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货币理论与政策解决方案手册，第 4 版，麻省理工学院出版社 2017 年*

  卡尔·沃尔什

  2017 年 3 月

  内容

  1 引言 ...1

2 第 2 章：Money-in-the-Utility 函数......2

3 第 3 章：货币和交易......15

4 第 4 章：货币与公共财政......35

5 第 5 章：信息和投资组合的刚性......46

6 第 6 章：自由裁量权政策和时间不一致......52

7 第七章：名义价格和工资刚性 ...82

8 第 8 章：新凯恩斯主义货币经济学 ...98

9 第 9 章：开放经济中的货币政策 ...127

10 第 10 章：金融市场和货币政策 ...146

11 第 11 章：有效的下限和资产负债表政策......163

12 第 12 章：货币政策操作程序 ...175

  1 引言


本手册包含《货币理论》和《政治 i c y , 4 t h i c y , 4 t h icy,4^(th)i c y, 4^{t h} 》版（剑桥，马萨诸塞州：麻省理工学院出版社，2017 年）中所有章末问题的解决方案。请向 walshc@ucsc.edu 报告任何错误或建议的更正。拼写错误和更正将发布在 http://people.ucsc.edu/~ walshc/mtp4e/。

敦促用户检查 http://people.ucsc.edu/~walshc/mtp4e/before 分配问题中发布的更正。对于与这些练习相关的 Matlab 程序，应发送至 walshc@ucsc.edu。


2 第 2 章：Money-in-the-Utility 函数


  1. 第 2.2 节的 MIU 模型意味着货币和消费之间的边际替代率被设定为等于 i t / ( 1 + i t ) i t / 1 + i t i_(t)//(1+i_(t))i_{t} /\left(1+i_{t}\right) （见 （2.12））。该模型假设代理人带着资源进入时期 t t tt ，并使用这些资源 ω t ω t omega_(t)\omega_{t} 来购买资本、消费、名义债券和货币。这些货币持有的实际价值在期间 t t tt 产生了效用。相反，假设在 period t t tt 中选择的货币持有量在 period t + 1 t + 1 t+1t+1 之前不会产生效用。效用和 ∑ β i U ( c t + i , M t + i / P t + i ) ∑ β i U c t + i , M t + i / P t + i sumbeta^(i)U(c_(t+i),M_(t+i)//P_(t+i))\sum \beta^{i} U\left(c_{t+i}, M_{t+i} / P_{t+i}\right) 以前一样，但预算限制采用
ω t = c t + M t + 1 P t + b t + k t ω t = c t + M t + 1 P t + b t + k t omega_(t)=c_(t)+(M_(t+1))/(P_(t))+b_(t)+k_(t)\omega_{t}=c_{t}+\frac{M_{t+1}}{P_{t}}+b_{t}+k_{t}

和家庭选择 c t , k t , b t c t , k t , b t c_(t),k_(t),b_(t)c_{t}, k_{t}, b_{t} ，并在 M t + 1 M t + 1 M_(t+1)M_{t+1} 时期 t t tt 。家庭的真正财富 ω t ω t omega_(t)\omega_{t} 是由
ω t = f ( k t − 1 ) + ( 1 − δ ) k t − 1 + ( 1 + r t − 1 ) b t − 1 + m t ω t = f k t − 1 + ( 1 − δ ) k t − 1 + 1 + r t − 1 b t − 1 + m t omega_(t)=f(k_(t-1))+(1-delta)k_(t-1)+(1+r_(t-1))b_(t-1)+m_(t)\omega_{t}=f\left(k_{t-1}\right)+(1-\delta) k_{t-1}+\left(1+r_{t-1}\right) b_{t-1}+m_{t}

推导住户选择的一阶条件， M t + 1 M t + 1 M_(t+1)M_{t+1} 并证明
U m ( c t + 1 , m t + 1 ) U c ( c t + 1 , m t + 1 ) = i t U m c t + 1 , m t + 1 U c c t + 1 , m t + 1 = i t (U_(m)(c_(t+1),m_(t+1)))/(U_(c)(c_(t+1),m_(t+1)))=i_(t)\frac{U_{m}\left(c_{t+1}, m_{t+1}\right)}{U_{c}\left(c_{t+1}, m_{t+1}\right)}=i_{t}

（由 Kevin Salyer 建议。设 value 函数为
V ( ω t , m t ) = max { U ( c t , m t ) + β V ( ω t + 1 , m t + 1 ) } V ω t , m t = max U c t , m t + β V ω t + 1 , m t + 1 V(omega_(t),m_(t))=max{U(c_(t),m_(t))+beta V(omega_(t+1),m_(t+1))}V\left(\omega_{t}, m_{t}\right)=\max \left\{U\left(c_{t}, m_{t}\right)+\beta V\left(\omega_{t+1}, m_{t+1}\right)\right\}
  受制于
ω t + 1 = f ( k t ) + ( 1 − δ ) k t + ( 1 + r t ) b t + m t + 1 ω t + 1 = f k t + ( 1 − δ ) k t + 1 + r t b t + m t + 1 omega_(t+1)=f(k_(t))+(1-delta)k_(t)+(1+r_(t))b_(t)+m_(t+1)\omega_{t+1}=f\left(k_{t}\right)+(1-\delta) k_{t}+\left(1+r_{t}\right) b_{t}+m_{t+1}
  和
ω t − c t − m t + 1 ( 1 + π t + 1 ) − b t − k t = 0 ω t − c t − m t + 1 1 + π t + 1 − b t − k t = 0 omega_(t)-c_(t)-m_(t+1)(1+pi_(t+1))-b_(t)-k_(t)=0\omega_{t}-c_{t}-m_{t+1}\left(1+\pi_{t+1}\right)-b_{t}-k_{t}=0

在这种设置中， r t r t r_(t)r_{t} 是债券的实际回报。让我们 λ t λ t lambda_(t)\lambda_{t} 表示最后一个约束的拉格朗日。 c t , m t + 1 , b t c t , m t + 1 , b t c_(t),m_(t+1),b_(t)c_{t}, m_{t+1}, b_{t} 的一阶条件 和 k t k t k_(t)k_{t} 加上包络定理得到
U c ( c t , m t ) = λ t β V m ( ω t + 1 , m t + 1 ) + β V ω ( ω t + 1 , m t + 1 ) = λ t ( 1 + π t + 1 ) β ( 1 + r t ) V ω ( ω t + 1 , m t + 1 ) = λ t V ω ( ω t , m t ) = λ t V m ( ω t , m t ) = U m ( c t , m t ) . U c c t , m t = λ t β V m ω t + 1 , m t + 1 + β V ω ω t + 1 , m t + 1 = λ t 1 + π t + 1 β 1 + r t V ω ω t + 1 , m t + 1 = λ t V ω ω t , m t = λ t V m ω t , m t = U m c t , m t . {:[U_(c)(c_(t),m_(t))=lambda_(t)],[betaV_(m)(omega_(t+1),m_(t+1))+betaV_(omega)(omega_(t+1),m_(t+1))=lambda_(t)(1+pi_(t+1))],[beta(1+r_(t))V_(omega)(omega_(t+1),m_(t+1))=lambda_(t)],[V_(omega)(omega_(t),m_(t))=lambda_(t)],[V_(m)(omega_(t),m_(t))=U_(m)(c_(t),m_(t)).]:}\begin{gathered} U_{c}\left(c_{t}, m_{t}\right)=\lambda_{t} \\ \beta V_{m}\left(\omega_{t+1}, m_{t+1}\right)+\beta V_{\omega}\left(\omega_{t+1}, m_{t+1}\right)=\lambda_{t}\left(1+\pi_{t+1}\right) \\ \beta\left(1+r_{t}\right) V_{\omega}\left(\omega_{t+1}, m_{t+1}\right)=\lambda_{t} \\ V_{\omega}\left(\omega_{t}, m_{t}\right)=\lambda_{t} \\ V_{m}\left(\omega_{t}, m_{t}\right)=U_{m}\left(c_{t}, m_{t}\right) . \end{gathered}

现在梳理其中的第二个和第三个以获得
β V m ( ω t + 1 , m t + 1 ) + β V ω ( ω t + 1 , m t + 1 ) = β ( 1 + π t + 1 ) ( 1 + r t ) V ω ( ω t + 1 , m t + 1 ) β V m ω t + 1 , m t + 1 + β V ω ω t + 1 , m t + 1 = β 1 + π t + 1 1 + r t V ω ω t + 1 , m t + 1 betaV_(m)(omega_(t+1),m_(t+1))+betaV_(omega)(omega_(t+1),m_(t+1))=beta(1+pi_(t+1))(1+r_(t))V_(omega)(omega_(t+1),m_(t+1))\beta V_{m}\left(\omega_{t+1}, m_{t+1}\right)+\beta V_{\omega}\left(\omega_{t+1}, m_{t+1}\right)=\beta\left(1+\pi_{t+1}\right)\left(1+r_{t}\right) V_{\omega}\left(\omega_{t+1}, m_{t+1}\right)
  或
β V m ( ω t + 1 , m t + 1 ) = [ ( 1 + π t + 1 ) ( 1 + r t ) − 1 ] β V ω ( ω t + 1 , m t + 1 ) = i t β V ω ( ω t + 1 , m t + 1 ) . β V m ω t + 1 , m t + 1 = 1 + π t + 1 1 + r t − 1 β V ω ω t + 1 , m t + 1 = i t β V ω ω t + 1 , m t + 1 . {:[betaV_(m)(omega_(t+1),m_(t+1))=[(1+pi_(t+1))(1+r_(t))-1]betaV_(omega)(omega_(t+1),m_(t+1))],[=i_(t)betaV_(omega)(omega_(t+1),m_(t+1)).]:}\begin{aligned} \beta V_{m}\left(\omega_{t+1}, m_{t+1}\right) & =\left[\left(1+\pi_{t+1}\right)\left(1+r_{t}\right)-1\right] \beta V_{\omega}\left(\omega_{t+1}, m_{t+1}\right) \\ & =i_{t} \beta V_{\omega}\left(\omega_{t+1}, m_{t+1}\right) . \end{aligned}
  因此
V m ( ω t + 1 , m t + 1 ) V ω ( ω t + 1 , m t + 1 ) = i t V m ω t + 1 , m t + 1 V ω ω t + 1 , m t + 1 = i t (V_(m)(omega_(t+1),m_(t+1)))/(V_(omega)(omega_(t+1),m_(t+1)))=i_(t)\frac{V_{m}\left(\omega_{t+1}, m_{t+1}\right)}{V_{\omega}\left(\omega_{t+1}, m_{t+1}\right)}=i_{t}

但是最后一个一阶条件暗示 V m ( ω t + 1 , m t + 1 ) = U m ( c t + 1 , m t + 1 ) V m ω t + 1 , m t + 1 = U m c t + 1 , m t + 1 V_(m)(omega_(t+1),m_(t+1))=U_(m)(c_(t+1),m_(t+1))V_{m}\left(\omega_{t+1}, m_{t+1}\right)=U_{m}\left(c_{t+1}, m_{t+1}\right) ，而第一个和第四个产生 V ω ( ω t + 1 , m t + 1 ) = λ t + 1 = U c ( c t + 1 , m t + 1 ) V ω ω t + 1 , m t + 1 = λ t + 1 = U c c t + 1 , m t + 1 V_(omega)(omega_(t+1),m_(t+1))=lambda_(t+1)=U_(c)(c_(t+1),m_(t+1))V_{\omega}\left(\omega_{t+1}, m_{t+1}\right)=\lambda_{t+1}=U_{c}\left(c_{t+1}, m_{t+1}\right) 。因此
U m ( c t + 1 , m t + 1 ) U c ( c t + 1 , m t + 1 ) = i t U m c t + 1 , m t + 1 U c c t + 1 , m t + 1 = i t (U_(m)(c_(t+1),m_(t+1)))/(U_(c)(c_(t+1),m_(t+1)))=i_(t)\frac{U_{m}\left(c_{t+1}, m_{t+1}\right)}{U_{c}\left(c_{t+1}, m_{t+1}\right)}=i_{t}

