TESTBlambda_(t)I∄beta((1+i_(t))/(1+pi_(t+1)))lambda_(t+1)\mathrm{TESTB} \lambda_{t} I \nexists \beta\left(\frac{1+i_{t}}{1+\pi_{t+1}}\right) \lambda_{t+1}
which differs from (2) in that i_(t)i_{t} rather than i_(t+1)i_{t+1} appears. This difference in timing is related to the dependence of utility on M_(t)//P_(t)M_{t} / P_{t} in the model of the text and on A_(t)//P_(t)=A_{t} / P_{t}=M_(t-1)//P_(t)+tau_(t)+[(1+i_(t-1))B_(t-1)-B_(t)]//P_(t)M_{t-1} / P_{t}+\tau_{t}+\left[\left(1+i_{t-1}\right) B_{t-1}-B_{t}\right] / P_{t} in the model of this problem. In the basic MIU model of section 2.2, the household can reduce current consumption at a cost of U_(c)(c_(t),m_(t))=lambda_(t)U_{c}\left(c_{t}, m_{t}\right)=\lambda_{t} and purchase bonds instead, yielding a real payoff of (1+i_(t))//(1+pi_(t+1))\left(1+i_{t}\right) /\left(1+\pi_{t+1}\right) in period t+1t+1. This gross interest income could be spend on consumption, but it did not serve to augment the period t+1t+1 holdings of money that enter the utility function. In the model of this problem, the same option to lower current consumption to purchase bonds is open, but purchasing bonds has an addition cost since purchasing additional bonds reduces A_(t)//P_(t)A_{t} / P_{t} which reduces current utility by U_(a)(c_(t),z_(t)-b_(t))=i_(t)lambda_(t)U_{a}\left(c_{t}, z_{t}-b_{t}\right)=i_{t} \lambda_{t} in period tt. Thus, the utility cost of purchasing a bond is U_(c)(c_(t),z_(t)-b_(t))+U_(a)(c_(t),z_(t)-b_(t))=(1+i_(t))lambda_(t)U_{c}\left(c_{t}, z_{t}-b_{t}\right)+U_{a}\left(c_{t}, z_{t}-b_{t}\right)=\left(1+i_{t}\right) \lambda_{t}. The bond pays off {:1+i_(t))//(1+pi_(t+1))\left.1+i_{t}\right) /\left(1+\pi_{t+1}\right) in period t+1t+1 and this return, since it is paid out before the goods market opens, yields the marginal utility of A_(t+1)//P_(t+1)A_{t+1} / P_{t+1} (equal to i_(t+1)lambda_(t+1)i_{t+1} \lambda_{t+1} ) plus the marginal utility of consumption (lambda_(t+1))\left(\lambda_{t+1}\right). That is, the gross interest income from the bond augments the real stock of money that enters time tt utility and can be spend on period tt consumption. Hence,
Assume the representative household’s utility function is given by (2.29). Show that (2.24) implies (2.30). Now suppose u(c,m)=(1-gamma)ln c+gamma ln mu(c, m)=(1-\gamma) \ln c+\gamma \ln m. Show that if consumption is constant in the steady state, there is a unique steady-state capital stock k^(ss)k^{s s} such that beta[f_(k)(k^(ss))+1-delta]=1\beta\left[f_{k}\left(k^{s s}\right)+1-\delta\right]=1. Explain why variations in the growth rate of the money supply do not affect the steady-state kk in this case but do when the utility function is (2.29). From (2.29), u(c,m)=(c^(1-gamma)m^(gamma))^(1-eta)//(1-eta)u(c, m)=\left(c^{1-\gamma} m^{\gamma}\right)^{1-\eta} /(1-\eta). This utility specification implies
To linearize this expression around the steady state, note that it can be rewritten as
(u_(l)(c_(t),m_(t),l_(t)))/(u_(c)(c_(t),m_(t),l_(t)))=Aac_(t)(1-n_(t))^(-1)+A(1-a)m_(t)^(1-b)c_(t)^(b)(1-n_(t))^(-1)\frac{u_{l}\left(c_{t}, m_{t}, l_{t}\right)}{u_{c}\left(c_{t}, m_{t}, l_{t}\right)}=A a c_{t}\left(1-n_{t}\right)^{-1}+A(1-a) m_{t}^{1-b} c_{t}^{b}\left(1-n_{t}\right)^{-1}
The right side of this expression can be approximated as
{:[Aac_(t)(1-n_(t))^(-1)+A(1-a)m_(t)^(1-b)c_(t)^(b)(1-n_(t))~~(Aac)/(1-n)(1+ hat(c)_(t)+ hat(n)_(t))],[+(A(1-a))/(1-n)m^(1-b)c^(b)(1+(1-b) hat(m)_(t)+b hat(c)_(t)+ hat(n)_(t))]:}\begin{aligned}
A a c_{t}\left(1-n_{t}\right)^{-1}+A(1-a) m_{t}^{1-b} c_{t}^{b}\left(1-n_{t}\right) \approx & \frac{A a c}{1-n}\left(1+\hat{c}_{t}+\hat{n}_{t}\right) \\
& +\frac{A(1-a)}{1-n} m^{1-b} c^{b}\left(1+(1-b) \hat{m}_{t}+b \hat{c}_{t}+\hat{n}_{t}\right)
\end{aligned}
Since u_(l)//u_(c)u_{l} / u_{c} is equal to the marginal product of labor, we have u_(l)//u_(c)=alphaY_(t)//N_(t)u_{l} / u_{c}=\alpha Y_{t} / N_{t}, or
which is the equivalent of (2.73).
