes decrease, fiscal deficits, and fiscal expenditures increase;

Multiplier effect or Ricardian equivalence

Keynes' "multiplier effect"

• Fiscal multiplier (fiscal multiplier): the ratio of changes in total social demand to changes in fiscal expenditures
  • The expansion of government fiscal spending has given unemployed workers jobs and income; these people spend their income again, allowing more people to have jobs and income...
  • The fiscal expenditure of 1 yuan by the government can drive the demand expansion of more than 1 yuan in the whole society - the multiplier effect
• Tax multiplier: the ratio of changes in aggregate demand to changes in taxes
  • Government tax cuts increase the disposable income of residents, which will increase residents' expenditure; the increase in residents' expenditure can increase the income of more people...
  • The tax cut of 1 yuan by the government may drive the demand expansion of more than 1 yuan in the whole society - the multiplier effect

"The Ministry of Finance can fill old bottles with banknotes, and then bury these old bottles in disused coal mines at an appropriate depth, then fill the coal mines with garbage, and then lease the mining rights of the banknote-producing areas to private companies. Dig up these banknotes again—if we can do this, the problem of unemployment will disappear;
moreover, the actual income and capital wealth of the society will probably be much greater than they are now. Of course, it is more reasonable to build large-scale construction projects. If there are political or practical difficulties which prevent the government from doing so, the above measures are better than nothing."
- John Maynard Keynes, The General Theory, 10(VI) 1936

Ricardian Equivalence

• government budget constraints
  • Single-period budget constraint: current government expenditure = current government revenue + government bond issuance
  • Multi-period budget constraint (government debt not defaulting): Discounted sum of government expenditure = discounted sum of government revenue
• Consequences of government tax cuts (bond issuance to finance spending)
  • Given the discounted sum of fiscal expenditures, the government's current tax cuts will inevitably lead to future tax increases (to ensure the discounted sum of government revenue)
  • Residents will expect that although current residents' income will increase due to tax cuts, future income will decrease due to tax increases
  • So residents will save the increased income from tax cuts to pay for future tax increases
• The driving effect of tax cuts on aggregate demand is 0 (the tax multiplier is 0) - Ricardian Equivalence

To judge whether it is Keynesian multiplier effect or Ricardian equivalence, the most direct way is to estimate the actual financial multiplier. In the actual research, the estimated value of fiscal multiplier is not 0 significantly, and quite a lot of estimated results exceed 1; this shows that the Ricardian equivalence generally does not hold in reality;

But this result does not mean that Keynes's theory is invulnerable. In Keynes's multiplier theory, the key to determining the fiscal multiplier is the consumer's marginal propensity to consume $c_1$, but after the rational expectations revolution, especially Friedman put forward the concept of permanent income;

However, Keynes's views cannot be rejected just because the theory is weak. Solo famously said

I remember reading that people still don't fully understand how a giraffe pumps enough blood all the way up to its head. However, it's hard to imagine anyone would conclude from this that giraffes don't have long necks. At least those who have been to the zoo don't think so.

"Multiplier Effect" or "Crowding Out"——Inspiration from "Broken Window Theory"

• The economy is in a state of excess capacity (insufficient demand), and fiscal spending has a multiplier effect (fiscal multiplier is greater than 0)
• The economy is in a state of full use of production capacity, and fiscal expenditure crowds out private expenditure (fiscal multiplier is 0)

Reasons why Ricardian equivalence does not hold

• Residents do not necessarily increase their future tax expectations because of current tax cuts
• Residents Face Mobility Constraints——Tax Cuts Relax Residents' Mobility Constraints

Monetary Policy

Prior to this, our analysis revolved around real economic variables. That is, in the model presented earlier, there is no money, and there are no nominal prices in money (but there are real prices in terms of consumer goods). Without introducing currency, we can already analyze a series of valuable issues such as economic growth, economic structure, consumption and saving decisions, and fiscal policy. Discussing these issues in a currency-free model allows us to bypass the interference of currency and see the essence of these issues. For example, some people see that residents' savings are in the form of bank deposits and cash in reality, so they mistakenly believe that it is monetary policy that determines the amount of savings. This is the mistake of failing to recognize the nature of saving – that saving is the exchange of future consumption at the expense of current consumption. In the no-money model, we can avoid such mistakes by clarifying the nature of savings first.