在第 2.2 节考虑的案例中，当时 t t tt 持有的现金在当时 t t tt 产生了效用，但机会成本在于本可以持有的债券的利息收入损失。由于这笔利息支付发生在 t + 1 t + 1 t+1t+1 ，因此必须将其贴现回去，以与当时持有货币的边际价值进行比较 t t tt 。

2. （Carlstrom 和 Fuerst （2001）。假设代表住户的效用取决于消费和可用于消费的实际货币余额水平。设 A t / P t A t / P t A_(t)//P_(t)A_{t} / P_{t} 为进入效用函数的真实货币存量。如果忽略资本，则家庭的目标是在预算约束下实现最大化 ∑ β i U ( c t + i , A t + i / P t + i ) ∑ β i U c t + i , A t + i / P t + i sumbeta^(i)U(c_(t+i),A_(t+i)//P_(t+i))\sum \beta^{i} U\left(c_{t+i}, A_{t+i} / P_{t+i}\right)
Y t + M t − 5 P t + τ t + ( 1 + i t − 1 ) B t 尺 1 P t = C t + M t P t + B t P t , Y t + M t − 5 P t + τ t + 1 + i t − 1 B t 尺 1 P t = C t + M t P t + B t P t , Y_(t)+(M_(t-5))/(P_(t))+tau_(t)+((1+i_(t-1))B_(t尺1))/(P_(t))=C_(t)+(M_(t))/(P_(t))+(B_(t))/(P_(t)),Y_{t}+\frac{M_{t-5}}{P_{t}}+\tau_{t}+\frac{\left(1+i_{t-1}\right) B_{t 尺 1}}{P_{t}}=C_{t}+\frac{M_{t}}{P_{t}}+\frac{B_{t}}{P_{t}},尺

其中收入 Y t Y t Y_(t)Y_{t} 被视为一个外生过程。假设产生效用的货币存量是购买债券之后但在收到收入或购买消费品之前持有的货币的实际价值：
A t P t = M t − 1 P t + τ t + ( 1 + i t − 1 ) B t − 1 P t − B t P t A t P t = M t − 1 P t + τ t + 1 + i t − 1 B t − 1 P t − B t P t (A_(t))/(P_(t))=(M_(t-1))/(P_(t))+tau_(t)+((1+i_(t-1))B_(t-1))/(P_(t))-(B_(t))/(P_(t))\frac{A_{t}}{P_{t}}=\frac{M_{t-1}}{P_{t}}+\tau_{t}+\frac{\left(1+i_{t-1}\right) B_{t-1}}{P_{t}}-\frac{B_{t}}{P_{t}}

（a） 推导 B t B t B_(t)B_{t} 和 的 A t A t A_(t)A_{t} 一阶条件。设 value 函数为
V ( z t ) = max { U ( c t , z t − b t ) + β V ( z t + 1 } , V z t = max U c t , z t − b t + β V z t + 1 , V(z_(t))=max{U(c_(t),z_(t)-b_(t))+beta V(z_(t+1)},:}V\left(z_{t}\right)=\max \left\{U\left(c_{t}, z_{t}-b_{t}\right)+\beta V\left(z_{t+1}\right\},\right.
  哪里
z t + 1 = M t P t + 1 + τ t + 1 + ( 1 + i t ) B t P t + 1 = ( P t P t + 1 ) m t + τ t + 1 + ( 1 + i t ) ( P t P t + 1 ) b t z t + 1 = M t P t + 1 + τ t + 1 + 1 + i t B t P t + 1 = P t P t + 1 m t + τ t + 1 + 1 + i t P t P t + 1 b t {:[z_(t+1)=(M_(t))/(P_(t+1))+tau_(t+1)+((1+i_(t))B_(t))/(P_(t+1))],[=((P_(t))/(P_(t+1)))m_(t)+tau_(t+1)+(1+i_(t))((P_(t))/(P_(t+1)))b_(t)]:}\begin{aligned} z_{t+1} & =\frac{M_{t}}{P_{t+1}}+\tau_{t+1}+\frac{\left(1+i_{t}\right) B_{t}}{P_{t+1}} \\ & =\left(\frac{P_{t}}{P_{t+1}}\right) m_{t}+\tau_{t+1}+\left(1+i_{t}\right)\left(\frac{P_{t}}{P_{t+1}}\right) b_{t} \end{aligned}
  和
Y t + z t − c t − m t − b t = 0 Y t + z t − c t − m t − b t = 0 Y_(t)+z_(t)-c_(t)-m_(t)-b_(t)=0Y_{t}+z_{t}-c_{t}-m_{t}-b_{t}=0

在时间 t t tt ，家庭选择 c t , m t , b t c t , m t , b t c_(t),m_(t),b_(t)c_{t}, m_{t}, b_{t} 。让我们 λ t λ t lambda_(t)\lambda_{t} 表示最后一个约束的拉格朗日。一阶条件加上包络定理得到
U c ( c t , z t − b t ) = λ t β ( P t P t + 1 ) V z ( z t + 1 ) = λ t − U a ( c t , a t ) + β ( 1 + i t ) ( P t P t + 1 ) V z ( z t + 1 ) = λ t V z ( z t ) = U a ( c t , z t − b t ) + λ t U c c t , z t − b t = λ t β P t P t + 1 V z z t + 1 = λ t − U a c t , a t + β 1 + i t P t P t + 1 V z z t + 1 = λ t V z z t = U a c t , z t − b t + λ t {:[U_(c)(c_(t),z_(t)-b_(t))=lambda_(t)],[beta((P_(t))/(P_(t+1)))V_(z)(z_(t+1))=lambda_(t)],[-U_(a)(c_(t),a_(t))+beta(1+i_(t))((P_(t))/(P_(t+1)))V_(z)(z_(t+1))=lambda_(t)],[V_(z)(z_(t))=U_(a)(c_(t),z_(t)-b_(t))+lambda_(t)]:}\begin{gathered} U_{c}\left(c_{t}, z_{t}-b_{t}\right)=\lambda_{t} \\ \beta\left(\frac{P_{t}}{P_{t+1}}\right) V_{z}\left(z_{t+1}\right)=\lambda_{t} \\ -U_{a}\left(c_{t}, a_{t}\right)+\beta\left(1+i_{t}\right)\left(\frac{P_{t}}{P_{t+1}}\right) V_{z}\left(z_{t+1}\right)=\lambda_{t} \\ V_{z}\left(z_{t}\right)=U_{a}\left(c_{t}, z_{t}-b_{t}\right)+\lambda_{t} \end{gathered}

现在将其中的第二个和第三个组合在一起以消除 value 函数，可以得到 − U a ( c t , z t − b t ) + ( 1 + i t ) λ t = λ t − U a c t , z t − b t + 1 + i t λ t = λ t -U_(a)(c_(t),z_(t)-b_(t))+(1+i_(t))lambda_(t)=lambda_(t)-U_{a}\left(c_{t}, z_{t}-b_{t}\right)+\left(1+i_{t}\right) \lambda_{t}=\lambda_{t} ，或
U a ( c t , a t ) = i t λ t . U a c t , a t = i t λ t . U_(a)(c_(t),a_(t))=i_(t)lambda_(t).U_{a}\left(c_{t}, a_{t}\right)=i_{t} \lambda_{t} .

在第四个方程 中使用它， V z ( z t ) = U a ( c t , a t ) + λ t = λ t ( 1 + i t ) V z z t = U a c t , a t + λ t = λ t 1 + i t V_(z)(z_(t))=U_(a)(c_(t),a_(t))+lambda_(t)=lambda_(t)(1+i_(t))V_{z}\left(z_{t}\right)=U_{a}\left(c_{t}, a_{t}\right)+\lambda_{t}=\lambda_{t}\left(1+i_{t}\right) .现在更新这一时期并使用它来消除 V z ( z t + 1 ) V z z t + 1 V_(z)(z_(t+1))V_{z}\left(z_{t+1}\right) 第二个 F O C F O C FOCF O C 时期，我们只剩下
U c ( c t , a t ) = λ t λ t = β ( 1 + i t + 1 ) ( P t P t + 1 ) λ t + 1 = β ( 1 + i t + 1 1 + π t + 1 ) λ t + 1 U c c t , a t = λ t λ t = β 1 + i t + 1 P t P t + 1 λ t + 1 = β 1 + i t + 1 1 + π t + 1 λ t + 1 {:[U_(c)(c_(t),a_(t))=lambda_(t)],[lambda_(t)=beta(1+i_(t+1))((P_(t))/(P_(t+1)))lambda_(t+1)=beta((1+i_(t+1))/(1+pi_(t+1)))lambda_(t+1)]:}\begin{gathered} U_{c}\left(c_{t}, a_{t}\right)=\lambda_{t} \\ \lambda_{t}=\beta\left(1+i_{t+1}\right)\left(\frac{P_{t}}{P_{t+1}}\right) \lambda_{t+1}=\beta\left(\frac{1+i_{t+1}}{1+\pi_{t+1}}\right) \lambda_{t+1} \end{gathered}

（乙）这些条件跟经文里的条件有什么不同？在第 2.2 节的基本 MIU 模型中，方程 （2.10） 和 （1） 相同。使用 （2.7） 和 （2.10），方程 （2.6） 可以写成（设置 n = 0 n = 0 n=0n=0 后）
TESTB λ t I ∄ β ( 1 + i t 1 + π t + 1 ) λ t + 1 TESTB λ t I ∄ β 1 + i t 1 + π t + 1 λ t + 1 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 i_(t)i_{t} rather than i t + 1 i t + 1 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 M_(t)//P_(t)M_{t} / P_{t} in the model of the text and on A t / P t = A t / P t = A_(t)//P_(t)=A_{t} / P_{t}= M t − 1 / P t + τ t + [ ( 1 + i t − 1 ) B t − 1 − B t ] / P t M t − 1 / P t + τ t + 1 + i t − 1 B t − 1 − B 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 ) = λ t U c c t , m t = λ t 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 + π t + 1 ) 1 + i t / 1 + π t + 1 (1+i_(t))//(1+pi_(t+1))\left(1+i_{t}\right) /\left(1+\pi_{t+1}\right) in period t + 1 t + 1 t+1t+1. This gross interest income could be spend on consumption, but it did not serve to augment the period t + 1 t + 1 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 A_(t)//P_(t)A_{t} / P_{t} which reduces current utility by U a ( c t , z t − b t ) = i t λ t U a c t , z t − b t = i t λ t 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 t t 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 ) λ t U c c t , z t − b t + U a c t , z t − b t = 1 + i t λ t 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 + π t + 1 ) 1 + i t / 1 + π t + 1 {:1+i_(t))//(1+pi_(t+1))\left.1+i_{t}\right) /\left(1+\pi_{t+1}\right) in period t + 1 t + 1 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 A_(t+1)//P_(t+1)A_{t+1} / P_{t+1} (equal to i t + 1 λ t + 1 i t + 1 λ t + 1 i_(t+1)lambda_(t+1)i_{t+1} \lambda_{t+1} ) plus the marginal utility of consumption ( λ t + 1 ) λ t + 1 (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 t t tt utility and can be spend on period t t tt consumption. Hence, 
( 1 + i t ) λ t = β ( 1 + i t 1 + π t + 1 ) ( 1 + i t + 1 ) λ t + 1 ⇒ λ t = β ( 1 + i t + 1 1 + π t + 1 ) λ t + 1 1 + i t λ t = β 1 + i t 1 + π t + 1 1 + i t + 1 λ t + 1 ⇒ λ t = β 1 + i t + 1 1 + π t + 1 λ t + 1 (1+i_(t))lambda_(t)=beta((1+i_(t))/(1+pi_(t+1)))(1+i_(t+1))lambda_(t+1)=>lambda_(t)=beta((1+i_(t+1))/(1+pi_(t+1)))lambda_(t+1)\left(1+i_{t}\right) \lambda_{t}=\beta\left(\frac{1+i_{t}}{1+\pi_{t+1}}\right)\left(1+i_{t+1}\right) \lambda_{t+1} \Rightarrow \lambda_{t}=\beta\left(\frac{1+i_{t+1}}{1+\pi_{t+1}}\right) \lambda_{t+1}
  1. 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 − γ ) ln ⁡ c + γ ln ⁡ m u ( c , m ) = ( 1 − γ ) ln ⁡ c + γ ln ⁡ m 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 s s k s s k^(ss)k^{s s} such that β [ f k ( k s s ) + 1 − δ ] = 1 β f k k s s + 1 − δ = 1 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 k k kk in this case but do when the utility function is (2.29). From (2.29), u ( c , m ) = ( c 1 − γ m γ ) 1 − η / ( 1 − η ) u ( c , m ) = c 1 − γ m γ 1 − η / ( 1 − η ) 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 
u c ( c , m ) = ( 1 − γ ) c − η ( 1 − γ ) c − γ m γ ( 1 − η ) . u c ( c , m ) = ( 1 − γ ) c − η ( 1 − γ ) c − γ m γ ( 1 − η ) . u_(c)(c,m)=(1-gamma)c^(-eta(1-gamma))c^(-gamma)m^(gamma(1-eta)).u_{c}(c, m)=(1-\gamma) c^{-\eta(1-\gamma)} c^{-\gamma} m^{\gamma(1-\eta)} .

