## 经济代写|宏观经济学代写Macroeconomics代考|Currency substitution

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## 经济代写|宏观经济学代写Macroeconomics代考|Currency substitution

The model is amenable to discussing the role of currency substitution, that is, the possibility of phasing out the currency and being replaced by a sounder alternative.

The issue of understanding how different currencies interact, has a long tradition in monetary economics. Not only because, in antiquity, many objects operated as monies, but also because, prior to the emergence of the Fed, currency in the U.S. were issued by commercial banks, so there was an innumerable number of currencies circulating at each time. A popular way to think this issue is Gresham’s Law; faced with a low quality currency and a high quality currency, Gresham’s Law argues that people will try to get rid of the low quality currency while hoarding the high quality currency, bad money displaces good money. Of course while this may be true at the individual level, it may not be so at the aggregate level because prices may increase faster when denominated in units of the bad-quality currency debasing its value. Sturzenegger (1994) discusses this issue and makes two points.

• When there are two or more currencies, it is more likely that the condition $v^{\prime}(m) m=0$ is satisfied (particularly for the low quality currency). Thus, the hyperinflation paths are more likely.
• If the dynamics of money continue are described by an analogous to (19.21) such as
$$\dot{m}_1=\left(\rho+\sigma_1\right) m_1-v^{\prime}\left(m_1, m_2\right) m_1,$$
notice that if the second currency $m_2$ reduces the marginal utility of the first one, then the inflation rate on the equilibrium path is lower: less inflation is needed to wipe out the currency.

This pattern seems to have occurred in a series of hyperinflations in Argentina in the late 80 s, each new wave coming faster but with lower inflation. Similarly, at the end of the 2000s, also in Argentina, very tight monetary conditions during the fixed exchange regime led to the development of multiple private currencies. Once the exchange rate regime was removed, these currencies suffered hyperinflations and disappeared in a wink (see Colacelli and Blackburn 2009).

## 经济代写|宏观经济学代写Macroeconomics代考|Superneutrality

How do these results extend to a model with capital accumulation? We can see this easily also in the context of the Sidrauski model (we assume no population growth), but where we give away the assumption of exogenous output and allow for capital accumulation. Consider now the utility function
$$\int_0^{\infty} u\left(c_t, m_t\right) e^{-\rho t} d t,$$
where $u_c, u_m>0$ and $u_{c c}, u_{m m}<0$. However, we’ll allow the consumer to accumulate capital now. Defining again $a=k+m$, the resource constraint can be written as
$$a_t=r_t a_t+w_t-\tau_t-c_t-i_t m_t$$
The Hamiltonian is
$$H=u\left(c_t, m_t\right)+\lambda_t\left[r a_t+w_t-\tau_t-c_t-i_t m_t\right]$$
The FOC are, as usual,
$$\begin{gathered} u_c\left(c_t, m_t\right)=\lambda_t, \ u_m\left(c_t, m_t\right)=\lambda_t i_t, \ \dot{\lambda}_t=(\rho-r) \lambda_t . \end{gathered}$$
The first two equations give, once again, a money demand function $u_m=u_c i$, but the important result is that because the interest rate now is the marginal product of capital, in steady state $r=\rho=f^{\prime}\left(k^*\right)$, where we use the * superscript to denote the steady state. We leave the computations to you, assuming $\tau=-\sigma m$, and using the fact that $w$ is the marginal product of labour, replacing in (19.28) we find that
$$c=f\left(k^*\right)$$
But this is the level of income that we would have had in the model with no money! This result is known as superneutrality: not only does the introduction of money not affect the equilibrium, neither does the inflation rate.

Later, we will see the motives for why we believe this is not a good description of the effects of inflation, which we believe in the real world are harmful for the economy.

# 宏观经济学代考

## 经济代写|宏观经济学代写Macroeconomics代考|Currency substitution

• 当有两种或两种以上货币时，更有可能出现这种 情况 $v^{\prime}(m) m=0$ 满意（特别是对于低质量的货 币)。因此，恶性通货膨胀路径更有可能。
• 如果货币的动态继续由类似于 (19.21) 的描述，例 如
$$\dot{m}_1=\left(\rho+\sigma_1\right) m_1-v^{\prime}\left(m_1, m_2\right) m_1$$
请注意，如果第二种货币 $m_2$ 减少第一个的边际效用，则均衡路径上的通货膨胀率较低: 消大货币 所需的通货膨胀率较低。
这种模式似乎发生在 80 年代后期阿根廷的一系列恶性通 货膨胀中，每一波新的浪潮都来得更快，但通货憉胀率 更低。同样，在 2000 年代末，也是在阿根廷，固定汇 率制度期间非常紧缩的货币条件导致了多种私人货币的 发展。一旦汇率制度被取消，这些货币就会遭受恶性通 货膨胀并在盿眼间消失（参见 Colacelli 和 Blackburn 2009).

## 经济代写|宏观经济学代写Macroeconomics代考|Superneutrality

$$\int_0^{\infty} u\left(c_t, m_t\right) e^{-\rho t} d t$$

$$a_t=r_t a_t+w_t-\tau_t-c_t-i_t m_t$$

$$H=u\left(c_t, m_t\right)+\lambda_t\left[r a_t+w_t-\tau_t-c_t-i_t m_t\right]$$

$$u_c\left(c_t, m_t\right)=\lambda_t, u_m\left(c_t, m_t\right)=\lambda_t i_t, \dot{\lambda}_t=(\rho-r) \lambda_t$$

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