IS-LM模型：从失衡到均衡的模拟

IS-LM模型：从失衡到均衡的模拟

$1IS-LM$模型

$IS-LM$是产品市场和货币市场共同均衡时的模型，它由两条曲线构成，分别是$IS$曲线和$LM$曲线。其中$IS$曲线是在产品市场均衡(产品服务供给等于需求、计划支出等于实际支出、计划投资等于储蓄、非计划存货等于0)条件下，均衡实际收入$Y$与实际利率$r$之间的反向变化关系；$LM$曲线是在货币市场均衡(货币供给等于货币需求)条件下，均衡实际利率$r$与实际收入$Y$之间的正向变化关系。用方程表示为
$$\{\begin{array}{l}Y=C(Y)+I(r)\\ L(r,Y)=M/P\end{array}$$
其中$Y=C(Y)+I(r)$为产品市场均衡条件(计划支出=实际支出)。消费$C(Y)$是关于收入$Y$的函数，假设是线性的：
$$C(Y)=\alpha +\beta Y$$
其中$\beta \in (0,1)$称为边际消费倾向，$\alpha >0$为自主消费，即没有收入时的消费。在资本边际效率不变时，投资$I(r)$是关于利率$r$的递减函数，假设也是线性的：
$$I(r)=e-dr$$
其中$e>0$是自发投资，$d$是投资对利率的敏感程度。于是产品市场均衡条件可记作
$$Y=\alpha +\beta Y+e-dr$$

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$L(r,Y)$为实际货币需求，它是由$L_1(Y)$需求和$L_2(r)$需求构成。$L_1(Y)$由交易性需求和预防性需求构成，随收入$Y$增加而增加，不妨假定为正比例函数
$$L_1(Y)=kY$$
其中$k$表示用于支付日常开支(交易性需求)和未来不确定性(预防性需求)占实际收入的比重。$L_2(r)$需求称为投机性需求，它是关于实际利率的递减函数，假设为负比例函数：
$$L_2(r)=A-hr$$
其中$A>0$是参数，$h$表示$L_2$对利率$r$变化的敏感程度。$M$表示名义货币供给，$P$表示价格水平，$M/P$表示实际货币供给。货币市场均衡条件可以记作
$$kY+A-hr=M/P$$
我们将上述两个模型重新写在一起
$$\{\begin{array}{l}Y=\alpha +\beta Y+e-dr\\ kY+A-hr=M/P\end{array}$$
将$r,Y$视为内生变量，两个方程组可以解出唯一均衡值，记作$(r^{\ast },Y^{\ast })$。其中$r^{\ast }$称为均衡实际利率，$Y^{\ast }$称为均衡实际收入，或均衡国民收入。从几何上看，也就是这两条直线的交点。

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然而，初始的实际收入和实际利率并不是均衡的，很有可能并不在上述两条直线的交点处，例如下图$E^{\prime }$,$E^{\prime \prime }$和$E^{\prime \prime \prime }$。

假设初始状态在$E^{\prime \prime \prime }$，此时计划投资大于储蓄$I>S$，实际收入$Y$增加，实际利率$r$增加，即$E^{\prime \prime \prime }$点即向右移动，又向上移动，合力为右上方，直至进入$II$区域。在$II$区域中，$I<S$，实际收入减少，于是向左移动；$L>M$，实际利率继续向上移动，合力为左上方，此时进入$I$区域。在$I$区域，$Y$减少，$r$降低，合力在左下方，进入$IV$区域。在$IV$区域，$Y$增加，$r$降低，进入$III$区域，于是重新回到$III$区域。但每次都与均衡点$E$不断接近。

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为了使上述模型动态化，引入时间因素$t$，于是
$$\{\begin{array}{l}{C}_t=\alpha +\beta Y_t\\ I_t=e-d{r}_t\\ Y_{t+1}=C_t+I_t\\ k{Y}_t+A-h{r}_{t+1}=M/P\end{array}$$
整理得到
$$\{\begin{array}{l}{C}_t=\alpha +\beta Y_t\\ I_t=e-d{r}_t\\ Y_{t+1}=C_t+I_t\\ r_{t+1}=(k{Y}_t+A-M/P)/h\end{array}$$