在 （2.24） 中使用它会得到
( 1 − γ ) ( c s s ) − η ( 1 − γ ) m t γ ( 1 − η ) = β [ f k ( k ∗ ) + 1 − δ ] ( 1 − γ ) ( c s s ) − η ( 1 − γ ) m t + 1 γ ( 1 − η ) , ( 1 − γ ) c s s − η ( 1 − γ ) m t γ ( 1 − η ) = β f k k ∗ + 1 − δ ( 1 − γ ) c s s − η ( 1 − γ ) m t + 1 γ ( 1 − η ) , (1-gamma)(c^(ss))^(-eta(1-gamma))m_(t)^(gamma(1-eta))=beta[f_(k)(k^(**))+1-delta](1-gamma)(c^(ss))^(-eta(1-gamma))m_(t+1)^(gamma(1-eta)),(1-\gamma)\left(c^{s s}\right)^{-\eta(1-\gamma)} m_{t}^{\gamma(1-\eta)}=\beta\left[f_{k}\left(k^{*}\right)+1-\delta\right](1-\gamma)\left(c^{s s}\right)^{-\eta(1-\gamma)} m_{t+1}^{\gamma(1-\eta)},
  或。
m t γ ( 1 − η ) = β [ f k ( k ∗ ) + 1 − δ ] m t + 1 γ ( 1 − η ) . m t γ ( 1 − η ) = β f k k ∗ + 1 − δ m t + 1 γ ( 1 − η ) . m_(t)^(gamma(1-eta))=beta[f_(k)(k^(**))+1-delta]m_(t+1)^(gamma(1-eta)).m_{t}^{\gamma(1-\eta)}=\beta\left[f_{k}\left(k^{*}\right)+1-\delta\right] m_{t+1}^{\gamma(1-\eta)} .

重新排列这个表达式，可以得到
( m t + 1 m t ) γ ( 1 − η ) = { 1 β [ f k ( k ∗ ) + 1 − δ ] } . m t + 1 m t γ ( 1 − η ) = 1 β f k k ∗ + 1 − δ . ((m_(t+1))/(m_(t)))^(gamma(1-eta))={(1)/(beta[f_(k)(k^(**))+1-delta])}.\left(\frac{m_{t+1}}{m_{t}}\right)^{\gamma(1-\eta)}=\left\{\frac{1}{\beta\left[f_{k}\left(k^{*}\right)+1-\delta\right]}\right\} .

将两侧加注到幂 1 / γ ( 1 − η ) 1 / γ ( 1 − η ) 1//gamma(1-eta)1 / \gamma(1-\eta) 能产生 （2.30）。如果效用不是 （2.29） 由 ( 1 − γ ) ln ⁡ c t + γ ln ⁡ m t ( 1 − γ ) ln ⁡ c t + γ ln ⁡ m t (1-gamma)ln c_(t)+gamma ln m_(t)(1-\gamma) \ln c_{t}+\gamma \ln m_{t} ，则 u c = ( 1 − γ ) / c u c = ( 1 − γ ) / c u_(c)=(1-gamma)//cu_{c}=(1-\gamma) / c ，它与 无关。 m m mm 因此，（2.24） 变为
( 1 − γ ) / c ∗ = β [ f k ( k ) + 1 − δ ] ( 1 − γ ) / c ∗ , ( 1 − γ ) / c ∗ = β f k ( k ) + 1 − δ ( 1 − γ ) / c ∗ , (1-gamma)//c^(**)=beta[f_(k)(k)+1-delta](1-gamma)//c^(**),(1-\gamma) / c^{*}=\beta\left[f_{k}(k)+1-\delta\right](1-\gamma) / c^{*},
  和
TESTBANKSETLER.COM β [ f k ( k s s ) + 1 − δ ] = 1 .  TESTBANKSETLER.COM  β f k k s s + 1 − δ = 1 . {:[" TESTBANKSETLER.COM "],[beta[f_(k)(k^(ss))+1-delta]=1.]:}\begin{gathered} \text { TESTBANKSETLER.COM } \\ \beta\left[f_{k}\left(k^{s s}\right)+1-\delta\right]=1 . \end{gathered}

货币供应量增长率的变化不会影响稳态 k k kk ，因为效用的可分性 和 c c cc m m mm 意味着消费的边际效用与 m m mm 无关。

4. 假设 W = ∑ β t ( ln ⁡ c t + m t e − γ m t ) , γ > 0 W = ∑ β t ln ⁡ c t + m t e − γ m t , γ > 0 W=sumbeta^(t)(ln c_(t)+m_(t)e^(-gammam_(t))),gamma > 0W=\sum \beta^{t}\left(\ln c_{t}+m_{t} e^{-\gamma m_{t}}\right), \gamma>0 ， 和 β = 0.95 β = 0.95 beta=0.95\beta=0.95 。假设 production 函数为 f ( k t ) = k t 5 f k t = k t 5 f(k_(t))=k_(t)^(5)f\left(k_{t}\right)=k_{t}^{5} 和 δ = 0.02 δ = 0.02 delta=0.02\delta=0.02 。什么样的通货膨胀率使稳态福利最大化？以福利最大化的通货膨胀率计算的实际货币余额如何取决于 γ γ gamma\gamma ？稳态福利最大化名义利率是 i ss = 0 i ss  = 0 i^("ss ")=0i^{\text {ss }}=0 （见第 2.3 节）此时 u m = 0 u m = 0 u_(m)=0u_{m}=0 。如果 R R RR 是总实际利率（1 加上实际利率）， 1 + i = R ( 1 + π ) 1 + i = R ( 1 + π ) 1+i=R(1+pi)1+i=R(1+\pi) 而产生零名义通货膨胀率的通货膨胀率是
π s s = 1 R − 1 π s s = 1 R − 1 pi^(ss)=(1)/(R)-1\pi^{s s}=\frac{1}{R}-1

在稳态中， R R RR 等于 1 / β 1 / β 1//beta1 / \beta ，或 1 / R = β = 0.95 1 / R = β = 0.95 1//R=beta=0.951 / R=\beta=0.95 （参见第 49 页的 2.19）。因此，最佳通货膨胀率是 0.95 − 1 = − 0.5 0.95 − 1 = − 0.5 0.95-1=-0.50.95-1=-0.5 或通 5 % 5 % 5%5 \% 货紧缩率。要确定货币需求如何取决于参数 γ γ gamma\gamma ，请使用代表代理的一阶条件（2.12） ，以稳态名义利率计算：
u m u c = i s s 1 + i s s u m u c = i s s 1 + i s s (u_(m))/(u_(c))=(i^(ss))/(1+i^(ss))\frac{u_{m}}{u_{c}}=\frac{i^{s s}}{1+i^{s s}}

给定效用函数的形式，这将变为
u m u c = c t ( 1 − γ m t ) e − γ m t = i s s 1 + i s s u m u c = c t 1 − γ m t e − γ m t = i s s 1 + i s s (u_(m))/(u_(c))=c_(t)(1-gammam_(t))e^(-gammam_(t))=(i^(ss))/(1+i^(ss))\frac{u_{m}}{u_{c}}=c_{t}\left(1-\gamma m_{t}\right) e^{-\gamma m_{t}}=\frac{i^{s s}}{1+i^{s s}}

在福利最大化通货膨胀率 下， i s s = 0 i s s = 0 i^(ss)=0i^{s s}=0 这需要 1 − γ m s s = 0 1 − γ m s s = 0 1-gammam^(ss)=01-\gamma m^{s s}=0 或
m s s = 1 γ m s s = 1 γ m^(ss)=(1)/(gamma)m^{s s}=\frac{1}{\gamma}

因此，实际 i s s = 0 i s s = 0 i^(ss)=0i^{s s}=0 货币需求在 中正在减少 γ γ gamma\gamma 。

5. 假设效用函数 （2.66） 替换为
u ( c t , m t , l t ) = ( 1 1 − Φ ) { [ a c t 1 − b + ( 1 − a ) m t 1 − b ] 1 1 − b l t 1 − η } 1 − Φ u c t , m t , l t = 1 1 − Φ a c t 1 − b + ( 1 − a ) m t 1 − b 1 1 − b l t 1 − η 1 − Φ u(c_(t),m_(t),l_(t))=((1)/(1-Phi)){[ac_(t)^(1-b)+(1-a)m_(t)^(1-b)]^((1)/(1-b))l_(t)^(1-eta)}^(1-Phi)u\left(c_{t}, m_{t}, l_{t}\right)=\left(\frac{1}{1-\Phi}\right)\left\{\left[a c_{t}^{1-b}+(1-a) m_{t}^{1-b}\right]^{\frac{1}{1-b}} l_{t}^{1-\eta}\right\}^{1-\Phi}

（a） 推导出家庭最佳货币持有量的一阶条件。在 MIU 模型中，最佳货币持有的基本条件是货币和消费之间的边际替代率等于持有货币的机会成本（见 2.26 或 2.57）。根据此问题中给出的效用函数的规范，此条件为
u m u c = ( 1 − a ) m t − b a c t − b = i t 1 + i t . u m u c = ( 1 − a ) m t − b a c t − b = i t 1 + i t . (u_(m))/(u_(c))=((1-a)m_(t)^(-b))/(ac_(t)^(-b))=(i_(t))/(1+i_(t)).\frac{u_{m}}{u_{c}}=\frac{(1-a) m_{t}^{-b}}{a c_{t}^{-b}}=\frac{i_{t}}{1+i_{t}} .

因为效用可以写成 v ¯ ( c , m ) ϕ ( l ) v ¯ ( c , m ) ϕ ( l ) bar(v)(c,m)phi(l)\bar{v}(c, m) \phi(l) ，所以货币和消费之间的边际替代率与闲暇无关。

（b） 说明如何使用此效用函数规范更改 （2.72） 和 （2.73）。方程 （2.72） 是用消费 λ t λ t lambda_(t)\lambda_{t} 的边际效用来表示的，这一点没有改变。但是，与 consumption 之间的关系 λ t λ t lambda_(t)\lambda_{t} 确实发生了变化。在不可分离效用的情况下，消费的边际效用现在取决于实际货币持有量和闲暇时间：
λ t ≡ u c ( c t , m t , l t ) = { [ a c t 1 − b + ( 1 − a ) m t 1 − b ] 1 1 − b } − Φ × [ a c t 1 − b + ( 1 − a ) m t 1 − b ] b 1 − b ( l t 1 − η ) 1 − Φ a c t − b = { [ a c t 1 − b + ( 1 − a ) m t 1 − b ] b − Φ 1 − b } ( l t 1 − η ) 1 − Φ a c t − b . λ t ≡ u c c t , m t , l t = a c t 1 − b + ( 1 − a ) m t 1 − b 1 1 − b − Φ × a c t 1 − b + ( 1 − a ) m t 1 − b b 1 − b l t 1 − η 1 − Φ a c t − b = a c t 1 − b + ( 1 − a ) m t 1 − b b − Φ 1 − b l t 1 − η 1 − Φ a c t − b . {:[lambda_(t)-=u_(c)(c_(t),m_(t),l_(t))={[ac_(t)^(1-b)+(1-a)m_(t)^(1-b)]^((1)/(1-b))}^(-Phi)xx],[[ac_(t)^(1-b)+(1-a)m_(t)^(1-b)]^((b)/(1-b))(l_(t)^(1-eta))^(1-Phi)ac_(t)^(-b)],[={[ac_(t)^(1-b)+(1-a)m_(t)^(1-b)]^((b-Phi)/(1-b))}(l_(t)^(1-eta))^(1-Phi)ac_(t)^(-b).]:}\begin{aligned} \lambda_{t} \equiv & u_{c}\left(c_{t}, m_{t}, l_{t}\right)=\left\{\left[a c_{t}^{1-b}+(1-a) m_{t}^{1-b}\right]^{\frac{1}{1-b}}\right\}^{-\Phi} \times \\ & {\left[a c_{t}^{1-b}+(1-a) m_{t}^{1-b}\right]^{\frac{b}{1-b}}\left(l_{t}^{1-\eta}\right)^{1-\Phi} a c_{t}^{-b} } \\ = & \left\{\left[a c_{t}^{1-b}+(1-a) m_{t}^{1-b}\right]^{\frac{b-\Phi}{1-b}}\right\}\left(l_{t}^{1-\eta}\right)^{1-\Phi} a c_{t}^{-b} . \end{aligned}

