Macro Models
Macroeconomic models help us trace how changes in policy, productivity, expectations, and other shocks affect an economy. This page develops three related systems: a long-run classical model of factor markets, output, saving, and prices; a three-equation IS–MP–PC model of short-run output, inflation, and interest rates; and an extended four-equation model that incorporates unemployment and additional shocks. For each model, we describe the economics, derive its equilibrium, and use an interactive simulator to examine comparative statics. The simulations are illustrative, not empirical forecasts.
Variables and Parameters
The classical model uses uppercase \(Y\) for real output and \(P\) for the price level. The short-run models use lowercase \(y\) for real output and \(\pi\) for the inflation rate. Unless otherwise noted, inflation, unemployment, and real interest rates are expressed in percent, while changes in these rates are measured in percentage points.
Endogenous variables
| Symbol | Meaning | Used in |
|---|---|---|
| \(N\) | Employment (labor input) | Classical |
| \(W/P\) | Real wage; \(W\) denotes the nominal wage | Classical |
| \(Y\) | Real GDP, determined by production and goods-market clearing | Classical |
| \(C\), \(I\), \(S\) | Real consumption, investment, and national saving | Classical |
| \(P\) | Aggregate price level (not the inflation rate) | Classical |
| \(r\) | Real short-term interest rate | All models |
| \(y_t\) | Real GDP in period \(t\) | Short-run models |
| \(\pi_t\) | Inflation rate in period \(t\) | Short-run models |
| \(UR_t\) | Unemployment rate | Four-equation model |
Exogenous quantities and shocks
| Symbol | Meaning | Used in |
|---|---|---|
| \(K\) | Capital stock | Classical |
| \(A\) | Total factor productivity | Classical |
| \(N^s\) | Exogenously supplied labor | Classical |
| \(M\), \(V\) | Money supply and velocity of money | Classical |
| \(G\), \(T\) | Government purchases and net taxes | Classical |
| \(y^*\) | Natural level of real GDP | Short-run models |
| \(r^*\) | Natural real interest rate | Short-run models |
| \(\pi^*\) | Central bank’s inflation target | Short-run models |
| \(UR^*\) | Natural unemployment rate (NAIRU) | Four-equation model |
| \(E_t[\pi_{t+1}]\) | Inflation expected at time \(t\) for time \(t+1\) | Short-run models |
| \(\epsilon_t^{IS}\) | Goods-market / IS shock, in output units | Short-run models |
| \(\epsilon_t^{MP}\) | Monetary-policy shock, in interest-rate percentage points | Four-equation model |
| \(\epsilon_t^{PC}\) | Phillips-curve shock, in inflation percentage points | Short-run models |
| \(\epsilon_t^{OL}\) | Okun’s Law shock, in unemployment percentage points | Four-equation model |
Structural parameters
| Symbol | Meaning | Restriction |
|---|---|---|
| \(\alpha_K\) | Capital’s share in Cobb–Douglas production | \(0<\alpha_K<1\) |
| \(c_0\), \(c_1\), \(c_2\) | Autonomous consumption; income sensitivity (MPC); interest-rate sensitivity of consumption | \(c_0,c_2\geq0\), \(0\leq c_1<1\) |
| \(i_0\), \(i_1\) | Autonomous investment; interest-rate sensitivity of investment | \(i_0\geq0\), \(i_1>0\) |
| \(\alpha\) | Sensitivity of IS output to the real interest-rate gap | \(\alpha>0\) |
| \(\beta\) | Monetary-policy response to deviations of inflation from target | \(\beta>0\) |
| \(\delta_3\) | Three-equation Phillips-curve slope with respect to the output gap | \(\delta_3\geq0\) |
| \(\delta_4\) | Four-equation Phillips-curve sensitivity to the unemployment gap | \(\delta_4\geq0\) |
| \(\gamma\) | Okun coefficient: response of unemployment to the output gap | \(\gamma\geq0\) |
A notation distinction. The classical simulator labels its Cobb–Douglas capital share alpha, while the short-run simulators use alpha for the IS coefficient. We write the former as \(\alpha_K\) to distinguish the two. Likewise, the short-run simulators both display a parameter delta, but the three-equation and four-equation Phillips curves use different independent variables, so we distinguish their coefficients as \(\delta_3\) and \(\delta_4\) in these notes.
