We have now looked at each of the pieces of the dynamic *AD*–*AS* model. As a quick summary, Table 15-1 lists the equations, variables, and parameters in the model. The variables are grouped according to whether they are *endogenous* (to be determined by the model) or *exogenous* (taken as given by the model).

Figure 15.2: TABLE 15-1: **The Equations, Variables, and Parameters in the Dynamic ***AD*–*AS* Model

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The model’s five equations determine the paths of five endogenous variables: output *Y _{t}*, the real interest rate

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We are almost ready to put these pieces together to see how various shocks to the economy influence the paths of these variables over time. Before doing so, however, we need to establish the starting point for our analysis: the economy’s long-run equilibrium.

The long-run equilibrium represents the normal state around which the economy fluctuates. It occurs when there are no shocks (*ε _{t}* =

Straightforward algebra applied to the model’s five equations can be used to determine the long-run values of the five endogenous variables:

In words, the long-run equilibrium is described as follows: output and the real interest rate are at their natural values, inflation and expected inflation are at the target rate of inflation, and the nominal interest rate equals the natural rate of interest plus target inflation.

The long-run equilibrium of this model reflects two related principles: the classical dichotomy and monetary neutrality. Recall that the classical dichotomy is the separation of real from nominal variables and that monetary neutrality is the property according to which monetary policy does not influence real variables. The equations immediately above show that the central bank’s inflation target
influences only inflation *π _{t}*, expected inflation

To study the behavior of this economy in the short run, it is useful to analyze the model graphically. Because graphs have two axes, we need to focus on two variables. We will use output *Y _{t}* and inflation

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To generate this graph, we need two equations that summarize the relationships between output *Y _{t}* and inflation

The first relationship between output and inflation comes almost directly from the Phillips curve equation. We can get rid of the one endogenous variable in the equation (*E _{t}*

This equation relates inflation *π _{t}* and output

Figure 15-2 graphs the relationship between inflation *π _{t}* and output

Figure 15.3: FIGURE 15-2: **The Dynamic Aggregate Supply Curve** The dynamic aggregate supply curve *DAS*_{t} shows a positive association between output *Y*_{t} and inflation *π*_{t}. Its upward slope reflects the Phillips curve relationship: other things equal, high levels of economic activity are associated with high inflation. The dynamic aggregate supply curve is drawn for given values of past inflation *π*_{t}_{−1}, the natural level of output
, and the supply shock *υ*_{t}. When these variables change, the curve shifts.

The *DAS* curve is drawn for given values of past inflation *π _{t}*

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The dynamic aggregate supply curve is one of the two relationships between output and inflation that determine the economy’s short-run equilibrium. The other relationship is (no surprise) the dynamic aggregate demand curve. We derive it by combining four equations from the model and then eliminating all the endogenous variables other than output and inflation. Once we have an equation with only two endogenous variables (*Y _{t}* and

We begin with the demand for goods and services:

To eliminate the endogenous variable *r _{t}*, the real interest rate, we use the Fisher equation to substitute

To eliminate another endogenous variable, the nominal interest rate *i _{t}*, we use the monetary-policy equation to substitute for

Next, to eliminate the endogenous variable of expected inflation *E _{t}π_{t}*

As was our goal, this equation has only two endogenous variables: output *Y _{t}* and inflation

If we now bring like terms together and solve for *Y*t, we obtain

This equation relates output *Y _{t}* to inflation

Figure 15-3 graphs the relationship between inflation *π _{t}* and output

Figure 15.4: FIGURE 15-3: **The Dynamic Aggregate Demand Curve** The dynamic aggregate demand curve shows a negative association between output and inflation. Its downward slope reflects monetary policy and the demand for goods and services: a high level of inflation causes the central bank to raise nominal and real interest rates, which in turn reduces the demand for goods and services. The dynamic aggregate demand curve is drawn for given values of the natural level of output *Y*_{t}, the inflation target
, and the demand shock *ε*_{t}. When these exogenous variables change, the curve shifts.

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It is tempting to think of this dynamic aggregate demand curve as nothing more than the standard aggregate demand curve from **Chapter 12** with inflation, rather than the price level, on the vertical axis. In some ways, they are similar: they both embody the link between the interest rate and the demand for goods and services. But there is an important difference. The conventional aggregate demand curve in **Chapter 12** is drawn for a given money supply. By contrast, because the monetary-policy rule is used to derive the dynamic aggregate demand equation, the dynamic aggregate demand curve is drawn for a given rule for monetary policy. Under that rule, the central bank sets the interest rate based on macroeconomic conditions, and it allows the money supply to adjust accordingly.

The dynamic aggregate demand curve slopes downward because of the following mechanism. When inflation rises, the central bank responds by following its rule and increasing the nominal interest rate. Because the rule specifies that the central bank raise the nominal interest rate by more than the increase in inflation, the real interest rate rises as well. The increase in the real interest rate reduces the quantity of goods and services demanded. This negative association between inflation and quantity demanded, working through central bank policy, makes the dynamic aggregate demand curve slope downward.

The dynamic aggregate demand curve shifts in response to changes in fiscal and monetary policy. As we noted earlier, the shock variable *ε _{t}* reflects changes in government spending and taxes (among other things). Any change in fiscal policy that increases the demand for goods and services means a positive value of

Monetary policy enters the dynamic aggregate demand curve through the target inflation rate
. The *DAD* equation shows that, other things equal, an increase in
raises the quantity of output demanded. (There are two negative signs in front of
, so the effect is positive.) Here is the mechanism that lies behind this mathematical result: When the central bank raises its target for inflation, it pursues a more expansionary monetary policy by reducing the nominal interest rate, as dictated by the monetary-policy rule. For any given rate of inflation, the lower nominal interest rate in turn means a lower real interest rate, and a lower real interest rate stimulates spending on goods and services. Thus, output is higher for any given inflation rate, so the dynamic aggregate demand curve shifts to the right. Conversely, when the central bank reduces its target for inflation, it raises nominal and real interest rates, thereby dampening demand for goods and services and shifting the dynamic aggregate demand curve to the left.

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The economy’s short-run equilibrium is determined by the intersection of the dynamic aggregate demand curve and the dynamic aggregate supply curve. The economy can be represented algebraically using the two equations we have just derived:

In any period *t*, these equations together determine two endogenous variables: inflation *π _{t}* and output

Taking these exogenous variables as given, we can illustrate the economy’s short-run equilibrium as the intersection of the dynamic aggregate demand curve and the dynamic aggregate supply curve, as in Figure 15-4. The short-run equilibrium level of output *Y _{t}* can be less than its natural level
, as it is in this figure, greater than its natural level, or equal to it. As we have seen, when the economy is in long-run equilibrium, output is at its natural level
.

Figure 15.5: FIGURE 15-4: **The Short-Run Equilibrium** The short-run equilibrium is determined by the intersection of the dynamic aggregate demand curve and the dynamic aggregate supply curve. This equilibrium determines the inflation rate and level of output that prevail in period *t*. In the equilibrium shown in this figure, the short-run equilibrium level of output *Y*_{t} falls short of the economy’s natural level of output
.

The short-run equilibrium determines not only the level of output *Y _{t}* but also the inflation rate

These linkages of economic outcomes across time periods will become clear as we work through a series of examples.

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