线性化后，这将变为
λ ^ t = Ω 1 c ^ t − Ω 2 m ^ t + ( 1 − Φ ) ( 1 − η ) l ^ t λ ^ t = Ω 1 c ^ t − Ω 2 m ^ t + ( 1 − Φ ) ( 1 − η ) l ^ t hat(lambda)_(t)=Omega_(1) hat(c)_(t)-Omega_(2) hat(m)_(t)+(1-Phi)(1-eta) hat(l)_(t)\hat{\lambda}_{t}=\Omega_{1} \hat{c}_{t}-\Omega_{2} \hat{m}_{t}+(1-\Phi)(1-\eta) \hat{l}_{t}

由于 l ^ = − n n ^ t / ( 1 − n ) l ^ = − n n ^ t / ( 1 − n ) hat(l)=-n hat(n)_(t)//(1-n)\hat{l}=-n \hat{n}_{t} /(1-n) ，我们可以写
λ ^ t = Ω 1 c ^ t − Ω 2 m ^ t − Ω 3 n ^ t λ ^ t = Ω 1 c ^ t − Ω 2 m ^ t − Ω 3 n ^ t hat(lambda)_(t)=Omega_(1) hat(c)_(t)-Omega_(2) hat(m)_(t)-Omega_(3) hat(n)_(t)\hat{\lambda}_{t}=\Omega_{1} \hat{c}_{t}-\Omega_{2} \hat{m}_{t}-\Omega_{3} \hat{n}_{t}

其中 Ω 3 = ( 1 − Φ ) ( 1 − η ) n / ( 1 − n ) Ω 3 = ( 1 − Φ ) ( 1 − η ) n / ( 1 − n ) Omega_(3)=(1-Phi)(1-eta)n//(1-n)\Omega_{3}=(1-\Phi)(1-\eta) n /(1-n) .将此与 （2.67） 进行比较。对于劳动力市场均衡，闲暇和消费之间的边际替代率必须等于劳动的边际产品。此条件采用以下形式
{ [ a c t 1 − b + ( 1 − a ) m t 1 − b ] 1 − Φ 1 − b } ( l t 1 − η ) − Φ ( 1 − η ) l t − η { [ a c t 1 − b + ( 1 − a ) m t 1 − b ] b − Φ 1 − b } ( l t 1 − η ) 1 − Φ a c t − b = α Y t N t a c t 1 − b + ( 1 − a ) m t 1 − b 1 − Φ 1 − b l t 1 − η − Φ ( 1 − η ) l t − η a c t 1 − b + ( 1 − a ) m t 1 − b b − Φ 1 − b l t 1 − η 1 − Φ a c t − b = α Y t N t ({[ac_(t)^(1-b)+(1-a)m_(t)^(1-b)]^((1-Phi)/(1-b))}(l_(t)^(1-eta))^(-Phi)(1-eta)l_(t)^(-eta))/({[ac_(t)^(1-b)+(1-a)m_(t)^(1-b)]^((b-Phi)/(1-b))}(l_(t)^(1-eta))^(1-Phi)ac_(t)^(-b))=alpha(Y_(t))/(N_(t))\frac{\left\{\left[a c_{t}^{1-b}+(1-a) m_{t}^{1-b}\right]^{\frac{1-\Phi}{1-b}}\right\}\left(l_{t}^{1-\eta}\right)^{-\Phi}(1-\eta) l_{t}^{-\eta}}{\left\{\left[a c_{t}^{1-b}+(1-a) m_{t}^{1-b}\right]^{\frac{b-\Phi}{1-b}}\right\}\left(l_{t}^{1-\eta}\right)^{1-\Phi} a c_{t}^{-b}}=\alpha \frac{Y_{t}}{N_{t}}
  或
X c t b a l t = α Y t N t X c t b a l t = α Y t N t (Xc_(t)^(b))/(al_(t))=alpha(Y_(t))/(N_(t))\frac{X c_{t}^{b}}{a l_{t}}=\alpha \frac{Y_{t}}{N_{t}}

其中 X = a c t 1 − b + ( 1 − a ) m t 1 − b X = a c t 1 − b + ( 1 − a ) m t 1 − b X=ac_(t)^(1-b)+(1-a)m_(t)^(1-b)X=a c_{t}^{1-b}+(1-a) m_{t}^{1-b} .线性化此表达式可得到
x ^ t + b c ^ t + ( n 1 − n ) n ^ t = y ^ t − n ^ t x ^ t + b c ^ t + n 1 − n n ^ t = y ^ t − n ^ t hat(x)_(t)+b hat(c)_(t)+((n)/(1-n)) hat(n)_(t)= hat(y)_(t)- hat(n)_(t)\hat{x}_{t}+b \hat{c}_{t}+\left(\frac{n}{1-n}\right) \hat{n}_{t}=\hat{y}_{t}-\hat{n}_{t}

这可以与 （2.73） 进行比较。请注意，与 （2.73） 不同，（3） 独立于 η η eta\eta 。从 的 X X XX 定义 ，
x ^ t = ( 1 − b ) γ c ^ t + ( 1 − b ) ( 1 − γ ) m ^ t x ^ t = ( 1 − b ) γ c ^ t + ( 1 − b ) ( 1 − γ ) m ^ t hat(x)_(t)=(1-b)gamma hat(c)_(t)+(1-b)(1-gamma) hat(m)_(t)\hat{x}_{t}=(1-b) \gamma \hat{c}_{t}+(1-b)(1-\gamma) \hat{m}_{t}
  其中 γ = a ( c s s ) 1 − b / [ a ( c s s ) 1 − b + ( 1 − a ) ( m s s ) 1 − b ] γ = a c s s 1 − b / a c s s 1 − b + ( 1 − a ) m s s 1 − b gamma=a(c^(ss))^(1-b)//[a(c^(ss))^(1-b)+(1-a)(m^(ss))^(1-b)]\gamma=a\left(c^{s s}\right)^{1-b} /\left[a\left(c^{s s}\right)^{1-b}+(1-a)\left(m^{s s}\right)^{1-b}\right] .因此
( 1 − b ) γ c ^ t + ( 1 − b ) ( 1 − γ ) m ^ t + b c ^ t + ( n 1 − n ) n ^ t = y ^ t − n ^ t TESTBANKSELIER.COM ( 1 − b ) γ c ^ t + ( 1 − b ) ( 1 − γ ) m ^ t + b c ^ t + n 1 − n n ^ t = y ^ t − n ^ t  TESTBANKSELIER.COM  {:[(1-b)gamma hat(c)_(t)+(1-b)(1-gamma) hat(m)_(t)+b hat(c)_(t)+((n)/(1-n)) hat(n)_(t)= hat(y)_(t)- hat(n)_(t)],[" TESTBANKSELIER.COM "]:}\begin{gathered} (1-b) \gamma \hat{c}_{t}+(1-b)(1-\gamma) \hat{m}_{t}+b \hat{c}_{t}+\left(\frac{n}{1-n}\right) \hat{n}_{t}=\hat{y}_{t}-\hat{n}_{t} \\ \text { TESTBANKSELIER.COM } \end{gathered}
  或
[ 1 + ( n s s 1 − n s s ) ] n ^ t = y ^ t − [ b + ( 1 − b ) γ ] c ^ t − ( 1 − b ) ( 1 − γ ) m ^ t 1 + n s s 1 − n s s n ^ t = y ^ t − [ b + ( 1 − b ) γ ] c ^ t − ( 1 − b ) ( 1 − γ ) m ^ t [1+((n^(ss))/(1-n^(ss)))] hat(n)_(t)= hat(y)_(t)-[b+(1-b)gamma] hat(c)_(t)-(1-b)(1-gamma) hat(m)_(t)\left[1+\left(\frac{n^{s s}}{1-n^{s s}}\right)\right] \hat{n}_{t}=\hat{y}_{t}-[b+(1-b) \gamma] \hat{c}_{t}-(1-b)(1-\gamma) \hat{m}_{t}

  1. 假设实用程序函数 （2.66） 被
u ( c t , m t , 1 − n t ) = [ a C t 1 − b + ( 1 − a ) m t 1 − b ] 1 − Φ 1 − b 1 − Φ [ ( 1 − n t ) 1 − η 1 − η ] u c t , m t , 1 − n t = a C t 1 − b + ( 1 − a ) m t 1 − b 1 − Φ 1 − b 1 − Φ 1 − n t 1 − η 1 − η u(c_(t),m_(t),1-n_(t))=([aC_(t)^(1-b)+(1-a)m_(t)^(1-b)]^((1-Phi)/(1-b)))/(1-Phi)[((1-n_(t))^(1-eta))/(1-eta)]u\left(c_{t}, m_{t}, 1-n_{t}\right)=\frac{\left[a C_{t}^{1-b}+(1-a) m_{t}^{1-b}\right]^{\frac{1-\Phi}{1-b}}}{1-\Phi}\left[\frac{\left(1-n_{t}\right)^{1-\eta}}{1-\eta}\right]

（a） 推导出家庭最佳货币持有量的一阶条件。在 MIU 模型中，最佳货币持有的基本条件是货币和消费之间的边际替代率等于持有货币的机会成本。根据此问题中给出的效用函数的规范，此条件为
u m u c = ( 1 − a ) m t − b a c t − b = i t 1 + i t u m u c = ( 1 − a ) m t − b a c t − b = i t 1 + i t (u_(m))/(u_(c))=((1-a)m_(t)^(-b))/(ac_(t)^(-b))=(i_(t))/(1+i_(t))\frac{u_{m}}{u_{c}}=\frac{(1-a) m_{t}^{-b}}{a c_{t}^{-b}}=\frac{i_{t}}{1+i_{t}}

这与使用 （2.57） 获得的相同。因为效用可以写成 v ( c , m ) ϕ ( l ) v ( c , m ) ϕ ( l ) v(c,m)phi(l)v(c, m) \phi(l) ，所以货币和消费之间的边际替代率与闲暇无关。

（b） 说明如何使用此效用函数规范更改 （2.72） 和 （2.73）。由 （2.72） 表示的基本 Euler 条件没有改变。如果 是 λ ^ t λ ^ t hat(lambda)_(t)\hat{\lambda}_{t} 稳态附近消费边际效用的百分比偏差，则仍然有
λ ^ t = E t λ ^ t + 1 + r ^ t . λ ^ t = E t λ ^ t + 1 + r ^ t . hat(lambda)_(t)=E_(t) hat(lambda)_(t+1)+ hat(r)_(t).\hat{\lambda}_{t}=\mathrm{E}_{t} \hat{\lambda}_{t+1}+\hat{r}_{t} .