Long-Run Classical Model
Economic intuition
In the classical model, prices and wages adjust so that markets clear. Labor supply determines employment, competitive firms hire until the real wage equals the marginal product of labor, and a Cobb–Douglas production function maps employment and capital into real output. Given output, the real interest rate adjusts to equalize national saving and investment. Finally, nominal spending (\(MV\)) and real production (\(Y\)) determine the price level (\(P\)).
This separation is useful for identifying which shocks have real effects and which have nominal effects. A change in capital or technology affects production; a change in government purchases affects the composition of spending; a change in money or velocity affects the price level when real output and velocity’s other effects are held fixed.
Model specification
We treat labor supply as exogenous (\(N=N^s\)). With capital share \(\alpha_K\), the production function and marginal product of labor are
\[ \begin{aligned} Y &= AK^{\alpha_K}N^{1-\alpha_K},\\ \frac{W}{P}=MPN &= (1-\alpha_K)AK^{\alpha_K}N^{-\alpha_K}. \end{aligned} \]
Consumption, desired investment, and national saving are
\[ \begin{aligned} C &= c_0+c_1(Y-T)-c_2r,\\ I &= i_0-i_1r,\\ S &= Y-C-G. \end{aligned} \]
Loanable-funds equilibrium requires \(S=I\). The long-run aggregate-supply curve is vertical at productive capacity, while aggregate demand follows the quantity equation:
\[ Y=AK^{\alpha_K}(N^s)^{1-\alpha_K}, \qquad MV=PY. \]
Graphical interpretation
The simulator connects four diagrams: labor market (\(W/P\) against \(N\)), production (\(Y\) against \(N\)), loanable funds (\(r\) against saving or investment), and AD–AS (\(P\) against \(Y\)). For example, increasing \(K\) shifts labor demand and the production function; output rises, the loanable-funds market adjusts, and the price level falls for unchanged \(MV\). Increasing \(M\) alone raises \(P\) without changing output or the real interest rate in this model.
Interactive simulator
Adjust the parameters above the plots, or open Preset shocks to try an isolated change. All four markets remain visible together; the equilibrium values appear below the plots. Black indicates the saved baseline and red indicates the new scenario. Use Set New Baseline to make the current scenario the point of comparison.
Comparative statics
Here are useful experiments, with all other inputs held fixed:
| Experiment | Main prediction |
|---|---|
| \(A\uparrow\) or \(K\uparrow\) | Higher output and marginal product of labor; greater saving tends to reduce \(r\); lower \(P\) with unchanged \(MV\). |
| \(N^s\uparrow\) | Employment and output rise; marginal product of labor falls, other things equal. |
| \(G\uparrow\) | National saving falls, \(r\) rises, and private investment is crowded out; full-employment \(Y\) remains fixed. |
| \(T\uparrow\) | At given \(Y\) and \(r\), consumption falls; saving rises, \(r\) falls, and investment rises. |
| \(M\uparrow\) or \(V\uparrow\) | \(P\) rises proportionally, holding \(Y\) fixed. |
With \(N=N^s\), real output and the real wage follow immediately from production and labor demand:
\[ \begin{aligned} Y^{eq}&=AK^{\alpha_K}(N^s)^{1-\alpha_K},\\ \left(\frac{W}{P}\right)^{eq}&=(1-\alpha_K)A\left(\frac{K}{N^s}\right)^{\alpha_K}. \end{aligned} \]
Set national saving equal to investment:
\[ Y-c_0-c_1(Y-T)+c_2r-G=i_0-i_1r. \]
Therefore, assuming \(i_1+c_2>0\),
\[ \boxed{r^{eq}=\frac{i_0-Y^{eq}+c_0+c_1(Y^{eq}-T)+G}{i_1+c_2}} \]
and
\[ \begin{aligned} C^{eq}&=c_0+c_1(Y^{eq}-T)-c_2r^{eq},\\ I^{eq}&=i_0-i_1r^{eq}=S^{eq},\\ \boxed{P^{eq}}&=\boxed{\frac{MV}{Y^{eq}}}. \end{aligned} \]
These values satisfy \(Y^{eq}=C^{eq}+I^{eq}+G\) in the model. The nominal wage can be recovered as \(W^{eq}=P^{eq}(W/P)^{eq}\).
