Measuring Shock Effects With Linear AD-AS Equations

Measuring Shocks With Linear Equations

An AD-AS diagram shows how aggregate demand and short-run aggregate supply jointly determine output and the price level. Linear equations make the same relationships measurable: given the slopes of the curves and the size of a shift, we can calculate the change in each equilibrium variable. This is useful because a shock’s effect depends not only on which curve moves, but also on how responsive the other curve is.

The calculations below compare a starting equilibrium with a new short-run equilibrium. Potential output is held fixed throughout. A change in the price level is reported as a change in the level of prices; it is not, by itself, an inflation rate.

Model and assumptions: Let \(Y\) denote real output, \(P\) the price level, and \(Y_p\) fixed potential output. Use the linear aggregate-demand equation \(Y=A-bP\) and the linear short-run aggregate-supply equation \(P=C+s(Y-Y_p)\), where \(b>0\) and \(s>0\). The parameters \(A\) and \(C\) are curve intercepts. A demand shock changes \(A\); a supply shock changes \(C\). The model describes short-run equilibria, assumes the two slopes and \(Y_p\) do not change during the comparison, and applies only over a range where the linear relationships are a reasonable approximation.

In this convention, a higher \(A\) shifts aggregate demand outward: at any given price level, the demanded quantity of output is higher. A higher \(C\) shifts short-run aggregate supply upward: at any given output gap, the price level is higher. Thus an increase in \(C\) is an adverse supply shift, while a decrease in \(C\) is a favorable one. The slopes \(b\) and \(s\) are positive magnitudes: AD slopes downward in a graph with \(P\) on the vertical axis, while short-run AS slopes upward.

The equations must use consistent units. For example, if output is measured in billions of dollars and the price level in index points, then \(b\) measures billions of output per price-index point, and \(s\) measures price-index points per billion units of output. The intercept changes \(\Delta A\) and \(\Delta C\) are measured in the corresponding units.

Equilibrium and the Change Formulas

At equilibrium, the output and price level satisfy both equations at once. Substituting the short-run supply equation into aggregate demand gives a single equation for equilibrium output:

Y=A-b\left[C+s(Y-Y_p)\right].

Collecting the terms in \(Y\) gives \((1+bs)Y=A-bC+bsY_p\). Since \(1+bs\) is positive, there is a unique equilibrium in this linear model. Once \(Y\) is found, substitute it into either original equation to obtain \(P\).

For shock calculations, compare the initial and new equilibria. Write \(\Delta A\) and \(\Delta C\) for the changes in the intercepts, and \(\Delta Y\) and \(\Delta P\) for the resulting equilibrium changes. Subtracting the initial equilibrium equation from the new one, with \(Y_p\), \(b\), and \(s\) held fixed, yields the key formulas.

Shock-effect formulas: With potential output and both slopes fixed, the equilibrium changes are

\Delta Y=\frac{\Delta A-b\Delta C}{1+bs}, \qquad \Delta P=\frac{s\Delta A+\Delta C}{1+bs}.

A positive \(\Delta A\) is an outward AD shift; a positive \(\Delta C\) is an upward SRAS shift.

Why the Formulas Hold

The formulas follow from equilibrium, rather than from a separate assumption about how much of a shock is “passed through.” In the initial equilibrium, \(Y=A-bP\); in the new equilibrium, \(Y+\Delta Y=A+\Delta A-b(P+\Delta P)\). Subtracting the initial equation gives:

\Delta Y=\Delta A-b\Delta P.

Similarly, subtracting the initial supply equation from the new one gives \(\Delta P=\Delta C+s\Delta Y\), because potential output is unchanged. Substituting this expression for \(\Delta P\) into the AD change equation gives \(\Delta Y=\Delta A-b(\Delta C+s\Delta Y)\). Collecting the output-change terms gives \((1+bs)\Delta Y=\Delta A-b\Delta C\), which establishes the output formula. Substituting that result into the supply change equation gives:

\Delta P=\Delta C+s\left(\frac{\Delta A-b\Delta C}{1+bs}\right) =\frac{s\Delta A+\Delta C}{1+bs}.

