Why Short-Run Aggregate Supply Slopes Upward

Why the Short-Run Supply Relationship Slopes Upward

In the aggregate demand–aggregate supply model, short-run aggregate supply (SRAS) describes the relationship between the overall price level and the quantity of real output firms are willing to produce over a period when some wages or prices have not fully adjusted. Its usual upward slope means that, with the relevant expectations and other conditions held fixed, a higher-than-expected price level is associated with greater real output in the short run.

The key is that firms make production decisions using costs as well as selling prices. If a firm’s selling price rises while its nominal wage is temporarily fixed, the firm’s product price has risen relative to its labor cost. Labor becomes more attractive to hire, so the firm increases production. A related mechanism arises when some firms have set prices in advance: as the overall price level changes, those firms’ prices do not immediately keep pace, and their customers may demand more or less of their output.

Definition: Short-run aggregate supply is the relationship between the price level \(P\) and real output \(Y\) that firms are willing to supply over a period in which some nominal wages or prices are slow to adjust. The standard upward-sloping SRAS relationship describes a change in \(P\) with expected prices, nominal wage arrangements, productivity, and other relevant conditions held fixed. It is not a claim that every price change always raises output.

“Unexpected” matters because contracts and plans are often set using an expected price level, denoted \(P^e\). If actual \(P\) differs from \(P^e\), some costs and prices may be temporarily out of line with the overall price level. The size and direction of the production response depend on which nominal amounts are fixed and on how firms respond to changed relative prices.

Sticky Nominal Wages and the Firm’s Hiring Decision

Consider a firm that sells output at price \(P\), pays each worker a nominal wage \(W\), and uses labor \(N\) to produce \(Y=F(N)\). Assume the firm chooses employment to maximize profit, \(PF(N)-WN\), and that additional workers increase output at a diminishing rate. The marginal product of labor is \(MPL=F'(N)\). A profit-maximizing firm hires labor up to the point where the value of the marginal product equals the nominal wage.

P \cdot MPL = W \qquad\text{or, equivalently,}\qquad MPL = \frac{W}{P}

The ratio \(W/P\) is the real wage: the nominal wage measured in units of the firm’s output. If \(W\) is fixed in the short run and \(P\) rises, \(W/P\) falls. With a lower real wage, the firm is willing to employ more labor. Because each additional worker has a positive marginal product, higher employment raises the firm’s output.

Assumptions for the sticky-wage result: The nominal wage is predetermined over the period being analyzed; the firm sells output at the prevailing price level; labor’s marginal product is positive and diminishes as employment rises; and the firm chooses employment to maximize profit. Under these assumptions, a higher \(P\) lowers \(W/P\), increases desired labor, and raises output. If the nominal wage adjusts in the same proportion as \(P\), this particular mechanism does not create that response.

Worked Example: A Firm Facing a Higher Output Price

A firm has production function \(Y=10\sqrt{N}\), where \(N\) is its number of workers. Its nominal wage is fixed at \(W=20\) dollars per worker over the period. The marginal product of labor is \(MPL=5/\sqrt{N}\). Compare the firm’s employment and output when its output price rises from \(P=8\) to \(P=12\). Assume the firm maximizes profit and that the wage remains fixed at both prices.

Method. Use the hiring condition \(P \cdot MPL=W\). The common mistake is to treat a higher product price as if it increased the real wage. With \(W\) fixed, the real wage instead falls.

At \(P=8\). The real wage is \(W/P=20/8=2.5\). Set marginal product equal to this real wage:

\frac{5}{\sqrt{N}}=2.5 \quad\Longrightarrow\quad \sqrt{N}=2 \quad\Longrightarrow\quad N=4.

Output is \(Y=10\sqrt{4}=20\). As a check on the hiring condition, \(P \cdot MPL=8(5/2)=20=W\).

At \(P=12\). The nominal wage is still 20, so the real wage is \(W/P=20/12=5/3\). Then:

\frac{5}{\sqrt{N}}=\frac{5}{3} \quad\Longrightarrow\quad \sqrt{N}=3 \quad\Longrightarrow\quad N=9.

Output is \(Y=10\sqrt{9}=30\). The hiring condition checks: \(P \cdot MPL=12(5/3)=20=W\). Thus the price increase from 8 to 12 is accompanied by employment rising from 4 to 9 and output rising from 20 to 30 in this model. The arithmetic is exact; no rounding is needed.