The Condition of the Chinese Currency

Monetary aggregates in three calibers

• M0 = cash in circulation
• Narrow money: M1 = M0 + corporate demand deposits
• Broad money: M2 = M1 + time deposits of enterprises + time deposits of residents

Trading equation
$$MV=PY$$
$PY$ is understood as nominal GDP,$M$ is M2 (or M1). In this way, the reciprocal of the velocity of money$\frac{1}{V}$It is equal to $\frac{M2}{GDP}$. We can observe that China's $\frac{M2}{GDP}$The proportion continues to increase;

Analyzing Currencies with the Ramsey Model

The Euler condition for intertemporal optimization of previous residents is
$$u^{\prime }(c_1)=u{\_}^{\prime }(c_2)(1+r_2^i)$$
$r_2^i$Indicates the return rate of the i-th asset in the second period. The above formula should be true for all assets, including currencies. So at equilibrium, all assets have the same rate of return. But the rate of return on currency is 1, which is strictly lower than the rate of return on capital, so rational residents should not hold currency. This shows that currency, as an asset, must have some other benefits, so people are willing to endure its lower rate of return and hold it.

This benefit is naturally the convenience brought by currency. In the primitive society of bartering, the occurrence of transactions requires the double coupling of needs (double coincidence of wants). Both parties to the transaction happen to need the goods in the hands of the other party. Obviously, this is very difficult to come across. Later, everyone gradually discovered that there are some goods that everyone needs. Therefore, even if you don't need it, it's best to replace what you have with these goods first. Because when you see what you need in the future, you can easily exchange these goods. These goods that everyone needs are the predecessors of money, and gradually evolved into paper money in modern society. In macroeconomic models, two modeling techniques, CIA or MIU, are usually used to reflect the convenience of money in the model.

1. Cash in advance (CIA)
   The so-called cash in advance (CIA for short) is to require consumers to pay in currency when purchasing goods (usually consumer goods). Therefore, the total amount of consumer goods purchased by consumers is constrained by their money holdings
   $$Pc\le M$$
   where c is the real amount of consumption, P is the price of consumer goods, and M is money holdings. When money is introduced into the model, the convention is to uselowercase letters for real variables(variables that use consumption goods as the unit of measurement) anduppercase letters fornominal variables (variables that use currency as the unit of measurement). With the addition of CIA constraints, consumers naturally have an incentive to accumulate money.
2. 
   Another more commonly used method of money entering the utility function (MIU ) is to let money directly enter the utility function (money in the utility, referred to as MIU), that is, it is assumed that money holdings can directly bring utility. Money holdings in the utility function are real money stocks deflated by nominal prices. Correspondingly, the consumer's utility function can be written as
   $$u(c,\frac{M}{P})$$
   When talking about economic methodology, we once said that rationality is the cornerstone of economic analysis. The utility function represents the preferences of rational people, and naturally directly determines the results of economic analysis. Therefore, economic analysis generally does not modify the utility function, but uses only a few common forms. The reason is simply that we can explain everything we want by simply choosing a different utility function. But that doesn't really explain anything. Therefore, any change in the form of the utility function must have a very reasonable explanation. Naturally, there is also a very reasonable story behind MIU.

MIU can be explained by "money is time". Usually we assume that consumers prefer both consumption goods and leisure. When there is no currency, consumers can only barter, and the transaction is very inconvenient. Only when the double coupling of demand occurs, the barter transaction can occur. Holding the currency can reduce the time to search for transactions, thereby increasing the leisure of consumers and improving its utility. In this way, the holding of currency can produce utility. MIU is widely used in macro models because it is relatively simple to deal with. We will also use MIU next to introduce currency into the model.