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在长期中，非均衡逐渐向均衡靠拢，$r_t\approx r_{t+1}\approx r^{\ast }$,$Y_t\approx Y_{t+1}\approx Y^{\ast }$，于是
$$\{\begin{array}{l}{C}^{\ast }=\alpha +\beta Y^{\ast }\\ I^{\ast }=e-d{r}^{\ast }\\ Y^{\ast }=C^{\ast }+I^{\ast }\\ r^{\ast }=(k{Y}^{\ast }+A-M/P)/h\end{array}$$
使用行列式求解得到长期均衡点为
$$\{\begin{array}{l}{r}^{\ast }=\frac{k(\alpha +e)+(1-\beta )(A-M/P)}{kd+h(1-\beta )}\\ Y^{\ast }=\frac{h(\alpha +e)-d(A-M/P)}{kd+h(1-\beta )}\end{array}$$

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2 数值模拟

2.1 长期均衡解

令参数$\alpha =500$,$\beta =0.5$，$e=1250$，$d=250$，$k=0.5$，$h=250$，$A=1000$，$M=1250$，$P=1$，代入上述均衡解得到

alpha = 500
beta = 0.5
e = 1250
d = 250
k = 0.5
h = 250
A = 1000
M = 1250
P = 1
r_star = (k*(alpha+e)+(1-beta)*(A-M/P))/(k*d+h*(1-beta))
Y_star =( h*(alpha+e)-d*(A-M/P))/(k*d+h*(1-beta))
r_star
Y_star
# 3
# 2000

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现在假设初始实际利率为$r_0=10$，$Y_0=5000$，基于下列公式
$$\{\begin{array}{l}{C}_t=\alpha +\beta Y_t\\ I_t=e-d{r}_t\\ Y_{t+1}=C_t+I_t\\ r_{t+1}=(k{Y}_t+A-M/P)/h\end{array}$$

rm(list = ls())
# 参数初始化
alpha = 500
beta = 0.5
e = 1250
d = 250
k = 0.5
h = 250
A = 1000
M = 1250
P = 1
T = 100  # 迭代次数r = numeric()
Y = numeric()
# 初始值
r[1] = 10
Y[1] = 5000
# 迭代
for (t in 1:T) {C = 500+0.5*Y[t]I = 1250-250*r[t]Y[t+1] = C+Ir[t+1] = (k*Y[t]+A-M/P)/h}
par(mfrow=c(1,2),mar = c(5,5,5,5))
plot(Y,type = "l",lwd=2,xlab = "实际收入Y",ylab = "实际利率r",main = "实际利率均衡过程",cex.axis = 2, cex.lab = 2,cex.main = 2,family = "ST")
grid(col = "black")
plot(r,type = "l",lwd=2,xlab = "实际收入Y",ylab = "实际利率r",main = "实际收入均衡过程",cex.axis = 2, cex.lab = 2,cex.main = 2,family = "ST")
grid(col = "black")par(mfrow=c(1,1))
plot(Y,r,typ="l",lwd=2,xlab = "实际收入Y",ylab = "实际利率r",main = "均衡点收敛过程",cex.axis = 2, cex.lab = 2,cex.main = 2,family = "ST")
grid(col = "black")

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2.2 政府部门引入

引入政府部门，政府决策变量包括政府支出$G$、税收$T$和转移支付$T_r$，此时均衡条件如下：
$$\{\begin{array}{l}{C}_t=\alpha +\beta (Y_t-T+T_r)\\ I_t=e-d{r}_t\\ Y_{t+1}=C_t+I_t+G\\ r_{t+1}=(k{Y}_t+A-M/P)/h\end{array}$$
令政府购买$G=500$，税收$T=20$，转移支付$T_r=5$，