Three-Equation IS–MP–PC Model
Economic intuition
The short-run system lets real output temporarily deviate from its natural level. The IS curve describes how output depends on the real interest rate; the MP rule describes how the central bank sets the real rate in response to inflation; and the Phillips curve links inflation to the output gap and inflation expectations. The three relationships hold simultaneously in each period. We treat expectations and shocks as given when comparing two contemporaneous equilibria.
Model specification
\[ \begin{aligned} \text{IS:}\quad y_t &= y^*-\alpha(r_t-r^*)+\epsilon_t^{IS},\\ \text{MP:}\quad r_t &= r^*+\beta(\pi_t-\pi^*),\\ \text{PC:}\quad \pi_t &= E_t[\pi_{t+1}]+\delta_3(y_t-y^*)+\epsilon_t^{PC}. \end{aligned} \]
The endogenous variables are \(y_t\), \(\pi_t\), and \(r_t\). The IS and MP equations jointly produce a downward-sloping aggregate-demand schedule in \((y,\pi)\) space; the Phillips curve is an upward-sloping short-run aggregate-supply schedule.
Graphical interpretation
A positive IS shock shifts the IS curve outward and AD rightward. With unchanged expectations and no other shocks, equilibrium output rises, inflation increases when \(\delta_3>0\), and monetary policy raises the real interest rate. The interest-rate response partially offsets the original expansion in spending. A Phillips-curve shock instead shifts AS, and changing a structural parameter changes a curve’s slope rather than merely its position.
Interactive simulator
Try increasing the IS shock first, then raise \(\delta_3\) or \(\beta\) to see how the output and inflation responses change. IS, MP, and derived AD–AS remain visible together, with parameter controls above and equilibrium values below.
Comparative statics
Define \(D_3=1+\alpha\beta\delta_3\). Holding other inputs fixed, the response to an IS shock is
\[ \boxed{\frac{\partial y^{eq}}{\partial\epsilon^{IS}}=\frac{1}{D_3}}, \qquad \boxed{\frac{\partial\pi^{eq}}{\partial\epsilon^{IS}}=\frac{\delta_3}{D_3}}, \qquad \boxed{\frac{\partial r^{eq}}{\partial\epsilon^{IS}}=\frac{\beta\delta_3}{D_3}}. \]
Larger \(\delta_3\) makes inflation more sensitive to output and, through the MP response, mutes the expansion in real output from an unchanged positive IS shock. Analytical derivatives describe a general relationship; numerical simulations show the size of the effect for a particular calibration.
Substitute MP into IS and solve for inflation to obtain the AD schedule:
\[ \pi_t=\pi^*+\frac{y^*+\epsilon_t^{IS}-y_t}{\alpha\beta}. \]
Equating AD with the Phillips curve and collecting terms gives
\[ \boxed{y_t^{eq}=y^*+\frac{\alpha\beta(\pi^*-E_t[\pi_{t+1}])+\epsilon_t^{IS}-\alpha\beta\epsilon_t^{PC}}{1+\alpha\beta\delta_3}}. \]
Substitute the solved output into PC and MP:
\[ \begin{aligned} \pi_t^{eq}&=E_t[\pi_{t+1}]+\delta_3(y_t^{eq}-y^*)+\epsilon_t^{PC},\\ r_t^{eq}&=r^*+\beta(\pi_t^{eq}-\pi^*). \end{aligned} \]
These are same-period equilibrium identities, not a sequential timeline. Notice also that when shocks are zero and \(E_t[\pi_{t+1}]=\pi^*\), equilibrium is \((y^*,\pi^*,r^*)\) for any \(\beta>0\); it is not necessary to set \(\beta=1\).
Four-Equation IS–MP–PC–Okun Model
Economic intuition
The extension adds an explicit unemployment outcome and allows a monetary-policy shock. Instead of relating inflation directly to the output gap, the Phillips curve now relates inflation to unemployment relative to its natural rate. Okun’s Law connects that unemployment gap to the output gap. This creates four simultaneous equations for output, inflation, the real interest rate, and unemployment.