然而，（2.73） 是闲暇和消费之间的边际替代率等于劳动边际产品的条件。在不可分离效用的情况下，闲暇的边际效用现在取决于实际货币持有量和闲暇时间：
u l ( c t , m t , l t ) = [ a c t 1 − b + ( 1 − a ) m t 1 − b ] 1 − Φ 1 − b 1 − Φ ( 1 − n t ) − η u l c t , m t , l t = a c t 1 − b + ( 1 − a ) m t 1 − b 1 − Φ 1 − b 1 − Φ 1 − n t − η u_(l)(c_(t),m_(t),l_(t))=([ac_(t)^(1-b)+(1-a)m_(t)^(1-b)]^((1-Phi)/(1-b)))/(1-Phi)(1-n_(t))^(-eta)u_{l}\left(c_{t}, m_{t}, l_{t}\right)=\frac{\left[a c_{t}^{1-b}+(1-a) m_{t}^{1-b}\right]^{\frac{1-\Phi}{1-b}}}{1-\Phi}\left(1-n_{t}\right)^{-\eta}
Since 
u c ( c t , m t , l t ) = a ( 1 − b ) c t − b [ a c t 1 − b + ( 1 − a ) m t 1 − b ] b − Φ 1 − b [ ( 1 − n t ) 1 − η 1 − η ] u l ( c t , m t , l t ) u c ( c t , m t , l t ) = [ a c t 1 − b + ( 1 − a ) m t 1 − b ] 1 − Φ ( 1 − η ) c t b a ( 1 − b ) ( 1 − n t ) = [ a c t + ( 1 − a ) m t 1 − b c t b ] ( 1 − n t ) ( 1 − η ) a ( 1 − b ) ( 1 − Φ ) u c c t , m t , l t = a ( 1 − b ) c t − b a c t 1 − b + ( 1 − a ) m t 1 − b b − Φ 1 − b 1 − n t 1 − η 1 − η u l c t , m t , l t u c c t , m t , l t = a c t 1 − b + ( 1 − a ) m t 1 − b 1 − Φ ( 1 − η ) c t b a ( 1 − b ) 1 − n t = a c t + ( 1 − a ) m t 1 − b c t b 1 − n t ( 1 − η ) a ( 1 − b ) ( 1 − Φ ) {:[u_(c)(c_(t),m_(t),l_(t))=a(1-b)c_(t)^(-b)[ac_(t)^(1-b)+(1-a)m_(t)^(1-b)]^((b-Phi)/(1-b))[((1-n_(t))^(1-eta))/(1-eta)]],[(u_(l)(c_(t),m_(t),l_(t)))/(u_(c)(c_(t),m_(t),l_(t)))=([ac_(t)^(1-b)+(1-a)m_(t)^(1-b)])/(1-Phi)((1-eta)c_(t)^(b))/(a(1-b)(1-n_(t)))],[=([ac_(t)+(1-a)m_(t)^(1-b)c_(t)^(b)])/((1-n_(t)))((1-eta))/(a(1-b)(1-Phi))]:}\begin{aligned} & u_{c}\left(c_{t}, m_{t}, l_{t}\right)=a(1-b) c_{t}^{-b}\left[a c_{t}^{1-b}+(1-a) m_{t}^{1-b}\right]^{\frac{b-\Phi}{1-b}}\left[\frac{\left(1-n_{t}\right)^{1-\eta}}{1-\eta}\right] \\ & \frac{u_{l}\left(c_{t}, m_{t}, l_{t}\right)}{u_{c}\left(c_{t}, m_{t}, l_{t}\right)}=\frac{\left[a c_{t}^{1-b}+(1-a) m_{t}^{1-b}\right]}{1-\Phi} \frac{(1-\eta) c_{t}^{b}}{a(1-b)\left(1-n_{t}\right)} \\ &=\frac{\left[a c_{t}+(1-a) m_{t}^{1-b} c_{t}^{b}\right]}{\left(1-n_{t}\right)} \frac{(1-\eta)}{a(1-b)(1-\Phi)} \end{aligned}
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 ) = A a c t ( 1 − n t ) − 1 + A ( 1 − a ) m t 1 − b c t b ( 1 − n t ) − 1 u l c t , m t , l t u c c t , m t , l t = A a c t 1 − n t − 1 + A ( 1 − a ) m t 1 − b c t b 1 − n t − 1 (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 
A a c t ( 1 − n t ) − 1 + A ( 1 − a ) m t 1 − b c t b ( 1 − n t ) ≈ A a c 1 − n ( 1 + c ^ t + n ^ t ) + A ( 1 − a ) 1 − n m 1 − b c b ( 1 + ( 1 − b ) m ^ t + b c ^ t + n ^ t ) A a c t 1 − n t − 1 + A ( 1 − a ) m t 1 − b c t b 1 − n t ≈ A a c 1 − n 1 + c ^ t + n ^ t + A ( 1 − a ) 1 − n m 1 − b c b 1 + ( 1 − b ) m ^ t + b c ^ t + n ^ t {:[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 u_(l)//u_(c)u_{l} / u_{c} is equal to the marginal product of labor, we have u l / u c = α Y t / N t u l / u c = α Y t / N t u_(l)//u_(c)=alphaY_(t)//N_(t)u_{l} / u_{c}=\alpha Y_{t} / N_{t}, or 
A a c 1 − n ( 1 + c ^ t + n ^ t ) + A ( 1 − a ) 1 − n m 1 − b c b ( 1 + ( 1 − b ) m ^ t + b c ^ t + n ^ t ) = α ( y n ) ( 1 + y ^ t − n ^ t ) A a c 1 − n 1 + c ^ t + n ^ t + A ( 1 − a ) 1 − n m 1 − b c b 1 + ( 1 − b ) m ^ t + b c ^ t + n ^ t = α y n 1 + y ^ t − 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))=alpha((y)/(n))(1+ hat(y)_(t)- hat(n)_(t))\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)=\alpha\left(\frac{y}{n}\right)\left(1+\hat{y}_{t}-\hat{n}_{t}\right)
But since in the steady state, 
A a c 1 − n + A ( 1 − a ) 1 − n m 1 − b c b = α ( y n ) A a c 1 − n + A ( 1 − a ) 1 − n m 1 − b c b = α y n (Aac)/(1-n)+(A(1-a))/(1-n)m^(1-b)c^(b)=alpha((y)/(n))\frac{A a c}{1-n}+\frac{A(1-a)}{1-n} m^{1-b} c^{b}=\alpha\left(\frac{y}{n}\right)
this becomes 
γ ( c ^ t + n ^ t ) + ( 1 − γ ) [ ( 1 − b ) m ^ t + b c ^ t + n ^ t ] = α ( y n ) ( y ^ t − n ^ t ) γ c ^ t + n ^ t + ( 1 − γ ) ( 1 − b ) m ^ t + b c ^ t + n ^ t = α y n y ^ t − n ^ t gamma( hat(c)_(t)+ hat(n)_(t))+(1-gamma)[(1-b) hat(m)_(t)+b hat(c)_(t)+ hat(n)_(t)]=alpha((y)/(n))( hat(y)_(t)- hat(n)_(t))\gamma\left(\hat{c}_{t}+\hat{n}_{t}\right)+(1-\gamma)\left[(1-b) \hat{m}_{t}+b \hat{c}_{t}+\hat{n}_{t}\right]=\alpha\left(\frac{y}{n}\right)\left(\hat{y}_{t}-\hat{n}_{t}\right)
where γ ≡ [ γ ≡ [ gamma-=[\gamma \equiv[ Aac/(1-n)] ( n / α y ) ( n / α y ) (n//alpha y)(n / \alpha y), or 
n ^ t + [ γ + ( 1 − γ ) b ] c ^ t + ( 1 − γ ) ( 1 − b ) m ^ t = α ( y n ) ( y ^ t − n ^ t ) n ^ t + [ γ + ( 1 − γ ) b ] c ^ t + ( 1 − γ ) ( 1 − b ) m ^ t = α y n y ^ t − n ^ t hat(n)_(t)+[gamma+(1-gamma)b] hat(c)_(t)+(1-gamma)(1-b) hat(m)_(t)=alpha((y)/(n))( hat(y)_(t)- hat(n)_(t))\hat{n}_{t}+[\gamma+(1-\gamma) b] \hat{c}_{t}+(1-\gamma)(1-b) \hat{m}_{t}=\alpha\left(\frac{y}{n}\right)\left(\hat{y}_{t}-\hat{n}_{t}\right)
which is the equivalent of (2.73). 
7. Suppose a nominal interest rate of i m i m i^(m)i^{m} is paid on money balances. These payments are financed by a combination of lump-sum taxes and printing money. Let a a aa be the fraction financed by lump-sum taxes. The government’s budget identity is τ t + v t = i m m t τ t + v t = i m m t tau_(t)+v_(t)=i^(m)m_(t)\tau_{t}+v_{t}=i^{m} m_{t}, with τ t = a i m m t τ t = a i m m t tau_(t)=ai^(m)m_(t)\tau_{t}=a i^{m} m_{t} and v = θ m t v = θ m t 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 + π − i m = i − i m r + π − i m = i − i m 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 ( τ τ tau\tau in 2.4) consists of two components, the first being the lumpsum transfer v v vv and the second being the lump-sum tax τ τ tau\tau. Thus, the budget constraint becomes 
f ( k t − 1 ) + ( 1 − δ ) k t − 1 + 1 + i m 1 + π m t − 1 − τ t + v t = c t + k t + m t f k t − 1 + ( 1 − δ ) k t − 1 + 1 + i m 1 + π m t − 1 − τ t + v t = c t + k t + m t f(k_(t-1))+(1-delta)k_(t-1)+(1+i^(m))/(1+pi)m_(t-1)-tau_(t)+v_(t)=c_(t)+k_(t)+m_(t)f\left(k_{t-1}\right)+(1-\delta) k_{t-1}+\frac{1+i^{m}}{1+\pi} m_{t-1}-\tau_{t}+v_{t}=c_{t}+k_{t}+m_{t}