The denominator \(1+bs\) captures the feedback between the curves. For example, an outward AD shift raises output, and the upward-sloping SRAS curve then raises the price level. That higher price level in turn limits the increase in output demanded. The equations calculate this adjustment jointly.

Worked Example: An Outward Demand Shift

A hypothetical economy has \(Y_p=600\), with output measured in billions of dollars and \(P\) measured in price-index points. Its initial equations are \(Y=1100-5P\) and \(P=100+0.1(Y-600)\). A change in private spending shifts AD outward by 60 units of output, so the new AD equation is \(Y=1160-5P\). Find the new equilibrium and the changes from the initial equilibrium.

Method: solve the initial equilibrium, then apply the demand-shock formulas. First, at the initial equilibrium, the supply equation is \(P=100+0.1(Y-600)\). At \(Y=600\), it gives \(P=100\). The AD equation gives \(Y=1100-5(100)=600\), so the initial equilibrium is \(Y=600\), \(P=100\).

Here, \(b=5\), \(s=0.1\), \(\Delta A=60\), and \(\Delta C=0\). The denominator is \(1+bs=1+(5)(0.1)=1.5\). Therefore:

\Delta Y=\frac{60-(5)(0)}{1.5}=40, \qquad \Delta P=\frac{(0.1)(60)+0}{1.5}=\frac{6}{1.5}=4.

The new equilibrium is \(Y=600+40=640\) and \(P=100+4=104\). Check both equations: new AD gives \(1160-5(104)=640\); SRAS gives \(100+0.1(640-600)=104\). Both agree. The method avoids the common mistake of treating the full 60-unit shift in AD as a 60-unit equilibrium increase in output.

Worked Example: An Adverse Supply Shift

Use the same hypothetical economy and initial equilibrium as in the preceding example. A hypothetical increase in production costs raises the SRAS intercept by 6 price-index points, while AD is unchanged. Calculate the new equilibrium and check it in the original model’s equations with the changed intercept.

Method: identify the supply-intercept change and apply the formulas. The initial equilibrium is \(Y=600\), \(P=100\). Here, \(\Delta A=0\) and \(\Delta C=6\), with \(b=5\), \(s=0.1\), and \(1+bs=1.5\). Thus:

\Delta Y=\frac{0-(5)(6)}{1.5}=-20, \qquad \Delta P=\frac{(0.1)(0)+6}{1.5}=4.

The new equilibrium is \(Y=600-20=580\), \(P=100+4=104\). Check AD: \(1100-5(104)=580\). The shifted SRAS equation is \(P=106+0.1(Y-600)\); at \(Y=580\), it gives \(106+0.1(-20)=104\). The checks agree. The method avoids assuming that a 6-point increase in the supply intercept must raise the equilibrium price level by 6 points: output falls as well, moderating the price increase.

Worked Example: Simultaneous Demand and Supply Shifts

Return to the same initial equilibrium. Suppose a hypothetical drop in planned spending lowers the AD intercept by 30 output units, while improved input availability lowers the SRAS intercept by 2 price-index points. Find the new equilibrium, rounding only the final output and price level to two decimals.

