Keynesian Economics (3) IS-LM Model

IS-LM 모형

• IS-LM 모형(Invest-Saving-Liquidity-Monetary Model): 단기적으로 물가가 경직되어 있을 때, 실물 시장(Invest-Saving)의 균형과 화폐 시장(Liquidity-Monetary)의 균형이 동시에 이루어지는 국민소득과 실질이자율 조합을 분석하는 일반균형 모형

• 실물 시장 균형 (IS Curve):

  \[\begin{aligned} Y &=\overline{A}-\alpha r \end{aligned}\]

  • $\overline{A}$: 국민소득 혹은 실질이자율과 무관하게 고정적으로 지출되는 규모로서 자율 지출(Autonomous Expenditure)

    \[\begin{aligned} \overline{A} &\equiv\frac{C_{0}+I_{0}+G_{0}-MPC\cdot\overline{T}}{1-MPC} \end{aligned}\]

  • $\alpha$: 국민소득의 실질이자율탄력성

    \[\begin{aligned} \alpha &\equiv\frac{b}{1-MPC} \end{aligned}\]

• 화폐 시장 균형 (LM Curve):

  \[\begin{aligned} Y &=\frac{h}{k}r+\frac{1}{k}\cdot\frac{M}{\overline{P}} \end{aligned}\]

  • 단기적으로는 하방 경직성으로 인하여 물가가 고정되어 물가상승률이 $0$ 에 수렴하므로 피셔 방정식(Fisher Equation)에 근거하여 명목이자율을 실질이자율로 대체 가능함:

    \[i =r, \quad\because\pi^{e}=0 \quad\text{where}\quad\mathrm{Short-Run}\]

• 실물 시장과 화폐 시장 간 균형점 도출:

  \[\begin{aligned} Y^{*} &=\frac{h}{h+\alpha k}\overline{A}+\frac{\alpha}{h+\alpha k}\cdot\frac{M}{\overline{P}}\\ r^{*} &=\frac{k}{h+\alpha k}\overline{A}-\frac{1}{h+\alpha k}\cdot\frac{M}{\overline{P}} \end{aligned}\]

재정 정책의 유효성

정부지출 확대 정책

• 정부지출 확대 정책 시행:

  \[G\to G+\Delta G \quad\Longleftrightarrow\quad \overline{A}\to\overline{A}+\frac{1}{1-MPC}\Delta G\]

• 균형실질이자율 상승:

  \[\begin{aligned} r^{\prime} &=\frac{k}{h+\alpha k}\overline{A}-\frac{1}{h+\alpha k}\cdot\frac{M}{\overline{P}}+\frac{k}{h+\alpha k}\cdot\frac{1}{1-MPC}\Delta G\\ &=r^{*}+\frac{k}{h+\alpha k}\cdot\frac{1}{1-MPC}\Delta G\\ \therefore\frac{\Delta r}{\Delta G} &=\frac{k}{h+\alpha k}\cdot\frac{1}{1-MPC} >0 \end{aligned}\]

• 균형국민소득 증가:

  \[\begin{aligned} Y^{\prime} &=\frac{h}{h+\alpha k}\overline{A}+\frac{\alpha}{h+\alpha k}\cdot\frac{M}{\overline{P}}+\frac{h}{h+\alpha k}\cdot\frac{1}{1-MPC}\Delta G\\ &=Y^{*}+\frac{h}{h+\alpha k}\cdot\frac{1}{1-MPC}\Delta G\\ \therefore\frac{\Delta Y}{\Delta G} &=\underbrace{\frac{h}{h+\alpha k}}_{\substack{\text{이자율 상승에 따른}\\\text{구축효과}}}\cdot\underbrace{\frac{1}{1-MPC}}_{\substack{\text{정부지출의}\\\text{순수 승수효과}}} >0 \end{aligned}\]

조세 감면 정책

• 조세 감면 정책 시행:

  \[\overline{T}\to \overline{T}+\Delta T \quad\Longleftrightarrow\quad \overline{A}\to\overline{A}+\frac{MPC}{1-MPC}\Delta T\]