Money Economy under Flexible Prices

model assumptions

1. Private economy model (enterprises are owned by residents), nominal variables are expressed in uppercase, real variables are expressed in lowercase;
2. In addition to labor and capital stock, residents also hold money $M_0$, after the end of the first phase of production, residents need to pay taxes to the government $T_1$, and at the same time choose to buy nominal bonds issued by the government $B_1$(at the nominal interest rate R b 2 R_{b2} of period 2$R_b$)
3. At the end of period 1, the government injects money supply, increasing the total money stock in society. The income from the government's additional currency is called seigniorage . Seigniorage, together with taxes and government bonds, is the source of government revenue and is used to support government fiscal expenditures.
4. At the end of the first period, the additional currency issued by the government does not enter the utility function of the first period consumers

For the optimization problem of residents , the utility function is
$$U=u(c_1,\frac{M_0}{P})+\delta u(c_2,\frac{M_1^d}{P})$$
among them,$\frac{M_1^d}{P}$Indicates the money demand of residents. For the convenience of analysis, our free utility function can be split into their respective functions
$$U=u(c_1)+v(\frac{M_0}{P})+d\left[u\right(c_2)+v(\frac{M_1^d}{P}\left)\right]$$
The budget constraints of the first and second periods of residents are
$$\begin{gathered} P_1{c}_1+P_1{k}_1+T_1+B_1+M_1^d\le M_0+P_1{k}_0+R_1{k}_0+W_{1}l \\ {P}_2{c}_2+T_2\le M_1^d+P_2{k}_1+R_2{k}_1+W_{2}l+(1+R_b)B_1 \end{gathered}$$
The left side of the inequality is the nominal total expenditure of residents in one period, and the right side is the nominal total income of residents in one period. Divide both sides of the first period budget constraint by $P_1$
$$c_1+k_1+t_1+b_1+\frac{M_1^d}{P_1}\le m_0+k_0+r_1{k}_0+w_{1}l$$
Since the total amount of money in the current period was determined in the previous period, but entered into the utility function in the current period, the time subscript of all currency holdings and the time subscript of the price level will be wrong by one period. Therefore, the shrinking money holdings in period 2 should be defined as $m_1^d\triangleq \frac{M_1^d}{P_2}$. At the same time, define the inflation rate from the first period to the second period ${Pi}_2$则有
$$\frac{M_1^d}{P_1}=\frac{M_1^d}{P_2}\cdot \frac{P_2}{P_1}=(1+{Pi}_2)m_1^d$$
In this way, the budget constraint of the consumer in period 1 can be further expressed as
$$c_1+k_1+t_1+b_1+(1+{Pi}_2)m_1^d\le m_0+k_0+r_1{k}_0+w_{1}l$$
Similarly, the budget constraint for period 2 can be constrained in terms of consumption goods of the form
$$c_2+t_2\le m_1^d+k_1+r_2{k}_1+w_{2}l+\frac{1+R_b}{1+{Pi}_2}{b}_1$$
Set the Lagrangian function
$$\begin{array}{rl}L& =u(c_1)+v(\frac{M_0}{P})+d\left[u\right(c_2)+v(\frac{M_1^d}{P})]\\ & +l_1[m_0+k_0+r_1{k}_0+w_{1}l-c_1-k_1-t_1-b_1-(1+{Pi}_2)m_1^d]\\ & +l_2[m_1^d+k_1+r_2{k}_1+w_{2}l+\frac{1+R_b}{1+{Pi}_2}{b}_1-c_2-t_2]\\ FOC:& \{\begin{array}{l}\frac{\partial L}{\partial c_1}=0\Rightarrow u^{\prime }(c_1)=l_1\\ \frac{\partial L}{\partial c_2}=0\Rightarrow u{\_}^{\prime }\left(c_2\right)=l_2\\ \frac{\partial L}{\partial m_1^d}=0\Rightarrow v{\_}^{\prime }(m_1^d)=l_1(1+{Pi}_2)-l_2\\ \frac{\partial L}{\partial k_1}=0\Rightarrow l_1=l_2(1+r_2)\\ \frac{\partial L}{\partial b_1}=0\Rightarrow l_1=l_2\frac{1+R_b}{1+p_2}\end{array}\Rightarrow \{\begin{array}{ll}{u}^{\prime }\left(c_1\right)=\frac{1}{1+p_2}d\left[u^{\prime }\right(c_2)+v^{\prime }(m_1^d\left)\right]& \\ u^{\prime }\left(c_1\right)=u{\_}^{\prime }(c_2)(1+r_2)& (1)\\ u^{\prime }\left(c_1\right)=u{\_}^{\prime }\left(c_2\right)\frac{1+R_b}{1+p_2}& (2)\end{array}\end{array}$$
Combined with (1)(2), we can see that
$$1+r_2=\frac{1+R_b}{1+{Pi}_2}$$
This is the Fisher equation , which states that the nominal interest rate equals the real interest rate plus the rate of inflation;