为了简单起见，人口增长被忽略了。问题的值函数仍然由 （2.5） 给出，但由于预算约束的变化，一阶条件发生了变化。特别是，（2.8） 变为
u m − β [ f k + 1 − δ ] V ω ( ω t + 1 ) + β ( 1 + i m ) 1 + π V ω ( ω t + 1 ) = 0 u m − β f k + 1 − δ V ω ω t + 1 + β 1 + i m 1 + π V ω ω t + 1 = 0 u_(m)-beta[f_(k)+1-delta]V_(omega)(omega_(t+1))+(beta(1+i^(m)))/(1+pi)V_(omega)(omega_(t+1))=0u_{m}-\beta\left[f_{k}+1-\delta\right] V_{\omega}\left(\omega_{t+1}\right)+\frac{\beta\left(1+i^{m}\right)}{1+\pi} V_{\omega}\left(\omega_{t+1}\right)=0
The first-order condition for consumption (see 2.6), together with the envelope condition (see 2.10) implies 
u c ( c t , m t ) = β [ f k + 1 − δ ] V ω ( ω t + 1 ) = β [ f k + 1 − δ ] u c ( c t + 1 , m t + 1 ) = β R u c ( c t + 1 , m t + 1 ) u c c t , m t = β f k + 1 − δ V ω ω t + 1 = β f k + 1 − δ u c c t + 1 , m t + 1 = β R u c c t + 1 , m t + 1 {:[u_(c)(c_(t),m_(t))=beta[f_(k)+1-delta]V_(omega)(omega_(t+1))=beta[f_(k)+1-delta]u_(c)(c_(t+1),m_(t+1))],[=beta Ru_(c)(c_(t+1),m_(t+1))]:}\begin{aligned} u_{c}\left(c_{t}, m_{t}\right) & =\beta\left[f_{k}+1-\delta\right] V_{\omega}\left(\omega_{t+1}\right)=\beta\left[f_{k}+1-\delta\right] u_{c}\left(c_{t+1}, m_{t+1}\right) \\ & =\beta R u_{c}\left(c_{t+1}, m_{t+1}\right) \end{aligned}
Using this result, (2.11) can be rearranged, resulting in 
u m u c = 1 − [ β ( 1 + i m ) 1 + π ] u c ( c t + 1 , m t + 1 ) u c ( c t , m t ) = 1 − [ β ( 1 + i m ) 1 + π ] ( 1 β R ) = 1 − ( 1 + i m ) R ( 1 + π ) = R ( 1 + π ) − ( 1 + i m ) R ( 1 + π ) = i − i m 1 + i u m u c = 1 − β 1 + i m 1 + π u c c t + 1 , m t + 1 u c c t , m t = 1 − β 1 + i m 1 + π 1 β R = 1 − 1 + i m R ( 1 + π ) = R ( 1 + π ) − 1 + i m R ( 1 + π ) = i − i m 1 + i {:[(u_(m))/(u_(c))=1-[(beta(1+i^(m)))/(1+pi)](u_(c)(c_(t+1),m_(t+1)))/(u_(c)(c_(t),m_(t)))=1-[(beta(1+i^(m)))/(1+pi)]((1)/(beta R))],[=1-((1+i^(m)))/(R(1+pi))=(R(1+pi)-(1+i^(m)))/(R(1+pi))=(i-i^(m))/(1+i)]:}\begin{aligned} \frac{u_{m}}{u_{c}} & =1-\left[\frac{\beta\left(1+i^{m}\right)}{1+\pi}\right] \frac{u_{c}\left(c_{t+1}, m_{t+1}\right)}{u_{c}\left(c_{t}, m_{t}\right)}=1-\left[\frac{\beta\left(1+i^{m}\right)}{1+\pi}\right]\left(\frac{1}{\beta R}\right) \\ & =1-\frac{\left(1+i^{m}\right)}{R(1+\pi)}=\frac{R(1+\pi)-\left(1+i^{m}\right)}{R(1+\pi)}=\frac{i-i^{m}}{1+i} \end{aligned}
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 i^(m)i^{m}, this opportunity cost is i − i m i − i m 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 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 = θ m = ( 1 − a ) i m m v = θ m = ( 1 − a ) i m m v=theta m=(1-a)i^(m)mv=\theta m=(1-a) i^{m} m, or the rate of money growth will equal θ = ( 1 − a ) i m θ = ( 1 − a ) i m theta=(1-a)i^(m)\theta=(1-a) i^{m}. In the steady-state, π = θ π = θ pi=theta\pi=\theta. This means that π = ( 1 − a ) i m π = ( 1 − a ) i m pi=(1-a)i^(m)\pi=(1-a) i^{m}. Hence, the opportunity cost of money is given by 
i − i m ≈ r + π − i m = r + ( 1 − a ) i m − i m = r − a i m i − i m ≈ r + π − i m = r + ( 1 − a ) i m − i m = r − a i m i-i^(m)~~r+pi-i^(m)=r+(1-a)i^(m)-i^(m)=r-ai^(m)i-i^{m} \approx r+\pi-i^{m}=r+(1-a) i^{m}-i^{m}=r-a i^{m}
where r = R − 1 r = R − 1 r=R-1r=R-1. Paying interest on money affects the opportunity cost of money only if a > 0 a > 0 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 i i 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 τ t τ t 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, τ = θ m τ = θ m 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 ) 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 / β 1 / β 1//beta1 / \beta. Hence, 
1 − δ + f k = 1 β 1 − δ + f k = 1 β 1-delta+f_(k)=(1)/(beta)1-\delta+f_{k}=\frac{1}{\beta}
where f k f k 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 m m mm. Thus, in the steady state, 
1 − δ + f k ( k s s , m s s ) = 1 β 1 − δ + f k k s s , m s s = 1 β 1-delta+f_(k)(k^(ss),m^(ss))=(1)/(beta)1-\delta+f_{k}\left(k^{s s}, m^{s s}\right)=\frac{1}{\beta}
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 s s , m s s ) = β − 1 − 1 + δ f k k s s , m s s = β − 1 − 1 + δ 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 η η 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 i_(0)i_{0} (so the inflation rate is i 0 − r i 0 − r i_(0)-ri_{0}-r ), (2.39) gives money demand as m = A i − η m = A i − η m=Ai^(-eta)m=A i^{-\eta}, so this area can be calculated as 
c s t = ∫ 0 i 0 A i − η d i = A ( 1 1 − η ) i 0 1 − η c s t = ∫ 0 i 0   A i − η d i = A 1 1 − η i 0 1 − η 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 η η 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 = A i 0 1 − η i 0 m = A i 0 1 − η i_(0)m=Ai_(0)^(1-eta)i_{0} m=A i_{0}^{1-\eta}, so the deadweight loss is A [ η / ( 1 − η ) ] i 0 1 − η A [ η / ( 1 − η ) ] i 0 1 − η 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 ξ ξ 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 i_(0)i_{0} (so the inflation rate is i 0 − r ) i 0 − r {:i_(0)-r)\left.i_{0}-r\right), (2.40) gives money demand as m = B exp ⁡ ( − ξ i ) m = B exp ⁡ ( − ξ i ) m=B exp(-xi i)m=B \exp (-\xi i), so this area can be calculated as 
c s t = ∫ 0 i 0 B e − ξ i d i = ( B ξ ) ( 1 − e − ξ i 0 ) c s t = ∫ 0 i 0   B e − ξ i d i = B ξ 1 − e − ξ i 0 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 ξ ξ 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 ⁡ ( − ξ i 0 ) i 0 m = i 0 B exp ⁡ − ξ i 0 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 / ξ ) [ 1 − ( 1 + ξ i 0 ) e − ξ i 0 ] ( B / ξ ) 1 − 1 + ξ i 0 e − ξ i 0 (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 
∑ β i u ( c t + i , m t + i , l t + i ) = ∑ β i ( c t + i m t + i ) b l t + i d ∑ β i u c t + i , m t + i , l t + i = ∑ β i c t + i m t + i b l t + i d sumbeta^(i)u(c_(t+i),m_(t+i),l_(t+i))=sumbeta^(i)(c_(t+i)m_(t+i))^(b)l_(t+i)^(d)\sum \beta^{i} u\left(c_{t+i}, m_{t+i}, l_{t+i}\right)=\sum \beta^{i}\left(c_{t+i} m_{t+i}\right)^{b} l_{t+i}^{d}
but not if they are given by 
∑ β i u ( c t ± i , m t ± i , l t + i ) = ∑ β i ( c t Ψ i + k m t + i ) b l t + i d ∑ β i u c t ± i , m t ± i , l t + i = ∑ β i c t Ψ i + k m t + i b l t + i d sumbeta^(i)u(c_(t+-i),m_(t+-i),l_(t+i))=sumbeta^(i)(c_(t Psi i)+km_(t+i))^(b)l_(t+i)^(d)\sum \beta^{i} u\left(c_{t \pm i}, m_{t \pm i}, l_{t+i}\right)=\sum \beta^{i}\left(c_{t \Psi i}+k m_{t+i}\right)^{b} l_{t+i}^{d}
Discuss. (Assume output depends on capital and labor, and the aggregate production function is Cobb-Douglas.) The steady-state values of c s s , k s s , l s s , y s s c s s , k s s , l s s , y s s 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: 
u l u c = f l ( k s s , 1 − l s s ) f k ( k s s , 1 − l s s ) = 1 β − 1 + δ c s s = f ( k s s , 1 − l s s ) − δ k s s y s s = f ( k s s , 1 − l s s ) u l u c = f l k s s , 1 − l s s f k k s s , 1 − l s s = 1 β − 1 + δ c s s = f k s s , 1 − l s s − δ k s s y s s = f k s s , 1 − l s s {:[(u_(l))/(u_(c))=f_(l)(k^(ss),1-l^(ss))],[f_(k)(k^(ss),1-l^(ss))=(1)/(beta)-1+delta],[c^(ss)=f(k^(ss),1-l^(ss))-deltak^(ss)],[y^(ss)=f(k^(ss),1-l^(ss))]:}\begin{gathered} \frac{u_{l}}{u_{c}}=f_{l}\left(k^{s s}, 1-l^{s s}\right) \\ f_{k}\left(k^{s s}, 1-l^{s s}\right)=\frac{1}{\beta}-1+\delta \\ c^{s s}=f\left(k^{s s}, 1-l^{s s}\right)-\delta k^{s s} \\ y^{s s}=f\left(k^{s s}, 1-l^{s s}\right) \end{gathered}
If the single period utility function is of the form ( c t + i m t + i ) b l t + i d c t + i m t + i b l t + i d (c_(t+i)m_(t+i))^(b)l_(t+i)^(d)\left(c_{t+i} m_{t+i}\right)^{b} l_{t+i}^{d}, then 