7. Suppose a nominal interest rate of i^(m)i^{m} is paid on money balances. These payments are financed by a combination of lump-sum taxes and printing money. Let aa be the fraction financed by lump-sum taxes. The government’s budget identity is tau_(t)+v_(t)=i^(m)m_(t)\tau_{t}+v_{t}=i^{m} m_{t}, with tau_(t)=ai^(m)m_(t)\tau_{t}=a i^{m} m_{t} and v=thetam_(t)v=\theta m_{t}. Using Sidrauski’s model, do the following:
(a) Show that the ratio of the marginal utility of money to the marginal utility of consumption equals r+pi-i^(m)=i-i^(m)r+\pi-i^{m}=i-i^{m}. Explain why. The budget constraint in the basic Sidrauski model must be modified to take into account the interest payments on money and that net transfers ( tau\tau in 2.4) consists of two components, the first being the lumpsum transfer vv and the second being the lump-sum tax tau\tau. Thus, the budget constraint becomes
The ratio of the marginal utility of money to consumption is set equal to the opportunity cost of money. Since money now pays a nominal rate of interest i^(m)i^{m}, this opportunity cost is i-i^(m)i-i^{m}, the difference between the nominal return on capital and the nominal return on money.
(b) Show how i-i^(m)i-i^{m} is affected by the method used to finance the interest payments on money. Explain the economics behind your result. From the government’s budget constraint, interest payments not financed through lump-sum taxes must be financed by printing more money. Hence, v=theta m=(1-a)i^(m)mv=\theta m=(1-a) i^{m} m, or the rate of money growth will equal theta=(1-a)i^(m)\theta=(1-a) i^{m}. In the steady-state, pi=theta\pi=\theta. This means that pi=(1-a)i^(m)\pi=(1-a) i^{m}. Hence, the opportunity cost of money is given by
where r=R-1r=R-1. Paying interest on money affects the opportunity cost of money only if a > 0a>0. Printing money to finance interest payments on money only results in
通货膨胀;这提高了名义利率 ii ,从而抵消了支付利息的影响。如果个人认为转账与她自己的货币持有量成正比,那么这相当于个人将货币视为支付象征性的利率。如果这是通过一次性税收筹集资金的,那么通货膨胀的变化不会改变持有货币的机会成本——通货膨胀率的上升使个人的货币持有量贬值,但个人预期收到的转账增加会抵消。
8. Suppose agents do not treat tau_(t)\tau_{t} as a lump-sum transfer but instead assume their transfers will be proportional to their own holdings of money (because in equilibrium, tau=theta m\tau=\theta m ). Solve for an agent’s demand for money. What is the welfare cost of inflation? If the transfer is viewed by the individual as proportional to her own money holdings, then this is equivalent to the individual viewing money as paying a nominal rate of interest. If this is financed via lump-sum taxes, changes in inflation do not change the opportunity cost of holding money a rise in inflation that depreciates the individual’s money holdings is offset by the increase in the transfer the individual anticipates receiving.