For the optimization problem of the enterprise , because the enterprise is completely owned by the residents, the behavior of the enterprise is equivalent to liquidation every period, and the enterprise maximization goal is the nominal profit
$$\begin{array}{rl}\underset{k_t^d,l_t^d}{\max }& P_{t}AF(k_t^d,l_t^d)-R_t{k}_t^d-W_t{l}_t^d\\ \iff \underset{k_t^d,l_t^d}{\max }& AF(k_t^d,l_t^d)-r_t{k}_t^d-w_t{l}_t^d\\ FOC:& \{\begin{array}{l}A{F}_1(k_t^d,l_t^d)=r_t\\ A{F}_2(k_t^d,l_t^d)=w_t\end{array}\end{array}$$

The government in the model has two macro-policy tools: fiscal policy and monetary policy. Monetary policy is determined by the growth rate of money chosen by the government $\mu$ decides. Therefore, the money supply at the end of period 1 is
$$M_1^s=(1+m)M_0$$
By issuing additional currency, the government can obtain $\mu M_0$seigniorage revenue.

Nominal expenditures of the government in two periods are $G_{1}and{G}_2$, Now it can be supported by taxation, bond issuance and seigniorage in three ways.
$$\begin{gathered} G_1=T_1+B_1+\mu M_0 \\ {G}_2+(1+R_b)B_1=T_2 \end{gathered}$$
Divide both sides by the price level
$$\begin{gathered} g_1=t_1+b_1+\mu m_0 \\ {g}_2+\frac{1+R_b}{1+{Pi}_2}{b}_1=t_2 \end{gathered}$$
Also divide both sides of the money supply by the price level
$$\begin{gathered} \frac{M_1^s}{P_1}=(1+m)\frac{M_0}{P_1}\Rightarrow \frac{M_1^s}{P_2}\frac{P_2}{P_1}=(1+m)\frac{M_0}{P_1} \\ \Rightarrow (1+{Pi}_2)m_1^s=(1+m)m_0 \end{gathered}$$
Among them, $m_1^s\triangleq \frac{M_1^s}{P_2}$

According to market clearing
$$\{\begin{array}{l}{l}_1^d+l_2^d=l\\ k_1^d=k_0\\ k_2^d=k_1\\ m_1^s=m_1^d=m_1\end{array}$$
则有
$$\{\begin{array}{l}A{F}_1(k_0,L)=r_1,A{F}_2(k_0,L)=w_1\\ A{F}_1(k_1,L)=r_2,A{F}_2(k_1,L)=w_2\end{array}$$
Then the residents' intertemporal optimization Euler equation is
$$u^{\prime }(c_1)=u{\_}^{\prime }(c_2)(1+A{F}_1(k_0,L))\left(3\right)$$
The budget constraint of the first period of the residential sector is rewritten as
$$c_1+k_1+t_1+b_1+(1+{Pi}_2)m_1=m_0+k_0+AF(k_0,L)$$
Substitute the government's first-period budget constraint into the above formula, and eliminate $t_{1}with{b}_1$
$$c_1+k_1+g_1+(1+{Pi}_2)m_1=(1+m)m_0+k_0+AF(k_0,L)$$
And according to the change equation of real money stock $(1+{Pi}_2)m_1=(1+m)m_0$,可知
$$c_1+k_1+g_1=k_0+AF(k_0,L)(4)$$

Then deduce the resource constraints of the second period of the residential sector, using the optimization conditions of the enterprise and the Fisher equation
$$\{\begin{array}{l}{c}_2+t_2=m_1+k_1+r_2{k}_1+w_{2}l+\frac{1+R_b}{1+p_2}{b}_1\\ AF(k_1,L)=r_2{k}_1+w_{2}l\\ 1+r_2=\frac{1+R_b}{1+p_2}\end{array}\Rightarrow c_2+t_2=m_1+k_1+AF(k_1,L)+(1+r_2)b_1$$
Since we are now analyzing the equilibrium situation and discussing the overall resource constraints in the economy, the holding of currency will not bring about an increase in consumption, so $m_1$Remove from the right side of the above formula. Then the budget constraints of the government in the second period $g_2+(1+r_2)b_1=t_2$
$$c_2+g_2=k_1+AF(k_1,L)(5)$$
Combining (3)(4)(5), we can solve $c_1,c_{2}with{k}_1$three unknowns so that the endogenous real variables in the model can be determined. (you can use $g$ as$c$ ), as in the economy before the introduction of money;

