IS-LM-PC Model¶

by Professor Throckmorton
for Intermediate Macro
W&M ECON 304
Slides

Summary¶

  1. Okun's Law: negative relationship between output, $Y$, and the unemployment rate, $u$

  2. Real vs. nominal interest rate and the medium-run IS-LM-PC equilibrium

  3. IS-LM-PC: short-run vs. medium-run equilibrium
    Q: What does a recession look like in the model?

  4. The ZLB and Deflationary Spirals

Okun's Law¶

  • IS-LM model is in the $(\pi,Y)$ space, but Phillips Curve is in the $(\pi,u)$ space.
  • To combine IS-LM and PC, we will first transform the PC into the $(\pi,Y)$ space.
  • To do that, we need Okun's law, which is a negative relationship between output, $Y$, and unemployment rate, $u$, relative to the medium-run equilibrium

  • Recall these labor market definitions: $L = U + N$ and $u = U/L$

  • For the labor market, we assumed the production function is $Y = N$, so we can also substitute $N$ out for $Y$, \begin{gather*} u = \frac{U}{L} = \frac{L - N}{L} = 1 - \frac{Y}{L} \end{gather*}

    i.e., unemployment rate and output are negatively related (by definition)

  • Rearrange and add time subscripts (assuming $L$ is constant in short run) \begin{gather*} u = 1 - Y/L \quad \rightarrow \quad Y_t = L(1 - u_t) \end{gather*}

  • Note that equation also holds in the medium run, $Y_n = L(1 - u_n)$, i.e., the natural unemployment rate implies a natural, or potential, output level.

  • Subtract the medium-run equation from the short-run equation: \begin{align*} \rightarrow Y_t - Y_n &= -L(u_t - u_n) \\ \textrm{or} \quad u_t - u_n &= - \frac{1}{L}(Y_t - Y_n) \end{align*} This is Okun's Law.

  • Substitute Okun's Law into Phillips Curve \begin{align*} \pi_t - \pi_t^e &= - \alpha (u_t - u_n) \\ \rightarrow \pi_t - \pi_t^e &= \frac{\alpha}{L} (Y_t - Y_n) \end{align*}

Inflation vs. Output¶

  • Again, suppose $\pi_t^e = \bar{\pi}$, i.e., expectations are anchored, and the central bank is credible \begin{gather*} \pi_t - \bar{\pi} &= \frac{\alpha}{L} (Y_t - Y_n) \end{gather*}

    • If $Y_t = Y_n$ (eqm), then $\pi_t$ is stable at the Fed's target

    • If $Y_t < Y_n$ (recession), then $\pi_t$ falls below the Fed's target

    • If $Y_t > Y_n$ (boom), then $\pi_t$ rises above the Fed's target

  • This version of the Phillips Curve appears in the IS-LM-PC model and is equation 9.4 in Blanchard, Macroeconomics (9th Edition, 2025).

Real vs. Nominal Interest Rate¶

  • Recall the simple loan example from the Money Market. Suppose you lend me $P_t$ dollars, and next year I will pay you back

    \begin{gather*} \textrm{principal} + \textrm{interest} = P_t + i_tP_t = (1+i_t)P_t \end{gather*}

    where the nominal interest rate, $i_t$, we agree on ahead of time

  • Q: What is the real value of that future payment today?
    A:

    \begin{gather*} \frac{\textrm{\$ received in future}}{\textrm{expected \$ price of goods}} = \frac{(1+i_t)P_t}{P_{t+1}^e} \equiv 1 + r_t \end{gather*}

    • $P_{t+1}^e$: expected price of goods and services next year
    • $r_t$: the real interest rate (in terms of goods, not dollars)

  • Problems: It's not in terms of the inflation rate. And it's a nonlinear relationship, which is difficult to use/interpret.

Substitute in inflation rate

  1. Recall the definition of the inflation rate

\begin{align*} \pi_t &= P_t/P_{t-1} - 1 \\ \rightarrow \pi_{t+1} &= P_{t+1}/P_t - 1 \\ \rightarrow \pi_{t+1}^e &= P_{t+1}^e/P_t - 1 \\ \rightarrow 1+ \pi_{t+1}^e &= P_{t+1}^e/P_t \\ \rightarrow \frac{P_t}{P_{t+1}^e} &= \frac{1}{1+ \pi_{t+1}^e} \end{align*}

  1. Combine that with real interest rate

\begin{gather*} 1 + r_t = \frac{1+i_t}{1+\pi_{t+1}^e} \end{gather*}

Nonlinear $\rightarrow$ linear

  1. Recall the properties of natural log \begin{align*} \log(X/Y) &= \log(X) - \log(Y) \\ \log(1+X) &\approx X \textrm{ if $X$ is small} \end{align*}

  2. Take natural log of both sides \begin{gather*} \log(1 + r_t) = \log (1+i_t) - \log (1+\pi_{t+1}^e) \end{gather*} Note $0 < r_t,i_t,\pi_{t+1}^e < 0.10$ are small

  3. Approximate $\log(1+X) \approx X$ (since $X$ is small) \begin{gather*} r_t \approx i_t - \pi_{t+1}^e \end{gather*} i.e., the real interest rate equals the nominal rate minus expected inflation

Fisher Relation Examples

  • Fisher Relation

    \begin{gather*} r_t = i_t - \pi_{t+1}^e \end{gather*}

    Note that $i_t \geq 0$ in the U.S.

  • 2008-2015: Suppose $i_t = 0\%$ and $\pi_{t+1}^e = 1.75\%$ (annual rates)

    \begin{gather*} r_t = 0\% - 1.75\% = - 1.75\% \end{gather*}

    Note the real interest rate can be negative!

  • Great Depression: $i_t = 0\%$ and $\pi_{t+1}^e < 0\%$ (deflation)

    \begin{gather*} r_t = 0\% - \textrm{ something negative} = \textrm{ something positive} \end{gather*}

    Note that the real interest rate can be greater than the nominal rate, and that really hurts borrowers! (debt deflation)

Data Example

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  • 10Y TIPS principal is indexed to inflation, so price is higher than 10Y (non-indexed) Treasury and yield is lower.
  • Difference in yields is a measure of expected inflation rate (which is an application of the Fisher relation)
  • Implied expected inflation rate is usually the Fed's target of 2%

Goods Market in medium-run¶

  • Medium-run equilibrium defined by (Labor Market, PC, and Okun's Law) \begin{gather*} \pi_t^e = \bar{\pi} \rightarrow u_t = u_n \rightarrow Y_t = Y_n \end{gather*}

  • Substitute into goods market short-run equilibrium

    \begin{gather*} Y_n = C(c_0,Y_n,T) + I(b_0,Y_n,r_t) + G \end{gather*}

    Note investment is function of real interest rate.

  • Assuming $c_0$, $b_0$, $G$, and $T$ are fixed, then the only other variable in equation is $r$. Thus,

    \begin{gather*} r_t = r_n \end{gather*}

    in a medium-run equilibrium.

  • $r_n$ is the natural real interest rate, often called the "neutral rate" since $\pi_t = \bar{\pi}$ when $r_t = r_n$.

IS-LM-PC Graph¶

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