Why the Firm’s Condition Implies More Output

The upward response follows from the firm’s choice, not simply from drawing an upward-sloping curve. To see why, start with profit \(\pi=PF(N)-WN\), taking the output price and nominal wage as given. Increasing employment by a small amount changes revenue by approximately \(P \cdot MPL\) and wage cost by \(W\). At an interior profit maximum, the change in profit from a small employment change must be zero, giving \(P \cdot MPL-W=0\).

Write the hiring condition.
Profit maximization gives \(P \cdot MPL=W\), or \(MPL=W/P\).

Change the price level, keeping the nominal wage fixed.
If \(P\) rises while \(W\) is unchanged, the ratio \(W/P\) falls.

Find the new desired employment.
With diminishing marginal product, a larger quantity of labor is needed to lower \(MPL\) to the new, lower value of \(W/P\).

Translate employment into output.
Since labor’s marginal product is positive, the higher employment level produces more output. This gives a positive short-run relationship between \(P\) and \(Y\) under the stated assumptions.

This reasoning also explains why the result is conditional. If the wage were immediately adjusted in proportion to the price level, the ratio \(W/P\) would not fall. If productivity or the firm’s production technology changed, the relationship between employment and output could also differ. The sticky-wage argument isolates one mechanism by holding such conditions fixed; it does not say that a price-level change alone determines output in every circumstance.

Worked Example: Aggregate Output with a Predetermined Wage

Suppose the economy’s production relationship is \(Y=12\sqrt{N}\), where \(N\) is total employment. The marginal product of labor is \(MPL=6/\sqrt{N}\), and the nominal wage is fixed at \(W=18\). Compare the output firms choose when the price level is \(P=6\) and \(P=9\). Assume firms hire labor until \(P \cdot MPL=W\).

Method. Find the real wage \(W/P\) at each price level, use it in the hiring condition, and then substitute employment into the production function. Do not use the nominal wage alone as the labor cost relevant to the production decision.

At \(P=6\), the real wage is \(18/6=3\). Therefore:

\frac{6}{\sqrt{N}}=3 \quad\Longrightarrow\quad \sqrt{N}=2 \quad\Longrightarrow\quad N=4.

The resulting output is \(Y=12\sqrt{4}=24\). Checking the condition gives \(P \cdot MPL=6(6/2)=18\), equal to the nominal wage.

At \(P=9\), the real wage is \(18/9=2\). Therefore:

\frac{6}{\sqrt{N}}=2 \quad\Longrightarrow\quad \sqrt{N}=3 \quad\Longrightarrow\quad N=9.

Output is \(Y=12\sqrt{9}=36\). The hiring condition checks: \(9(6/3)=18\). Thus, with the same nominal wage and production relationship, the higher price level is associated with output rising from 24 to 36. No rounding is needed.

Imperfectly Adjusted Prices

Wages are not the only nominal amounts that may adjust slowly. Some firms set prices in advance, for example because changing a price is costly or because prices are reviewed only periodically. When the overall price level changes, the prices of these firms may temporarily lag behind. Their output then becomes relatively inexpensive or expensive compared with other goods, which can change the quantity customers want to buy from them.

A simple model makes this mechanism explicit. Let \(\lambda\) be the fraction of firms whose prices remain preset at \(P^e\) during the period, where \(0\leq\lambda<1\). Suppose the remaining fraction adjusts prices according to \(P_f=P^e+c(Y-Y_0)\), where \(Y_0\) is a reference output level and \(c>0\) describes how much those firms’ prices respond to output. For this illustrative model, treat the aggregate price level as the weighted average of the preset and adjusted prices:

P=\lambda P^e+(1-\lambda)P_f.

Substituting the price-setting rule for \(P_f\) gives \(P-P^e=(1-\lambda)c(Y-Y_0)\). Since \((1-\lambda)c>0\), solving for output shows that a higher price level relative to the preset expected level is associated with higher output. The equation is a simplified illustration of the mechanism, not a universal empirical formula. Its conclusion depends on the stated price-setting assumptions.

Worked Example: Output When Some Prices Are Preset

In a stylized economy, \(75\%\) of firms keep their prices preset at \(P^e=100\) for the period. The remaining firms set \(P_f=P^e+0.04(Y-Y_0)\), where output \(Y\) and reference output \(Y_0=2{,}000\) are measured in units of output. The aggregate price level is the weighted average \(P=0.75P^e+0.25P_f\). Find output when the observed price level is \(P=101.2\).

Method. First use the aggregate price equation to identify the adjusted firms’ price. Then use their price-setting rule to solve for output. The common mistake is to apply the price-setting rule to \(P\) directly, even though it describes \(P_f\), not the aggregate price level.