Model specification
\[ \begin{aligned} \text{IS:}\quad y_t &= y^*-\alpha(r_t-r^*)+\epsilon_t^{IS},\\ \text{MP:}\quad r_t &= r^*+\beta(\pi_t-\pi^*)+\epsilon_t^{MP},\\ \text{PC:}\quad \pi_t &= E_t[\pi_{t+1}]-\delta_4(UR_t-UR^*)+\epsilon_t^{PC},\\ \text{OL:}\quad UR_t &= UR^*-\gamma(y_t-y^*)+\epsilon_t^{OL}. \end{aligned} \]
Substituting Okun’s Law into the Phillips curve gives
\[ \underbrace{\pi_t=E_t[\pi_{t+1}]+\delta_4\gamma(y_t-y^*)-\delta_4\epsilon_t^{OL}+\epsilon_t^{PC}}_{\text{Derived short-run aggregate supply}}. \]
Connection to the three-equation model: when \(\epsilon^{OL}=0\) and \(\epsilon^{MP}=0\), the models share the same output-gap Phillips-curve slope if \(\delta_3=\delta_4\gamma\). Thus their delta slider values should not automatically be treated as interchangeable.
Graphical interpretation
The four structural diagrams show IS, MP, Phillips curve (inflation versus unemployment), and Okun’s Law (unemployment versus output). A fifth, derived AD–AS diagram shows where all four relationships jointly settle. A contractionary monetary-policy shock directly raises the interest rate at a given inflation rate; the resulting equilibrium typically features lower output and higher unemployment. An Okun shock shifts both the unemployment–output relationship and the derived AS curve.
Interactive simulator
Open Preset shocks to try the MP shock and Okun shock experiments. All five diagrams remain visible together, so you can see which structural schedules shift, which do not, and how the new equilibrium moves along the unchanged relationships.
Comparative statics
Define \(D_4=1+\alpha\beta\delta_4\gamma\). Holding expectations and other shocks fixed, an increase in the monetary-policy shock reduces equilibrium output:
\[ \frac{\partial y^{eq}}{\partial\epsilon^{MP}}=-\frac{\alpha}{D_4}<0. \]
A positive Okun shock has two linked effects: it shifts the unemployment relation upward and changes inflation through the Phillips curve. Although equilibrium output can rise in this particular model, unemployment still increases overall:
\[ \frac{\partial y^{eq}}{\partial\epsilon^{OL}}=\frac{\alpha\beta\delta_4}{D_4}, \qquad \boxed{\frac{\partial UR^{eq}}{\partial\epsilon^{OL}}=\frac{1}{D_4}>0}. \]
Combining IS and MP gives the AD schedule:
\[ \pi_t=\pi^*+\frac{y^*+\epsilon_t^{IS}-\alpha\epsilon_t^{MP}-y_t}{\alpha\beta}. \]
Combining PC and OL gives the AS schedule shown above. Equating the two produces
\[ \boxed{y_t^{eq}=y^*+\frac{\alpha\beta(\pi^*-E_t[\pi_{t+1}])+\epsilon_t^{IS}-\alpha\epsilon_t^{MP}-\alpha\beta\epsilon_t^{PC}+\alpha\beta\delta_4\epsilon_t^{OL}}{D_4}}. \]
The remaining equilibria follow by substitution into the three structural relationships:
\[ \begin{aligned} UR_t^{eq}&=UR^*-\gamma(y_t^{eq}-y^*)+\epsilon_t^{OL},\\ \pi_t^{eq}&=E_t[\pi_{t+1}]-\delta_4(UR_t^{eq}-UR^*)+\epsilon_t^{PC},\\ r_t^{eq}&=r^*+\beta(\pi_t^{eq}-\pi^*)+\epsilon_t^{MP}. \end{aligned} \]
For a given calibration these expressions exactly satisfy all four contemporaneous equations. The AD–AS diagram in this model has inflation (\(\pi\)) on the vertical axis, unlike the classical AD–AS diagram, which has the price level (\(P\)) on the vertical axis.
Lecture Slides
The slides Fitting the IS–MP–PC Model: Comparative Statics, Calibration, Structural Decomposition, Dynamics and Simulation extend the discussion to calibration, parameter sensitivity, expectations, persistence, and simulation. They serve as a companion to the three- and four-equation models above, not as a replacement for the interactive exercises.
Open or download the lecture slides (PDF)
The downloadable PDF retains the original presentation overlays, so a single numbered slide may occupy several PDF pages. For reading online, the link above generally works better than navigating through an embedded presentation.