At equilibrium, $(1+{Pi}_2)m_1^s=(1+m)m_0$Can be reduced to
$$(1+{Pi}_2)m_1=(1+m)m_0$$
It can be seen that the growth rate of the monetary aggregate $\mu$ affects both the inflation rate and the stock of real money.

Notice:

• The key step is that we remove the real money holdings $m_1$. This is possible because here we are analyzing the state of economic equilibrium. When discussing equilibrium, we are considering issues from the perspective of the whole society. In the micro-budget constraint of the household, at the beginning of the second period, hold more money $m_1$, you can buy more consumer goods. Because households will gain utility from holding real money at the end of the second period, the currency in the second period also has value, and residents also have the motivation to hold it. Therefore, in the second-period budget constraints of households at the micro level, it is necessary to include the money holdings $m_1$。
• However, using money to purchase consumer goods at the micro level will only bring about changes in the distribution of consumer goods among residents, and will not increase the total amount of consumer goods out of thin air. Therefore, when we think about equilibrium from the perspective of the whole society, what we care about is the budget constraint of consumer goods in the whole society. At this time, the budget constraint should not include the money stock m 1 $m_1$up. So, delete $m_1$The formulas before and after seem to be the budget constraints of the second period, but the former is the budget constraint of residents at the micro level, and the latter is the constraint formula of the total amount of consumer goods at the macro level, which are different issues considered at different levels, so the former includes $m_1$, which does not contain .

classical dichotomy

将发全推对的字方写出
$$\begin{gathered} \{\begin{array}{l}{u}^{\prime }(c_1)=u{\_}^{\prime }(c_2)(1+A{F}_1(k_0,L))\\ c_1+k_1+g_1=k_0+AF(k_0,L)\\ c_2+g_2=k_1+AF(k_1,L)\end{array} \\ (1+{Pi}_2)m_1=(1+m)m_0 \end{gathered}$$
These four equations can be divided into two parts. Only real variables are included in the first three equations, and they can completely determine the real endogenous variables in the model. The last equation includes money growth rates, but it only affects nominal variables. Therefore, in this model, the real economy (real variables) and the monetary economy (nominal variables) can be analyzed separately. In macroeconomics, this is called classical dichotomy . Many classical economists hold this point of view, believing that money is just a curtain over the real economy and does not affect the activities of the real economy, so they believe that the real economy and the monetary economy can be analyzed separately.

In the previous model, none of the real variables in the economy are affected by the stock of money and the growth rate of money. The property that the stock of money does not affect the variables of the real economy is called neutrality of money. The property that the growth rate of money does not affect real economic variables is called super neutrality of money.

The reason why the previous model presents the characteristics of this classical dichotomy, or that the currency is neutral and super-neutral, is because the nominal price in the model changes flexibly, so changes in the total amount of the currency market are only reflected in the nominal price , without affecting the real variables. But this conclusion may not hold if prices are sticky (nominal rigidity exists), or if there are other frictions in the economy. It can be visualized in this way, thinking of money as water, flowing in the real economy. If there is no friction between the currency and the real economy, the flow of money water will not affect the operation of the real economy. On the contrary, if there is friction between the two, then the water of money will drive the real economy.

optimal amount of money

Since the government can regulate the monetary aggregate by changing the growth rate of the currency, we naturally want to know how much monetary aggregate is the best. As before, the measure of good and bad here is the well-being of the residents. This is the optimal quantity of money problem posed by Friedman. Since the inflation rate is related to the growth rate of money, this question is equivalent to asking what is the optimal inflation rate. Since the inflation rate plus the real interest rate equals the nominal interest rate, and the real interest rate is determined by the real economy and has nothing to do with money, the question of the optimal amount of money is equivalent to asking what is the optimal nominal interest rate.