9. Suppose money is a productive input into production, so that the aggregate production function becomes y=f(k,m)y=f(k, m). Incorporate this modification into the model of section 2.2. Is money still superneutral? Explain. In the steady state, the household’s Euler condition implies that the gross real rate of return will equal 1//beta1 / \beta. Hence,
where f_(k)f_{k} is the marginal produce of capital. When reat money balances enters directly into the production function, the marginal product of capital will depend on mm. Thus, in the steady state,
and the steady-state capital stock will depend on the steady-state level of real money balances. Money will not be superneutral; a change in the money growth rate will affect inflation and the equilibrium level of money holdings. The level of capital in the steady-state will need to adjust to ensure f_(k)(k^(ss),m^(ss))=beta^(-1)-1+deltaf_{k}\left(k^{s s}, m^{s s}\right)=\beta^{-1}-1+\delta.
10. Consider the following two alternative specifications for the demand for money given by (2.39) and (2.40).
(a) Using (2.39), calculate the welfare cost as a function of eta\eta. We want to find the area under the money demand curve that represents to lost consumer surplus that arises when the nominal rate of interest is positive. When the nominal interest rate is i_(0)i_{0} (so the inflation rate is i_(0)-ri_{0}-r ), (2.39) gives money demand as m=Ai^(-eta)m=A i^{-\eta}, so this area can be calculated as
cs_(t)=int_(0)^(i_(0))Ai^(-eta)di=A((1)/(1-eta))i_(0)^(1-eta)c s_{t}=\int_{0}^{i_{0}} A i^{-\eta} d i=A\left(\frac{1}{1-\eta}\right) i_{0}^{1-\eta}
This gives the welfare cost as a function of eta\eta. Note that the government collects inflation tax revenue when the nominal rate of interest is positive (see chapter 4), so
we should actually subtract this seigniorage revenue from the reduction in consumer surplus to obtain the deadweight loss due to inflation. Seigniorage is i_(0)m=Ai_(0)^(1-eta)i_{0} m=A i_{0}^{1-\eta}, so the deadweight loss is A[eta//(1-eta)]i_(0)^(1-eta)A[\eta /(1-\eta)] i_{0}^{1-\eta}.
(b) Using (2.40), calculate the welfare cost as a function of xi\xi. We want to find the area under the money demand curve that represents to lost consumer surplus that arises when the nominal rate of interest is positive. When the nominal interest rate is i_(0)i_{0} (so the inflation rate is {:i_(0)-r)\left.i_{0}-r\right), (2.40) gives money demand as m=B exp(-xi i)m=B \exp (-\xi i), so this area can be calculated as
cs_(t)=int_(0)^(i_(0))Be^(-xi i)di=((B)/( xi))(1-e^(-xii_(0)))c s_{t}=\int_{0}^{i_{0}} B e^{-\xi i} d i=\left(\frac{B}{\xi}\right)\left(1-e^{-\xi i_{0}}\right)
This gives the welfare cost as a function of xi\xi. Note that the government collects inflation tax revenue when the nominal rate of interest is positive (see chapter 4), so we should actually subtract this seigniorage revenue from the reduction in consumer surplus to obtain the deadweight loss due to inflation. Seigniorage is i_(0)m=i_(0)B exp(-xii_(0))i_{0} m=i_{0} B \exp \left(-\xi i_{0}\right), so the deadweight loss is (B//xi)[1-(1+xii_(0))e^(-xii_(0))](B / \xi)\left[1-\left(1+\xi i_{0}\right) e^{-\xi i_{0}}\right].
11. In Sidrauski’s MIU model augmented to include a variable labor supply, money is superneutral if the representative agent’s preferences are given by
Discuss. (Assume output depends on capital and labor, and the aggregate production function is Cobb-Douglas.) The steady-state values of c^(ss),k^(ss),l^(ss),y^(ss)c^{s s}, k^{s s}, l^{s s}, y^{s s} must satisfy the following four equations:
is independent of mm and (5) - (8) involve only the four unknowns c^(ss),k^(ss),l^(ss),y^(ss)c^{s s}, k^{s s}, l^{s s}, y^{s s}. These can be solved for c^(ss),k^(ss),l^(ss)c^{s s}, k^{s s}, l^{s s}, and y^(ss)y^{s s} independently of mm or inflation. Superneutrality holds. If the utility function is (c_(t+i)+km_(t+i))^(b)l_(t+i)^(d)\left(c_{t+i}+k m_{t+i}\right)^{b} l_{t+i}^{d}, then
which is not independent of mm. Thus, (5) - (8) will involve 5 unknowns ( c^(ss),k^(ss),l^(ss),y^(ss)c^{s s}, k^{s s}, l^{s s}, y^{s s}, and m^("ss ")\mathrm{m}^{\text {ss }} ) and cannot be solved independently of the money demand condition and inflation. Superneutrality does not hold.