rm(list = ls())
# 参数初始化
alpha = 500
beta = 0.5
e = 1250
d = 250
k = 0.5
h = 250
A = 1000
M = 1250
P = 1
T = 20
Tr = 5
G = 500T = 100  # 迭代次数r = numeric()
Y = numeric()
# 初始值
r[1] = 10
Y[1] = 5000
# 迭代
for (t in 1:T) {C = 500+0.5*(Y[t]-T+Tr)I = 1250-250*r[t]Y[t+1] = C+I+Gr[t+1] = (k*Y[t]+A-M/P)/h}
par(mfrow=c(1,2),mar = c(5,5,5,5))
plot(Y,type = "l",lwd=2,xlab = "实际收入Y",ylab = "实际利率r",main = "实际利率均衡过程",cex.axis = 2, cex.lab = 2,cex.main = 2,family = "ST")
grid(col = "black")
plot(r,type = "l",lwd=2,xlab = "实际收入Y",ylab = "实际利率r",main = "实际收入均衡过程",cex.axis = 2, cex.lab = 2,cex.main = 2,family = "ST")
grid(col = "black")par(mfrow=c(1,1))
plot(Y,r,typ="l",lwd=2,xlab = "实际收入Y",ylab = "实际利率r",main = "均衡点收敛过程",cex.axis = 2, cex.lab = 2,cex.main = 2,family = "ST")
grid(col = "black")

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2.3 价格水平影响

使价格水平$P$不断下降，实际货币供给不断增加，实际货币供给增加又导致均衡实际利率不断降低，进而导致投资不断增加，均衡国民收入不断增加。

rm(list = ls())
# 参数初始化
alpha = 500
beta = 0.5
e = 1250
d = 250
k = 0.5
h = 250
A = 1000
M = 1250
T = 20
Tr = 5
G = 500
T = 100  # 迭代次数
r = numeric()
Y = numeric()
# 初始值
r[1] = 10
Y[1] = 5000
# 迭代P = c(1,0.8,0.6,0.4)
par(mfrow=c(2,2),mar = c(5,5,5,5))
for(j in P){for (t in 1:T) {C = 500+0.5*(Y[t]-T+Tr)I = 1250-250*r[t]Y[t+1] = C+I+Gr[t+1] = (k*Y[t]+A-M/j)/h}plot(Y,r,typ="l",lwd=2,xlab = "实际收入Y",ylab = "实际利率r",main = paste("价格水平P=",j),cex.axis = 2, cex.lab = 2,cex.main = 2,family = "ST")grid(col = "black")
}

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2.4 随机扰动因素

除收入外，还有其他一些因素也会影响消费；同理，除了利率，也有其他因素也会影响投资大小；货币需求和货币供给之间也存在随机误差。因此，均衡条件进一步改进为
$$\{\begin{array}{l}{C}_t=\alpha +\beta (Y_t-T+T_r)+{\epsilon }_t\\ I_t=e-d{r}_t+v_t\\ Y_{t+1}=C_t+I_t+G\\ r_{t+1}=(k{Y}_t+A-M/P+w_t)/h\\ {\epsilon }_t,v_t,w_t\sim N(0,1)\end{array}$$
其中${\epsilon }_t,v_t,w_t$假定服从标准正态分布。

#------------------------随机扰动影响-----------------------------
rm(list = ls())
# 参数初始化
alpha = 500
beta = 0.5
e = 1250
d = 250
k = 0.5
h = 250
A = 1000
M = 1250
P = 1
T = 20
Tr = 5
G = 500
T = 100  # 迭代次数
r = numeric()
Y = numeric()
# 初始值
r[1] = 4
Y[1] = 2450
# 迭代
for (t in 1:T) {C = 500+0.5*(Y[t]-T+Tr)+rnorm(1,0,1)I = 1250-250*r[t]+rnorm(1,0,1)Y[t+1] = C+I+Gr[t+1] = (k*Y[t]+A-M/P+rnorm(1,0,1) )/h}
par(mfrow=c(1,2),mar = c(5,5,5,5))
plot(Y,type = "l",lwd=2,xlab = "实际收入Y",ylab = "实际利率r",main = "实际利率均衡过程",cex.axis = 2, cex.lab = 2,cex.main = 2,family = "ST")
grid(col = "black")
plot(r,type = "l",lwd=2,xlab = "实际收入Y",ylab = "实际利率r",main = "实际收入均衡过程",cex.axis = 2, cex.lab = 2,cex.main = 2,family = "ST")
grid(col = "black")par(mfrow=c(1,1))
plot(Y,r,typ="l",lwd=2,xlab = "实际收入Y",ylab = "实际利率r",main = "均衡点收敛过程",cex.axis = 2, cex.lab = 2,cex.main = 2,family = "ST")
grid(col = "black")

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