u l u c = d c b m b l d − 1 b c b − 1 m b l d = d c b l u l u c = d c b m b l d − 1 b c b − 1 m b l d = d c b l (u_(l))/(u_(c))=(dc^(b)m^(b)l^(d-1))/(bc^(b-1)m^(b)l^(d))=(dc)/(bl)\frac{u_{l}}{u_{c}}=\frac{d c^{b} m^{b} l^{d-1}}{b c^{b-1} m^{b} l^{d}}=\frac{d c}{b l}
is independent of m m mm and (5) - (8) involve only the four unknowns c s s , k s s , l s s , y s s c s s , k s s , l s s , y s s 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 s s , k s s , l s s c s s , k s s , l s s c^(ss),k^(ss),l^(ss)c^{s s}, k^{s s}, l^{s s}, and y s s y s s y^(ss)y^{s s} independently of m m mm or inflation. Superneutrality holds. If the utility function is ( c t + i + k m t + i ) b l t + i d c t + i + k m t + i b l t + i d (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 
u l u c = d ( c + k m ) b l d − 1 b ( c + k m ) b − 1 l d = d ( c + k m ) a l u l u c = d ( c + k m ) b l d − 1 b ( c + k m ) b − 1 l d = d ( c + k m ) a l (u_(l))/(u_(c))=(d(c+km)^(b)l^(d-1))/(b(c+km)^(b-1)l^(d))=(d(c+km))/(al)\frac{u_{l}}{u_{c}}=\frac{d(c+k m)^{b} l^{d-1}}{b(c+k m)^{b-1} l^{d}}=\frac{d(c+k m)}{a l}
which is not independent of m m mm. Thus, (5) - (8) will involve 5 unknowns ( c s s , k s s , l s s , y s s c s s , k s s , l s s , y s s c^(ss),k^(ss),l^(ss),y^(ss)c^{s s}, k^{s s}, l^{s s}, y^{s s}, and m ss m ss  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 = (u=(u= a ln ⁡ c t + ( 1 − a ) ln ⁡ m t + Ψ l t 1 − η / ( 1 − η ) ) a ln ⁡ c t + ( 1 − a ) ln ⁡ m t + Ψ l t 1 − η / ( 1 − η ) {: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 Φ = b = 1 , Ω 1 = [ ( b − Φ ) γ − b ] = − b Φ = b = 1 , Ω 1 = [ ( b − Φ ) γ − b ] = − b Phi=b=1,Omega_(1)=[(b-Phi)gamma-b]=-b\Phi=b=1, \Omega_{1}=[(b-\Phi) \gamma-b]=-b and Ω 2 = ( b − Φ ) ( 1 − γ ) = 0 Ω 2 = ( b − Φ ) ( 1 − γ ) = 0 Omega_(2)=(b-Phi)(1-gamma)=0\Omega_{2}=(b-\Phi)(1-\gamma)=0. The linearized expression for λ ^ t λ ^ t hat(lambda)_(t)\hat{\lambda}_{t} is 
λ ^ t = − b 1 c ^ t λ ^ t = − b 1 c ^ t hat(lambda)_(t)=-b_(1) hat(c)_(t)\hat{\lambda}_{t}=-b_{1} \hat{c}_{t}
Then, in linearized form, the equilibrium conditions are: 
y ^ t = α k ^ t − 1 + ( 1 − α ) n ^ t + z t ( x s s k s s ) x ^ t = k ^ t − ( 1 − δ ) k ^ t − 1 , ( y s s k s s ) y ^ t = ( c s s k s s ) c ^ t + δ x ^ t TES r ^ t = α ( y s s k s s ) ( E t y ^ t + 1 − C k ^ t ) c ^ t = E t c ^ t + 1 − ( 1 b ) r ^ t b c ^ t + η ( n s s 1 − n s s ) n ^ t = y ^ t − n ^ t ı ^ t = r ^ t + E t π ^ t + 1 m ^ t − c ^ t = − ( 1 b ) ( 1 − i s s i s s ) ı ^ t m ^ t = m ^ t − 1 − π ^ t + u t . u t = ρ u u t − 1 + ϕ z t − 1 + φ t , 0 ≤ γ < 1 , z t = ρ z z t − 1 + e t y ^ t = α k ^ t − 1 + ( 1 − α ) n ^ t + z t x s s k s s x ^ t = k ^ t − ( 1 − δ ) k ^ t − 1 , y s s k s s y ^ t = c s s k s s c ^ t + δ x ^ t  TES  r ^ t = α y s s k s s E t y ^ t + 1 − C k ^ t c ^ t = E t c ^ t + 1 − 1 b r ^ t b c ^ t + η n s s 1 − n s s n ^ t = y ^ t − n ^ t ı ^ t = r ^ t + E t π ^ t + 1 m ^ t − c ^ t = − 1 b 1 − i s s i s s ı ^ t m ^ t = m ^ t − 1 − π ^ t + u t . u t = ρ u u t − 1 + ϕ z t − 1 + φ t , 0 ≤ γ < 1 , z t = ρ z z t − 1 + e t {:[ hat(y)_(t)=alpha hat(k)_(t-1)+(1-alpha) hat(n)_(t)+z_(t)],[((x^(ss))/(k^(ss))) hat(x)_(t)= hat(k)_(t)-(1-delta) hat(k)_(t-1)","],[((y^(ss))/(k^(ss))) hat(y)_(t)=((c^(ss))/(k^(ss))) hat(c)_(t)+delta hat(x)_(t)],[" TES "( hat(r)_(t))/()=alpha((y^(ss))/(k^(ss)))(E_(t) hat(y)_(t+1)-C hat(k)_(t))],[ hat(c)_(t)=E_(t) hat(c)_(t+1)-((1)/(b)) hat(r)_(t)],[b hat(c)_(t)+eta((n^(ss))/(1-n^(ss))) hat(n)_(t)= hat(y)_(t)- hat(n)_(t)],[ hat(ı)_(t)= hat(r)_(t)+E_(t) hat(pi)_(t+1)],[ hat(m)_(t)- hat(c)_(t)=-((1)/(b))((1-i^(ss))/(i^(ss))) hat(ı)_(t)],[ hat(m)_(t)= hat(m)_(t-1)- hat(pi)_(t)+u_(t).],[u_(t)=rho_(u)u_(t-1)+phiz_(t-1)+varphi_(t)","0 <= gamma < 1","],[z_(t)=rho_(z)z_(t-1)+e_(t)]:}\begin{gathered} \hat{y}_{t}=\alpha \hat{k}_{t-1}+(1-\alpha) \hat{n}_{t}+z_{t} \\ \left(\frac{x^{s s}}{k^{s s}}\right) \hat{x}_{t}=\hat{k}_{t}-(1-\delta) \hat{k}_{t-1}, \\ \left(\frac{y^{s s}}{k^{s s}}\right) \hat{y}_{t}=\left(\frac{c^{s s}}{k^{s s}}\right) \hat{c}_{t}+\delta \hat{x}_{t} \\ \text { TES } \frac{\hat{r}_{t}}{}=\alpha\left(\frac{y^{s s}}{k^{s s}}\right)\left(\mathrm{E}_{t} \hat{y}_{t+1}-\mathrm{C} \hat{k}_{t}\right) \\ \hat{c}_{t}=\mathrm{E}_{t} \hat{c}_{t+1}-\left(\frac{1}{b}\right) \hat{r}_{t} \\ b \hat{c}_{t}+\eta\left(\frac{n^{s s}}{1-n^{s s}}\right) \hat{n}_{t}=\hat{y}_{t}-\hat{n}_{t} \\ \hat{\imath}_{t}=\hat{r}_{t}+\mathrm{E}_{t} \hat{\pi}_{t+1} \\ \hat{m}_{t}-\hat{c}_{t}=-\left(\frac{1}{b}\right)\left(\frac{1-i^{s s}}{i^{s s}}\right) \hat{\imath}_{t} \\ \hat{m}_{t}=\hat{m}_{t-1}-\hat{\pi}_{t}+u_{t} . \\ u_{t}=\rho_{u} u_{t-1}+\phi z_{t-1}+\varphi_{t}, 0 \leq \gamma<1, \\ z_{t}=\rho_{z} z_{t-1}+e_{t} \end{gathered}
Notice that the first six equations can be solved for y ^ t , c ^ t , x ^ t , k ^ t , n ^ t y ^ t , c ^ t , x ^ t , k ^ t , n ^ t 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 r ^ t r ^ t hat(r)_(t)\hat{r}_{t}, i.e., all the real variables. Since m ^ m ^ hat(m)\hat{m} is equal to the deviation of real money balances around its steady-state value, we can write it as m ^ t n − p ^ t m ^ t n − p ^ t hat(m)_(t)^(n)- hat(p)_(t)\hat{m}_{t}^{n}-\hat{p}_{t}, where m ^ n m ^ n hat(m)^(n)\hat{m}^{n} represents nominal money balances. Then, the final four equations (the monetary sector) can be rewritten as 
ı ^ t = r ^ t + ( E t p ^ t + 1 − p ^ t ) ı ^ t = r ^ t + E t p ^ t + 1 − p ^ t hat(ı)_(t)= hat(r)_(t)+(E_(t) hat(p)_(t+1)- hat(p)_(t))\hat{\imath}_{t}=\hat{r}_{t}+\left(\mathrm{E}_{t} \hat{p}_{t+1}-\hat{p}_{t}\right)
m ^ t n − p ^ t = c ^ t − ( 1 b ) ( 1 − i s s i s s ) ı ^ t ( 1 − ρ u L ) m ^ t n = ( 1 − ρ u L ) h̸ t − 1 n + ϕ z t − 1 + φ m ^ t n − p ^ t = c ^ t − 1 b 1 − i s s i s s ı ^ t 1 − ρ u L m ^ t n = 1 − ρ u L h̸ t − 1 n + ϕ z t − 1 + φ {:[ hat(m)_(t)^(n)- hat(p)_(t)= hat(c)_(t)-((1)/(b))((1-i^(ss))/(i^(ss))) hat(ı)_(t)],[(1-rho_(u)L) hat(m)_(t)^(n)=(1-rho_(u)L)h_(t-1)^(n)+phiz_(t-1)+varphi]:}\begin{aligned} \hat{m}_{t}^{n}-\hat{p}_{t} & =\hat{c}_{t}-\left(\frac{1}{b}\right)\left(\frac{1-i^{s s}}{i^{s s}}\right) \hat{\imath}_{t} \\ \left(1-\rho_{u} L\right) \hat{m}_{t}^{n} & =\left(1-\rho_{u} L\right) \not h_{t-1}^{n}+\phi z_{t-1}+\varphi \end{aligned}
where L L LL is the lag operator (so L m ^ t n = m ^ t − 1 n L m ^ t n = m ^ t − 1 n 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 c ^ t c ^ t hat(c)_(t)\hat{c}_{t} and r ^ t r ^ t 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 b b 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 = Φ b = Φ b=Phib=\Phi, neither Ω 1 Ω 1 Omega_(1)\Omega_{1} nor Ω 2 Ω 2 Omega_(2)\Omega_{2} depend on b b bb and variations in m ^ t m ^ t hat(m)_(t)\hat{m}_{t} do not affect the linearized equilibrium conditions for the real variables. For the baseline calibration, Φ = 2 Φ = 2 Phi=2\Phi=2 and b = 9 b = 9 b=9b=9, so b − Φ < 0 b − Φ < 0 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 < Φ b < Φ 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 , 2 b = 1 , 2 b=1,2b=1,2, and 3 are shown in the figure. With b = 2 b = 2 b=2b=2, money growth shocks have no real effects as in this case, b = Φ b = Φ b=Phib=\Phi and Ω 2 = 0 Ω 2 = 0 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 a a 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 γ γ gamma\gamma in the linear approximation rises. This reduces the impact of money on marginal utility (see (2.67) and the definitions of γ γ gamma\gamma and Ω 2 Ω 2 Omega_(2)\Omega_{2} ). When a is small, changes in m m mm have a larger impact on the marginal utility of consumption and so have a larger effect 