12. Suppose preferences over consumption, money holdings, and leisure are given be (u=(u={:a ln c_(t)+(1-a)ln m_(t)+Psil_(t)^(1-eta)//(1-eta))\left.a \ln c_{t}+(1-a) \ln m_{t}+\Psi l_{t}^{1-\eta} /(1-\eta)\right). Fischer (1979) showed that the transition paths are independent of the money supply in this case because the marginal rate of substitution between leisure and consumption is independent of real money balances. Write the equilibrium conditions for this case, and show that the model dichotomizes into a real sector that determines output, consumption, and investment, and a monetary sector that determines the price level and the nominal interest rate. The linearized version of the model used to study dynamics was given by (2.68) - (2.75). When Phi=b=1,Omega_(1)=[(b-Phi)gamma-b]=-b\Phi=b=1, \Omega_{1}=[(b-\Phi) \gamma-b]=-b and Omega_(2)=(b-Phi)(1-gamma)=0\Omega_{2}=(b-\Phi)(1-\gamma)=0. The linearized expression for hat(lambda)_(t)\hat{\lambda}_{t} is
Notice that the first six equations can be solved for hat(y)_(t), hat(c)_(t), hat(x)_(t), hat(k)_(t), hat(n)_(t)\hat{y}_{t}, \hat{c}_{t}, \hat{x}_{t}, \hat{k}_{t}, \hat{n}_{t}, and hat(r)_(t)\hat{r}_{t}, i.e., all the real variables. Since hat(m)\hat{m} is equal to the deviation of real money balances around its steady-state value, we can write it as hat(m)_(t)^(n)- hat(p)_(t)\hat{m}_{t}^{n}-\hat{p}_{t}, where hat(m)^(n)\hat{m}^{n} represents nominal money balances. Then, the final four equations (the monetary sector) can be rewritten as
where LL is the lag operator (so L hat(m)_(t)^(n)= hat(m)_(t-1)^(n)L \hat{m}_{t}^{n}=\hat{m}_{t-1}^{n} ). These three equations only involves the nominal money supply, the price level, and the nominal rate of interest, given that hat(c)_(t)\hat{c}_{t} and hat(r)_(t)\hat{r}_{t} have been determined by the real side of the model.
13. For the model of section 2.5 , is the response of output and employment to a money growth rate shock increasing or decreasing in b? Explain. This questions was essentially answered in the text (see the discussion related to figure 2.5). To understand the role of bb in affecting the dynamics of the MIU model, notice that it only appears in (2.67), the equation for the marginal utility of consumption. If b=Phib=\Phi, neither Omega_(1)\Omega_{1} nor Omega_(2)\Omega_{2} depend on bb and variations in hat(m)_(t)\hat{m}_{t} do not affect the linearized equilibrium conditions for the real variables. For the baseline calibration, Phi=2\Phi=2 and b=9b=9, so b-Phi < 0b-\Phi<0. In this case, a rise in real money balances reduces the marginal utility of consumption and thereby also reduces labor supply; output falls. If b < Phib<\Phi, the opposite happens (see figure 2.5). To illustrate more clearly than was done in figure 2.5, the effects when b=1,2b=1,2, and 3 are shown in the figure. With b=2b=2, money growth shocks have no real effects as in this case, b=Phib=\Phi and Omega_(2)=0\Omega_{2}=0. (The program associated with this exercise is available on request.)
14. For the model of section 2.5 , is the response of output and employment to a money growth rate shock increasing or decreasing in aa ? Explain. The value of a affects the importance of money holding in the utility function. As a increases, money becomes less important, and the parameter gamma\gamma in the linear approximation rises. This reduces the impact of money on marginal utility (see (2.67) and the definitions of gamma\gamma and Omega_(2)\Omega_{2} ). When a is small, changes in mm have a larger impact on the marginal utility of consumption and so have a larger effect
Suppose the production function for shopping takes the form psi=c=e^(x)(n^(s))^(a)m^(b)\psi=c=e^{x}\left(n^{s}\right)^{a} m^{b}, where aa and bb are both positive but less than 1 and xx is a productivity factor. The agent’s utility is given by v(c,l)=c^(1-Phi)//(1-Phi)+l^(1-eta)//(1-eta)v(c, l)=c^{1-\Phi} /(1-\Phi)+l^{1-\eta} /(1-\eta), where l=1-n-n^(s)l=1-n-n^{s}, and nn is time spent in market employment.