Substitute the known values into the aggregate equation:

$$ 101.2=0.75(100)+0.25P_f =75+0.25P_f. $$

Subtract 75: \(26.2=0.25P_f\), so \(P_f=26.2/0.25=104.8\). Now apply the adjusted-firm rule:

104.8=100+0.04(Y-2{,}000) \quad\Longrightarrow\quad 4.8=0.04(Y-2{,}000) \quad\Longrightarrow\quad Y-2{,}000=120.

Therefore \(Y=2{,}120\). Check by recovering the aggregate price level: \(0.75(100)+0.25(104.8)=75+26.2=101.2\). The stated price level and calculated output are consistent; no rounding is needed.

Reading the Relationship Correctly

The upward slope describes a change in output associated with a change in the price level under a fixed set of short-run conditions. In the sticky-wage model, those conditions include the predetermined nominal wage. In the price-setting model, they include the preset-price arrangements and the expected price level. The curve summarizes the production responses of firms operating under such constraints; it is not a statement that all firms face identical contracts or respond in exactly the same way.

A useful exam check is to ask what is being held fixed. If a problem changes the actual price level but keeps the nominal wage or preset prices unchanged, it is describing a movement along the short-run relationship in the model. If the assumptions themselves change—for example, if the nominal wage is revised—then the original calculation may no longer apply. Determining how such changes alter the curve is a separate question.

Common Mistake: Reversing the Real-Wage Effect

A frequent error is to reason that a higher price level makes labor more expensive to the firm. With a fixed nominal wage, this reverses the relevant ratio. If \(W=18\), then at \(P=6\) the real wage is \(18/6=3\), while at \(P=9\) it is \(18/9=2\). The real wage falls, not rises. In the aggregate example above, using the mistaken real wage of 4 at \(P=9\) would produce \(6/\sqrt{N}=4\), hence \(N=2.25\) and \(Y=18\). That calculation is internally consistent with the wrong real wage, but the wage calculation is wrong: \(18/9=2\), not 4. Correcting it gives \(N=9\) and \(Y=36\).

Key takeaway: With some nominal wages or prices fixed temporarily, an unexpected change in the overall price level can alter firms’ relative prices and costs. Under the standard assumptions, a higher price level lowers the real cost of sticky-wage labor or makes preset-price firms’ goods relatively attractive, leading firms to supply more output in the short run.

Check Your Understanding

Use the assumptions stated in each question. Answers are included for checking.

A firm has \(Y=10\sqrt{N}\), \(MPL=5/\sqrt{N}\), and a fixed nominal wage \(W=24\). Find its employment and output at \(P=6\) and at \(P=10\). (Answer: At \(P=6\), \(N=25/16\) and \(Y=25/2=12.5\); at \(P=10\), \(N=625/144\) and \(Y=125/6\approx20.8333\).)
An economy has \(Y=15\sqrt{N}\), \(MPL=7.5/\sqrt{N}\), and fixed nominal wage \(W=30\). Calculate output at \(P=10\) and \(P=15\), and verify the hiring condition at each price. (Answer: At \(P=10\), \(N=6.25\), \(Y=37.5\), and \(P\cdot MPL=30\); at \(P=15\), \(N=14.0625\), \(Y=56.25\), and \(P\cdot MPL=30\).)
In a preset-price model, \(\lambda=0.60\), \(P^e=120\), \(Y_0=3{,}000\), and flexible firms set \(P_f=P^e+0.025(Y-Y_0)\). The aggregate price is \(P=0.60P^e+0.40P_f\). Find output if \(P=121.5\). (Answer: \(P_f=123.75\), so \(Y=3{,}150\).)
A firm has \(Y=8\sqrt{N}\), \(MPL=4/\sqrt{N}\), and a nominal wage that rises from \(W=20\) to \(W=25\) at the same time that \(P\) rises from 10 to 12.5. Assuming profit maximization, compare real wages and output. (Answer: \(W/P=2\) in both cases; \(N=4\) and \(Y=16\) in both cases, so this proportional wage adjustment eliminates the sticky-wage real-wage channel.)
In a short-run model with \(Y=20\sqrt{N}\), \(MPL=10/\sqrt{N}\), and fixed \(W=40\), a candidate claims that raising \(P\) from 10 to 20 doubles the real wage. Identify the error and calculate output at both prices. (Answer: The real wage falls from \(40/10=4\) to \(40/20=2\); employment rises from \(6.25\) to \(25\), and output rises from \(50\) to \(100\).)

What You'll Learn