在产出和就业方面，从显示对货币供应量冲击的冲动反应的图中可以看出。（可根据要求提供与此练习相关的程序。

3 Chapter 3: Money and Transactions 

  1. Suppose the production function for shopping takes the form ψ = c = e x ( n s ) a m b ψ = c = e x n s a m b psi=c=e^(x)(n^(s))^(a)m^(b)\psi=c=e^{x}\left(n^{s}\right)^{a} m^{b}, where a a aa and b b bb are both positive but less than 1 and x x xx is a productivity factor. The agent’s utility is given by v ( c , l ) = c 1 − Φ / ( 1 − Φ ) + l 1 − η / ( 1 − η ) v ( c , l ) = c 1 − Φ / ( 1 − Φ ) + l 1 − η / ( 1 − η ) 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 l=1-n-n^(s)l=1-n-n^{s}, and n n nn is time spent in market employment. 
    (a) Derive the transaction time function g ( c , m ) = n s g ( c , m ) = n s g(c,m)=n^(s)g(c, m)=n^{s}. From the shopping production function, 
g ( c , m ) = n s = ( c e x m b ) 1 / a g ( c , m ) = n s = c e x m b 1 / a g(c,m)=n^(s)=((c)/(e^(x)m^(b)))^(1//a)g(c, m)=n^{s}=\left(\frac{c}{e^{x} m^{b}}\right)^{1 / a}
(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 a a aa and b b bb ? How does it depend on x x xx ? From the definition of the agent’s utility, 
v ( c , l ) = c 1 − Φ 1 − Φ + ( 1 − n − n s ) 1 − η 1 − η = c 1 − Φ 1 − Φ + ( 1 − n − ( c e x m b ) 1 / a ) 1 − η 1 − η ≡ u ( c , n , m ) v ( c , l ) = c 1 − Φ 1 − Φ + 1 − n − n s 1 − η 1 − η = c 1 − Φ 1 − Φ + 1 − n − c e x m b 1 / a 1 − η 1 − η ≡ u ( c , n , m ) {:[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}
The marginal utility of money is 
∂ u ( c , n , m ) ∂ m = ∂ v ( c , l ) ∂ l ( ∂ l ∂ m ) TESTBANKSEITER.COM = ( 1 − n − n s ) − η ( b n s a m ) . ∂ u ( c , n , m ) ∂ m = ∂ v ( c , l ) ∂ l ∂ l ∂ m  TESTBANKSEITER.COM  = 1 − n − n s − η b n s a m . {:[(del u(c,n,m))/(del m)=(del v(c,l))/(del l)((del l)/(del m))],[" TESTBANKSEITER.COM "],[=(1-n-n^(s))^(-eta)((bn^(s))/(am)).]:}\begin{aligned} & \frac{\partial u(c, n, m)}{\partial m}=\frac{\partial v(c, l)}{\partial l}\left(\frac{\partial l}{\partial m}\right) \\ & \text { TESTBANKSEITER.COM } \\ &=\left(1-n-n^{s}\right)^{-\eta}\left(\frac{b n^{s}}{a m}\right) . \end{aligned}
The time spend shopping can be written as c 1 / a e − x / a m − b / a c 1 / a e − x / a m − b / a c^(1//a)e^(-x//a)m^(-b//a)c^{1 / a} e^{-x / a} m^{-b / a}. The marginal productivity of money in reducing shopping time is given by ( b / a ) ( n s / m ) ( b / a ) n s / m (b//a)(n^(s)//m)(b / a)\left(n^{s} / m\right), so an increase in b / a b / a b//ab / a increases the effect additional money holdings have in reducing the time needed for shopping. Additional money holdings result in more leisure (and more utility) when b / a b / a b//ab / a is large, thus acting to increase the marginal utility of money. In terms of (10), ∂ l / ∂ m ∂ l / ∂ m del l//del m\partial l / \partial m rises with b / a b / a b//ab / a. But the marginal utility of leisure is a decreasing function of total leisure, so ∂ v ( c , l ) / ∂ l ∂ v ( c , l ) / ∂ l del v(c,l)//del l\partial v(c, l) / \partial l declines. The effect of x x xx on the marginal utility of money, for given c c cc and n n nn, operates through n s n s n^(s)n^{s} and represents a productivity shift; an increase in x x xx reduces the time needed for shopping for given values of c c cc and m m mm. This affects the productivity of m m mm in the shopping time production function. The marginal product of money in reducing shopping time is ( b / a ) c 1 / a e − x / a m − ( 1 + b / a ) ( b / a ) c 1 / a e − x / a m − ( 1 + b / a ) (b//a)c^(1//a)e^(-x//a)m^(-(1+b//a))(b / a) c^{1 / a} e^{-x / a} m^{-(1+b / a)}. This is decreasing in x x xx; a higher x x xx decreases the marginal effect of m m mm in reducing shopping time, so money is less “productive.” 
© Is the marginal utility of consumption increasing or decreasing in m m mm ? An increase in consumption affects utility in two ways. First, consumption directly yields utility; v c > v c > v_(c) >v_{c}> 0. This represents the effect of consumption on utility, holding leisure constant. Since leisure is being held constant, v c v c v_(c)v_{c} is independent of m m mm. Second, higher consumption increases the time devoted to shopping, as this reduces the time available for leisure. 
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 c c cc on n s n s n^(s)n^{s}. This means that with higher money holdings, an increase in c c 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 
V a ( a t , k t − 1 ) P t = − ∑ i = 0 ∞ β i ( v l ( a t + i , k t + i − 1 ) g m ( c t + i , m t + i ) P t + i ) V a a t , k t − 1 P t = − ∑ i = 0 ∞   β i v l a t + i , k t + i − 1 g m c t + i , m t + i P t + i (V_(a)(a_(t),k_(t-1)))/(P_(t))=-sum_(i=0)^(oo)beta^(i)((v_(l)(a_(t+i),k_(t+i-1))g_(m)(c_(t+i),m_(t+i)))/(P_(t+i)))\frac{V_{a}\left(a_{t}, k_{t-1}\right)}{P_{t}}=-\sum_{i=0}^{\infty} \beta^{i}\left(\frac{v_{l}\left(a_{t+i}, k_{t+i-1}\right) g_{m}\left(c_{t+i}, m_{t+i}\right)}{P_{t+i}}\right)
Interpret this equation. How does it compare to (3.31)? Equations (3.5) and (3.8) were 
− v l g m + β V a ( a t + 1 , k t ) 1 + π t + 1 − β V k ( a t + 1 , k t ) = 0 − v l g m + β V a a t + 1 , k t 1 + π t + 1 − β V k a t + 1 , k t = 0 -v_(l)g_(m)+beta(V_(a)(a_(t+1),k_(t)))/(1+pi_(t+1))-betaV_(k)(a_(t+1),k_(t))=0-v_{l} g_{m}+\beta \frac{V_{a}\left(a_{t+1}, k_{t}\right)}{1+\pi_{t+1}}-\beta V_{k}\left(a_{t+1}, k_{t}\right)=0
and 
V a ( a t , k t − 1 ) = β V k ( a t + 1 , k t ) V a a t , k t − 1 = β V k a t + 1 , k t V_(a)(a_(t),k_(t-1))=betaV_(k)(a_(t+1),k_(t))V_{a}\left(a_{t}, k_{t-1}\right)=\beta V_{k}\left(a_{t+1}, k_{t}\right)
Using the (3.8), we can rewrite (3.5) as 
V a ( a t , k t − 1 ) = − v l g m + β V a ( a t + 1 , k t ) 1 + π t + 1 = − v l g m + β V a ( a t + 1 , k t ) ( P t P t + 1 ) V a a t , k t − 1 = − v l g m + β V a a t + 1 , k t 1 + π t + 1 = − v l g m + β V a a t + 1 , k t P t P t + 1 V_(a)(a_(t),k_(t-1))=-v_(l)g_(m)+beta(V_(a)(a_(t+1),k_(t)))/(1+pi_(t+1))=-v_(l)g_(m)+betaV_(a)(a_(t+1),k_(t))((P_(t))/(P_(t+1)))V_{a}\left(a_{t}, k_{t-1}\right)=-v_{l} g_{m}+\beta \frac{V_{a}\left(a_{t+1}, k_{t}\right)}{1+\pi_{t+1}}=-v_{l} g_{m}+\beta V_{a}\left(a_{t+1}, k_{t}\right)\left(\frac{P_{t}}{P_{t+1}}\right)
or 
V a ( a t , k t − 1 ) P t B A N K ¯ v l g m P t + β V a ( a t + 1 , k t ) C P t + 1 V a a t , k t − 1 P t B A N K ¯ v l g m P t + β V a a t + 1 , k t C P t + 1 (V_(a)(a_(t),k_(t-1)))/(P_(t)BAN) bar(K)(v_(l)g_(m))/(P_(t))+beta(V_(a)(a_(t+1),k_(t)))/(CP_(t+1))\frac{V_{a}\left(a_{t}, k_{t-1}\right)}{P_{t} B A N} \bar{K} \frac{v_{l} g_{m}}{P_{t}}+\beta \frac{V_{a}\left(a_{t+1}, k_{t}\right)}{C P_{t+1}}
Recursively solving this forward yields 
V a ( a t , k t − 1 ) P t = − ∑ i − 0 ∞ β i ( v l ( a t + i , k t + i − 1 ) g m ( c t + i , m t + i ) P t + i ) + lim T → ∞ β T V a ( a T + 1 , k T ) P T + 1 = − ∑ i − 0 ∞ β i ( v l ( a t + i , k t + i − 1 ) g m ( c t + i , m t + i ) P t + i ) V a a t , k t − 1 P t = − ∑ i − 0 ∞   β i v l a t + i , k t + i − 1 g m c t + i , m t + i P t + i + lim T → ∞   β T V a a T + 1 , k T P T + 1 = − ∑ i − 0 ∞   β i v l a t + i , k t + i − 1 g m c t + i , m t + i P t + i {:[(V_(a)(a_(t),k_(t-1)))/(P_(t))=-sum_(i-0)^(oo)beta^(i)((v_(l)(a_(t+i),k_(t+i-1))g_(m)(c_(t+i),m_(t+i)))/(P_(t+i)))+lim_(T rarr oo)beta^(T)(V_(a)(a_(T+1),k_(T)))/(P_(T+1))],[=-sum_(i-0)^(oo)beta^(i)((v_(l)(a_(t+i),k_(t+i-1))g_(m)(c_(t+i),m_(t+i)))/(P_(t+i)))]:}\begin{aligned} \frac{V_{a}\left(a_{t}, k_{t-1}\right)}{P_{t}} & =-\sum_{i-0}^{\infty} \beta^{i}\left(\frac{v_{l}\left(a_{t+i}, k_{t+i-1}\right) g_{m}\left(c_{t+i}, m_{t+i}\right)}{P_{t+i}}\right)+\lim _{T \rightarrow \infty} \beta^{T} \frac{V_{a}\left(a_{T+1}, k_{T}\right)}{P_{T+1}} \\ & =-\sum_{i-0}^{\infty} \beta^{i}\left(\frac{v_{l}\left(a_{t+i}, k_{t+i-1}\right) g_{m}\left(c_{t+i}, m_{t+i}\right)}{P_{t+i}}\right) \end{aligned}
Equation (3.31) took the form 
λ t P t = ∑ i = 0 ∞ β i ( μ t + i P t + i ) λ t P t = ∑ i = 0 ∞   β i μ t + i P t + i (lambda_(t))/(P_(t))=sum_(i=0)^(oo)beta^(i)((mu_(t+i))/(P_(t+i)))\frac{\lambda_{t}}{P_{t}}=\sum_{i=0}^{\infty} \beta^{i}\left(\frac{\mu_{t+i}}{P_{t+i}}\right)
Both the shopping time and the CIA models imply the utility value of money ( V a / P V a / P V_(a)//PV_{a} / P or λ / P ) λ / P ) 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 ) g m / P (g_(m)//P)\left(g_{m} / P\right) times the utility value of the time saving ( v l ) v l (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 
− ( i t 1 + i t ) ( g c g m ) − i t 1 + i t g c g m -((i_(t))/(1+i_(t)))((g_(c))/(g_(m)))-\left(\frac{i_{t}}{1+i_{t}}\right)\left(\frac{g_{c}}{g_{m}}\right)
(Recall that money reduced shopping time, so g m ≤ 0 g m ≤ 0 g_(m) <= 0g_{m} \leq 0.) Provide an intuitive interpretation for this expression. Equations (3.10) and (3.11) imply 
u c ( c t , l t ) = V a ( a t , k t − 1 ) [ 1 + w t g c ( c t , m t ) ] = V a ( a t , k t − 1 ) [ 1 − ( i t 1 + i t ) ( g c g m ) ] u c c t , l t = V a a t , k t − 1 1 + w t g c c t , m t = V a a t , k t − 1 1 − i t 1 + i t g c g m u_(c)(c_(t),l_(t))=V_(a)(a_(t),k_(t-1))[1+w_(t)g_(c)(c_(t),m_(t))]=V_(a)(a_(t),k_(t-1))[1-((i_(t))/(1+i_(t)))((g_(c))/(g_(m)))]u_{c}\left(c_{t}, l_{t}\right)=V_{a}\left(a_{t}, k_{t-1}\right)\left[1+w_{t} g_{c}\left(c_{t}, m_{t}\right)\right]=V_{a}\left(a_{t}, k_{t-1}\right)\left[1-\left(\frac{i_{t}}{1+i_{t}}\right)\left(\frac{g_{c}}{g_{m}}\right)\right]
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 ) ) g c / g m (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 w g c w g c 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 μ t = 0 μ t = 0 mu_(t)=0\mu_{t}=0. Use (3.41) and (3.42) to show that money still has value at time t t tt (that is, the price level at time t t tt is finite) as long as the CIA constraint is expected to bind in the future. By noting that 1 / ( 1 + π t + 1 ) = P t / P t + 1 1 / 1 + π t + 1 = P t / P t + 1 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 
λ t P t = β E t ( λ t + 1 P t + 1 + μ t + 1 P t + 1 ) λ t P t = β E t λ t + 1 P t + 1 + μ t + 1 P t + 1 (lambda_(t))/(P_(t))=betaE_(t)((lambda_(t+1))/(P_(t+1))+(mu_(t+1))/(P_(t+1)))\frac{\lambda_{t}}{P_{t}}=\beta \mathrm{E}_{t}\left(\frac{\lambda_{t+1}}{P_{t+1}}+\frac{\mu_{t+1}}{P_{t+1}}\right)
Recursively solving this equation forward, one can obtain (3.31): 
λ t P t = E t ∑ i = 1 ∞ β i ( μ t + i P t + i ) + lim T → ∞ β T E t ( λ t + T P t + T ) = E t ∑ i = 1 ∞ β i ( μ t + i P t + i ) λ t P t = E t ∑ i = 1 ∞   β i μ t + i P t + i + lim T → ∞   β T E t λ t + T P t + T = E t ∑ i = 1 ∞   β i μ t + i P t + i (lambda_(t))/(P_(t))=E_(t)sum_(i=1)^(oo)beta^(i)((mu_(t+i))/(P_(t+i)))+lim_(T rarr oo)beta^(T)E_(t)((lambda_(t+T))/(P_(t+T)))=E_(t)sum_(i=1)^(oo)beta^(i)((mu_(t+i))/(P_(t+i)))\frac{\lambda_{t}}{P_{t}}=\mathrm{E}_{t} \sum_{i=1}^{\infty} \beta^{i}\left(\frac{\mu_{t+i}}{P_{t+i}}\right)+\lim _{T \rightarrow \infty} \beta^{T} \mathrm{E}_{t}\left(\frac{\lambda_{t+T}}{P_{t+T}}\right)=\mathrm{E}_{t} \sum_{i=1}^{\infty} \beta^{i}\left(\frac{\mu_{t+i}}{P_{t+i}}\right)
Thus, the current value of money, 1 / P t 1 / P t 1//P_(t)1 / P_{t}, depends on the entire future path of μ t + i μ t + i mu_(t+i)\mu_{t+i}. If μ t + i > 0 μ t + i > 0 mu_(t+i) > 0\mu_{t+i}>0 for i ≥ 0 i ≥ 0 i >= 0i \geq 0, then money will have value at time t t tt even if μ t = 0 μ t = 0 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 
    • © Carl E. Walsh, 2017. This version: March 21, 2017. I would like to thank Akatsuki Sukeda for comments on chapter 11 exercises.