Friedman has a very ingenious answer to this question. In order to obtain the benefits of holding currency, people would rather accept the zero return of currency than obtain the nominal interest rate brought by holding financial assets. So Friedman said that the nominal interest rate measures the marginal benefit that money brings to people (this benefit is not presented in the form of money). The most basic intuition of economics is that only when marginal cost equals marginal revenue is optimal. Under the paper currency system, the production cost of currency can be regarded as zero. Since the marginal cost of money is 0, the optimal amount of money should make the marginal revenue of money also 0, that is, the nominal interest rate is also 0.

Therefore, the optimal amount of money is the amount of money that makes the nominal interest rate zero. Since the real interest rate is determined by the marginal rate of return on capital and is generally positive, a nominal interest rate of 0 means that the inflation rate needs to be negative, which naturally means that the money growth rate should be negative. Therefore, if the government wants to achieve the optimal amount of money, it needs to continuously recycle money and let prices grow negatively.

Define the structural equation:
$$\{\begin{array}{l}{u}^{\prime }(c_1)=\frac{1}{1+p_2}d\left[u^{\prime }\right(c_2)+v^{\prime }(m_1^d)]\\ u^{\prime }\left(c_1\right)=u{\_}^{\prime }\left(c_2\right)\frac{1+R_b}{1+p_2}\\ (1+{Pi}_2)m_1=(1+m)m_0\end{array}$$
Extending two periods to multiple periods is still valid. When a wirelessly continuous economy reaches a steady state, the above relationship is also satisfied. In a steady state, consumption and real money stock are constant, and the above mark ss $ss$名生安态
$$\begin{gathered} (1+p)m^{ss}=(1+m)m^{ss}\Rightarrow Pi=m \\ {u}^{\prime }(c^{ss})=\frac{1}{1+Pi}d\left[u^{\prime }\right(c^{ss})+v^{\prime }(m^{ss})] \\ \Rightarrow \frac{}{1+Pi}{v}^{\prime }(m^{ss})=(1-\frac{}{1+Pi})u^{\prime }(c^{ss}) \end{gathered}$$
when the optimal amount of money is reached, the marginal utility brought by the real money stock is equal to 0, that is,$v^{\prime }\left(m^{ss}\right)=0$ . It can be seen from the above formula that this is only possible at$Pi=d-1$ is possible. Since$\delta$ is between 0 and 1, so the inflation rate$Pi<0$

在安态时是
$$u^{\prime }\left(c^{ss}\right)=u{\_}^{\prime }\left(c^{ss}\right)\frac{1+R}{1+Pi}\Rightarrow R=\frac{}{1+Pi}-1=0$$

We make two comments on Friedman's optimal quantity of money

1. It tells us that what really matters is the stock of real money, not the stock of nominal money. When the government creates deflation and presses the nominal interest rate to 0, the real money stock is infinite (the marginal utility brought by the money stock is 0). That is to say, when the total amount of nominal money decreases, the real money stock increases instead. This is because the price falls faster than the money decreases. Therefore, when studying currency, don't be deceived by nominal currency and create illusions, but to see the real currency stock that is reduced by price is the key to pay attention to.
2. Friedman's optimal amount of money is not feasible in the real world. This is related to the nominal rigidity we will talk about later. In an inflationary environment where the overall price level rises, price changes are more flexible. But in a deflationary situation where prices are falling, prices are more difficult to adjust. For example, in an environment where overall prices are rising by 5%, it is relatively easy for a business owner to give workers a 3% wage increase even though workers' real earnings actually fall. But in the context of a 5% decline in overall prices, it is difficult to cut workers' wages by 3%, even though workers' real incomes have actually increased after the wage cuts. The asymmetric nature of this price adjustment in the real world makes it difficult for the economy to get out of it through spontaneous adjustment once it falls into a deflationary economy. In addition, debt is usually denominated in nominal monetary units. In a deflationary environment, the real price of debt rises, increasing the debt burden of borrowers. In order to repay debts with ever-increasing value, borrowers can only continuously reduce their expenditures and cut down on food and clothing to repay debts. When many borrowers are caught in this situation, it can lead to a long-term depression caused by debt deflation. There are many reasons why the government is unwilling to create deflation (such as the zero lower bound of the nominal interest rate, etc.). Therefore, no country in the real world takes the optimal amount of money as the goal of monetary policy regulation.