(a) Derive the transaction time function g(c,m)=n^(s)g(c, m)=n^{s}. From the shopping production function,
(b) Derive the money-in-the-utility function specification implied by the shopping production function. How does the marginal utility of money depend on the parameters aa and bb ? How does it depend on xx ? From the definition of the agent’s utility,
{:[v(c","l)=(c^(1-Phi))/(1-Phi)+((1-n-n^(s))^(1-eta))/(1-eta)],[=(c^(1-Phi))/(1-Phi)+((1-n-((c)/(e^(x)m^(b)))^(1//a))^(1-eta))/(1-eta)-=u(c","n","m)]:}\begin{aligned}
v(c, l) & =\frac{c^{1-\Phi}}{1-\Phi}+\frac{\left(1-n-n^{s}\right)^{1-\eta}}{1-\eta} \\
& =\frac{c^{1-\Phi}}{1-\Phi}+\frac{\left(1-n-\left(\frac{c}{e^{x} m^{b}}\right)^{1 / a}\right)^{1-\eta}}{1-\eta} \equiv u(c, n, m)
\end{aligned}
This effect will depend on the level of money holding. From (9), consumption and money are complements in producing shopping time, and higher money holdings reduce the effect of higher cc on n^(s)n^{s}. This means that with higher money holdings, an increase in cc has less of an effect in reducing leisure time and will therefore cause a rise in consumption to have a larger overall positive effect on utility.
2. Using (3.5) and (3.8), show that
Both the shopping time and the CIA models imply the utility value of money ( V_(a)//PV_{a} / P or lambda//P)\lambda / P) is equal to the discounted future return to money. This return is measured by the Lagrangian multiplier on the CIA constraint in (3.31) and is measured by the labor time saved in shopping by holding additional money (g_(m)//P)\left(g_{m} / P\right) times the utility value of the time saving (v_(l))\left(v_{l}\right).
3. Show that, for the shopping-time model (section 3.2.1), the tax on consumption is given by
(Recall that money reduced shopping time, so g_(m) <= 0g_{m} \leq 0.) Provide an intuitive interpretation for this expression. Equations (3.10) and (3.11) imply
showing that the tax on consumption (the wedge between the marginal utility of consumption and the marginal value of wealth) is (i//(1+i))(g_(c)//g_(m))(i /(1+i))\left(g_{c} / g_{m}\right). Intuitively, this tax is measured by the lost income due to the additional spending time shopping arising from a marginal increase in consumption, or wg_(c)w g_{c}. At the margin, the value of time is also equal to the opportunity cost of holding money times the marginal productivity of money in reducing shopping time.
4. In the model of section 3.3.2, suppose the current CIA constraint is not binding. This implies mu_(t)=0\mu_{t}=0. Use (3.41) and (3.42) to show that money still has value at time tt (that is, the price level at time tt is finite) as long as the CIA constraint is expected to bind in the future. By noting that 1//(1+pi_(t+1))=P_(t)//P_(t+1)1 /\left(1+\pi_{t+1}\right)=P_{t} / P_{t+1}, (3.42) can be rewritten as
Thus, the current value of money, 1//P_(t)1 / P_{t}, depends on the entire future path of mu_(t+i)\mu_{t+i}. If mu_(t+i) > 0\mu_{t+i}>0 for i >= 0i \geq 0, then money will have value at time tt even if mu_(t)=0\mu_{t}=0.
5. MIU and CIA models are alternative approaches to constructing models in which money has positive value in equilibrium.
(a) What strengths and weaknesses do you see in each of these approaches? Both the money-in-the-utility function approach and the cash-in-advance approach are best viewed as convenient short-cuts for generating a role for money. If we believe that the major role money plays is to facilitate transactions, then in some ways the CIA approach has an advantage in making this transactions role more explicit. It forces one to think more about the exact nature of the transactions technology and the timing of payments (e.g., can current period income be used to purchase current period consumption?). On the other hand, the rather rigid restrictions the CIA typically places on transactions are certainly unattractive. In modern economies we normally have multiple means that can be used to facilitate the transactions we undertake. Also, the generally exogenous distinction between cash and credit goods is troublesome, since most things are a bit