Algebraic analysis of the aggregate demand and supply model allows us to directly obtain all of the results in Chapter 22, as well as a number of additional results.

The Algebra of the Aggregate Demand and Supply Model

AggregATe DeMAnD CurveIn Chapter 20, we derived the IS curve Equation 12, which we repeat as Equation 1:

Y = 3C + I - df + G + NX - mpc * T4 *1

1 - mpc-

d + x

1 - mpc* r (1)

Substituting r = r + lp from the monetary policy curve Equation 1 in Chapter 21 into the above equation, we get the aggregate demand curve:1

Y = 3C + I - df + G + NX - mpc * T4 *1

1 - mpc-

d + x

1 - mpc* (r + lp) (2)

1Investment, taxes, and net exports are responsive to income, so more realistic investment, tax, and net export functions are as follows:

I = I - dr + iY

because investment should rise with income, with the responsiveness of investment to income represented by the parameter, i.

T = T + tY

because taxes rise with income, with the responsiveness of taxes to income represented by the parameter, t.

NX = NX - xr - nY

because net imports fall with income since imports rise with income, with the responsiveness of net exports to income represented by the parameter, n.

We modify Equation 2 to become

Y = 3C + I - df + G + NX - mpc * T4 *1

1 - mpc(1 - t) - i + n-

d + x

1 - mpc(1 - t) - i + n

* (r + lp)

Thus the aggregate demand curve is identical to Equation 2 except that we replace

[1 - mpc] by [1 - mpc(1 - t) - i + n].

Appendix 3 to chApter

22

1

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2 appendix 3 to chapter 22 The Algebra of the Aggregate Demand and Supply Model

implicationsThere are four immediate implications from the aggregate demand curve Equation 2.

1. The inflation rate p is multiplied by l(d + x)>(1 - mpc) and so equilibrium out-put is negatively related to the inflation rate. That is, the aggregate demand curve slopes downward.

2. The more monetary policymakers raise real interest rates as inflation goes up (i.e., the larger l is) the more responsive equilibrium output is to the inflation rate, as is indicated by the larger coefficient on inflation in Equation 1. In other words, the more willing monetary policymakers are to raise interest rates when faced by infla-tion, the steeper the AD curve is.

3. Since C, - mpc * T, I, - df, G, and NX all are multiplied by the same term, they all shift the aggregate demand curve exactly the same amount and in the same direction as they shift the IS curve.

4. Since r is multiplied by - (d + x)>(1 - mpc), an autonomous tightening of mon-etary policy (i.e., a rise in r) leads to a lower level of equilibrium output, shifting the aggregate demand curve to the left. An autonomous easing of monetary policy (a fall in r) results in a higher level of equilibrium output, shifting the aggregate demand curve to the right.

AggregATe Supply CurveSAs we saw in Chapter 22, there are two types of aggregate supply curves, the short run and the long run.

Short-run aggregate Supply curveTo generate an algebraic short-run AS curve, we need to specify how expectations of inflation are formed. We do this by assuming that households form their expectations about inflation by looking at past inflation, so pe = p-1, where p-1 is the inflation rate last year. Substituting pe = p-1 into Equation 2 in Chapter 22, we write the short-run aggregate supply curve as follows:

p = p-1 + g(Y - YP) + r (3)

Long-run aggregate Supply curveThe long-run aggregate supply curve is a vertical line at potential output:

Y = YP (4)

where YP = potential output.

AggregATe DeMAnD AnD Supply MoDelThe aggregate demand and supply model has two equilibria, the short run and the long run.

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appendix 3 to chapter 22 The Algebra of the Aggregate Demand and Supply Model 3

Short-run equilibriumSolving for the short-run equilibrium involves substituting the short-run aggregate sup-ply curve Equation 3 into the aggregate demand curve Equation 2:

Y = 3C + I - df + G + NX - mpc * T4 *1

1 - mpc-

d + x

1 - mpc

* (r + l 3p-1 + g (Y - YP) + r4 ) (5)

Multiplying both sides of Equation 5 by (1 - mpc) and collecting the Y terms on the left-hand side results in the following:

Y 31 - mpc + lg (d + x)4 = C + I - df + G + NX - mpc * T -

(d + x) * (r + l3p-1 - gYP + r4 ) (6)

Dividing both sides by 31 - mpc + lg (d + x)4 solves for aggregate output in the short run:

Y = [C + I - df + G + NX - mpc * T - (d + x)

* (r + l3p-1 - gYP + r4 )4 *1

1 - mpc + lg (d + x)

(7)

We solve for the inflation rate in the short run by substituting in Equation 7 into the short-run aggregate supply curve Equation 3:

p = p-1 - gYP + r + [C + I - df + G + NX - mpc * T - (d + x)

* (r + l[p-1 - gYP + r])] *g

1 - mpc + lg (d + x)

(8)

Collecting terms in p-1, YP, and r:

p = [C + I - df + G + NX - mpc * T - (d + x) * r]

*g

1 - mpc + lg (d + x)+ (p-1 - gYP + r) (9)

* c1 -lg (d + x)

1 - mpc + lg(d + x)d

which we can rewrite as follows:

p = 3C + I - df + G + NX - mpc * T - (d + x) * r4 *

g

1 - mpc + lg (d + x)+ (p-1 - gYP + r) (10)

* c 1 - mpc

1 - mpc + lg(d + x)d

Long-run equilibriumWe have already seen from the long-run aggregate supply curve that in long-run equilibrium,

Y = YP

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4 appendix 3 to chapter 22 The Algebra of the Aggregate Demand and Supply Model

We then solve for the long-run inflation rate by substituting YP for Y into the aggregate demand curve Equation 2:

YP = 3C + I - df + G + NX - mpc * T - (d + x)(r + lp)4 *1

1 - mpc (11)

Multiplying by (1 - mpc) and collecting terms in p,

(d + x)lp = C + I - df + G + NX - mpc * T - (d + x) r - (1 - mpc)YP (12)

Dividing by (d + e)l,

p = 3C + I - df + G + NX - mpc * T - (d + x) r4 - (1 - mpc)YP4 (13)

*1

(d + x)l

implicationsThere are five implications for aggregate output from the short- and long-run equilib-rium output Equations 4 and 7:

1. Increases in C, I, G, and NX, or decreases in T, f , and r, lead to higher aggregate output in the short run, but not in the long run.

2. Because l enters positively in the denominator in Equation 7, the more aggressively monetary policymakers respond to inflation (a higher l), the less impact demand

shocks (C, I, f ,G, NX, T, and r) have on aggregate output in the short run. 3. Positive price shocks, r, and increases in expected inflation, p-1, lead to a decline

in aggregate output in the short run, but not in the long run. 4. A rise in potential output leads to a rise in actual aggregate output in both the short

and long run. 5. The more aggressively monetary policymakers respond to inflation (a higher l), the

larger are the effects of price shocks, r, and changes in expected inflation, p-1, on aggregate output in the short run.

There are five implications for inflation from the short- and long-run inflation Equations 10 and 13:

6. Increases in C, I, G, and NX, or decreases in T, f , and r, lead to higher inflation in both the short and the long run.

7. Positive price shocks, r, and increases in expected inflation, p-1, lead to higher inflation in the short run, but not in the long run.

8. A rise in potential output leads to a decline in inflation in both the short and long runs.

9. The more aggressively monetary policymakers respond to inflation (a higher l), the less impact demand shocks (C, I, f ,G, NX, T, and r) have on inflation in the

short run. 10. The more aggressively monetary policymakers respond to inflation (a higher l),

the smaller are the effect of price shocks and changes in expected inflation, p-1, on inflation in the short run.

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