• 균형실질이자율 상승:

  \[\begin{aligned} r^{\prime} &=\frac{k}{h+\alpha k}\overline{A}-\frac{1}{h+\alpha k}\cdot\frac{M}{\overline{P}}+\frac{k}{h+\alpha k}\cdot\frac{MPC}{1-MPC}\Delta T\\ &=r^{*}+\frac{k}{h+\alpha k}\cdot\frac{MPC}{1-MPC}\Delta T\\ \therefore\frac{\Delta r}{\Delta T} &=\frac{k}{h+\alpha k}\cdot\frac{MPC}{1-MPC} >0 \end{aligned}\]

• 균형국민소득 증가:

  \[\begin{aligned} Y^{\prime} &=\frac{h}{h+\alpha k}\overline{A}+\frac{\alpha}{h+\alpha k}\cdot\frac{M}{\overline{P}}+\frac{h}{h+\alpha k}\cdot\frac{MPC}{1-MPC}\Delta T\\ &=Y^{*}+\frac{h}{h+\alpha k}\cdot\frac{MPC}{1-MPC}\Delta T\\ \therefore\frac{\Delta Y}{\Delta T} &=\underbrace{\frac{h}{h+\alpha k}}_{\substack{\text{이자율 상승에 따른}\\\text{구축효과}}}\cdot\underbrace{\frac{MPC}{1-MPC}}_{\substack{\text{조세 감면의}\\\text{순수 승수효과}}} >0 \end{aligned}\]

통화 정책의 유효성

통화 긴축 정책

• 통화 긴축 정책 시행:

  \[M\to M-\Delta M\]

• 균형실질이자율 상승:

  \[\begin{aligned} r^{\prime} &=\frac{h}{h+\alpha k}\overline{A}-\frac{1}{h+\alpha k}\cdot\frac{M}{\overline{P}}+\frac{1}{h+\alpha k}\cdot\frac{1}{\overline{P}}\Delta M\\ &=r^{*}+\frac{1}{h+\alpha k}\cdot\frac{1}{\overline{P}}\Delta M\\ \therefore\frac{\Delta r}{\Delta M} &=\frac{1}{h+\alpha k}\cdot\frac{1}{\overline{P}} \end{aligned}\]

• 균형국민소득 하락:

  \[\begin{aligned} Y^{\prime} &=\frac{h}{h+\alpha k}\overline{A}+\frac{\alpha}{h+\alpha k}\cdot\frac{M}{\overline{P}}-\frac{\alpha}{h+\alpha k}\cdot\frac{1}{\overline{P}}\Delta M\\ &=Y^{*}-\frac{\alpha}{h+\alpha k}\cdot\frac{1}{\overline{P}}\Delta M\\ \therefore\frac{\Delta Y}{\Delta M} &=-\frac{\alpha}{h+\alpha k}\cdot\frac{1}{\overline{P}} \end{aligned}\]

통화 완화 정책

• 통화 완화 정책 시행:

  \[M\to M+\Delta M\]

• 균형실질이자율 하락:

  \[\begin{aligned} r^{\prime} &=\frac{h}{h+\alpha k}\overline{A}-\frac{1}{h+\alpha k}\cdot\frac{M}{\overline{P}}-\frac{1}{h+\alpha k}\cdot\frac{1}{\overline{P}}\Delta M\\ &=r^{*}-\frac{1}{h+\alpha k}\cdot\frac{1}{\overline{P}}\Delta M\\ \therefore\frac{\Delta r}{\Delta M} &=-\frac{1}{h+\alpha k}\cdot\frac{1}{\overline{P}} \end{aligned}\]

• 균형국민소득 상승:

  \[\begin{aligned} Y^{\prime} &=\frac{h}{h+\alpha k}\overline{A}+\frac{\alpha}{h+\alpha k}\cdot\frac{M}{\overline{P}}+\frac{\alpha}{h+\alpha k}\cdot\frac{1}{\overline{P}}\Delta M\\ &=Y^{*}+\frac{\alpha}{h+\alpha k}\cdot\frac{1}{\overline{P}}\Delta M\\ \therefore\frac{\Delta Y}{\Delta M} &=\frac{\alpha}{h+\alpha k}\cdot\frac{1}{\overline{P}} \end{aligned}\]