Method: enter each intercept change with its algebraic sign. The changes are \(\Delta A=-30\) and \(\Delta C=-2\), and \(1+bs=1.5\). The output change is:

\Delta Y=\frac{-30-(5)(-2)}{1.5} =\frac{-30+10}{1.5} =\frac{-20}{1.5} =-13.333\ldots

The price-level change is:

\Delta P=\frac{(0.1)(-30)+(-2)}{1.5} =\frac{-3-2}{1.5} =\frac{-5}{1.5} =-3.333\ldots

Adding these changes to the initial equilibrium gives \(Y=600-13.333\ldots=586.67\) and \(P=100-3.333\ldots=96.67\), rounded to two decimals. Check using the new equations, \(Y=1070-5P\) and \(P=98+0.1(Y-600)\). At the unrounded output, \(P=98+0.1(-13.333\ldots)=96.666\ldots\), and AD gives \(1070-5(96.666\ldots)=586.666\ldots\). The results agree. This method avoids treating the favorable supply shift as if it compounded the contraction in demand: the negative \(\Delta C\) partly offsets it.

Interpreting Results and Avoiding Sign Errors

The formulas separate the role of each shock. An outward AD shift raises output and the price level when \(s>0\). An upward SRAS shift lowers output and raises the price level. When shocks occur together, their effects add algebraically. It is possible for one shock to dominate the other, so determine the signs from the formulas rather than naming the combined outcome from one shift alone.

For the simultaneous-shock example, a sign error might treat the supply contribution to \(\Delta Y\) as \(+b\Delta C\), giving the wrong calculation:

\Delta Y_{\mathrm{wrong}} =\frac{-30+(5)(-2)}{1.5} =\frac{-40}{1.5} =-26.667\ldots

This is incorrect because the output formula is \(\Delta Y=(\Delta A-b\Delta C)/(1+bs)\). With \(\Delta C=-2\), the term \(-b\Delta C\) is positive: the favorable supply shift supports output, partly offsetting the demand contraction. The correct change is \((-30+10)/1.5=-13.333\ldots\), not \(-26.667\ldots\).

Another useful check is whether the proposed changes satisfy both change equations, \(\Delta Y=\Delta A-b\Delta P\) and \(\Delta P=\Delta C+s\Delta Y\). A result that satisfies only one curve is not an equilibrium response. Finally, a move from one price level to another is not automatically an inflation rate. Inflation measures a rate of change in the price level over a stated period; the equations here calculate a difference between equilibria.

Key takeaway: First identify the intercept changes and keep their signs. Then calculate both equilibrium changes using the same positive denominator \(1+bs\), and verify the results against both AD and SRAS. The formulas assume fixed slopes and fixed potential output; they do not directly describe a change in potential output or a change in the curves’ slopes.

Check Your Understanding

For each problem, use the stated linear model and show the shock changes with their signs before calculating.

An economy has \(Y_p=600\), AD \(Y=900-3P\), and SRAS \(P=100+\frac{1}{3}(Y-600)\). Verify the initial equilibrium, then calculate the changes in output and the price level when the AD intercept rises by 40 and SRAS is unchanged. (Answer: Initial \(Y=600,\ P=100\); \(\Delta Y=20,\ \Delta P=6.67\).)
In a model with \(b=2\) and \(s=0.25\), an adverse supply shock raises \(C\) by 4 while \(A\) is unchanged. Calculate the equilibrium changes. (Answer: \(\Delta Y=-5.33,\ \Delta P=2.67\).)
For \(b=4\), \(s=0.25\), and fixed potential output, an outward AD shift raises \(A\) by 24 while a favorable supply shift lowers \(C\) by 3. Calculate both equilibrium changes. (Answer: \(\Delta Y=18,\ \Delta P=1.50\).)
An economy has \(Y_p=400\), AD \(Y=1000-5P\), and SRAS \(P=90+0.2(Y-400)\). Find its initial equilibrium. Then let \(A\) fall by 50 and \(C\) rise by 4. Calculate the new equilibrium. (Answer: Initial \(Y=475,\ P=105\); new \(Y=440,\ P=102\).)
With \(b=2\) and \(s=0.5\), an adverse supply shock raises \(C\) by 5. What change in \(A\) would keep equilibrium output unchanged, and what would then happen to the price level? (Answer: \(\Delta A=10\) keeps \(\Delta Y=0\); \(\Delta P=5\).)

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