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A model where a fraction of firms cannot adjust prices each period, generating a short-run aggregate supply curve that slopes upward and monetary non-neutrality.

Derivation

Step-by-step mathematical derivation with typeset equations and expandable detail.

Sections

A fraction of firms with sticky prices

The sticky-price model divides firms into two groups: a fraction $\theta$ that keeps prices fixed at the previous period's level (sticky-price firms), and a fraction $1 - \theta$ that sets prices optimally each period (flexible-price firms). All firms are otherwise identical, producing differentiated goods under monopolistic competition. The key parameter $\theta$ measures the degree of price rigidity in the economy. When $\theta = 0$ , all prices are flexible and the classical dichotomy holds; when $\theta = 1$ , all prices are fixed and monetary policy has maximum real effects.

This setup captures a fundamental insight: even partial price stickiness generates aggregate supply behavior that differs qualitatively from the flexible-price benchmark. When some firms cannot adjust, an increase in the money supply raises nominal spending, but not all firms raise prices to match. Those with sticky prices sell more at their fixed price, expanding real output. The aggregate price level rises less than proportionally to the money supply, so real balances increase and real activity expands. The degree of this non-neutrality depends directly on $\theta$ .

P = \theta\, P_{-1} + (1 - \theta)\, p^*

Aggregate price level (log-linear approximation): a weighted average of the previous price level (kept by sticky firms) and the optimal reset price $p^*$ (set by flexible firms).

\theta \in [0, 1]

The stickiness parameter: $\theta = 0$ is full flexibility (classical), $\theta = 1$ is complete rigidity (fixed prices).

Optimal price setting and the aggregate price level

Flexible-price firms set their price $p^*$ as a markup over marginal cost, which in the simplest specification is proportional to the overall price level and the output gap. In log-linear form, $p^* = P + a(Y - \bar{Y})$ , where $a > 0$ reflects how marginal cost rises with output. When output exceeds potential, marginal cost is high, and flexible firms set prices above the current average. When output is below potential, they set prices below average. The parameter $a$ captures the sensitivity of desired prices to real activity.

Substituting $p^* = P + a(Y - \bar{Y})$ into the price aggregation equation $P = \theta P_{-1} + (1-\theta) p^*$ and solving for $P$ yields the aggregate price level as a function of the previous price level and the output gap. The algebra requires collecting terms in $P$ on one side, which produces a coefficient $1/(1-(1-\theta)) = 1/\theta$ on the $P_{-1}$ and output-gap terms. This intermediate step leads directly to the short-run aggregate supply curve.

p^* = P + a(Y - \bar{Y})

Optimal reset price: flexible firms set prices above the average price level when output exceeds potential, and below when output falls short.

P = \theta P_{-1} + (1 - \theta)\bigl[P + a(Y - \bar{Y})\bigr]

Price-level equation after substituting the optimal price: the aggregate price level depends on last period's level, itself, and the output gap.

Deriving the upward-sloping SRAS

Solving the price aggregation equation for $P$ , collect the $P$ terms: $P - (1-\theta)P = \theta P_{-1} + (1-\theta)a(Y - \bar{Y})$ , which gives $\theta P = \theta P_{-1} + (1-\theta)a(Y - \bar{Y})$ . Dividing by $\theta$ and rearranging in terms of the price level (or equivalently inflation $\pi = P - P_{-1}$ ), we obtain the short-run aggregate supply curve: $P = P_{-1} + \frac{(1-\theta)a}{\theta}(Y - \bar{Y})$ . This is an upward-sloping relationship between the price level and output.

The slope $\frac{(1-\theta)a}{\theta}$ has intuitive comparative statics. When $\theta$ is large (many sticky firms), the SRAS curve is flat: a large output expansion produces only a small price increase because most firms do not adjust. When $\theta$ is small (few sticky firms), the SRAS is steep: nearly all firms raise prices in response to demand, so output barely moves. In the limit $\theta \to 0$ , the SRAS is vertical (classical aggregate supply), and in the limit $\theta \to 1$ , it is horizontal (fixed price level). This microfounded SRAS curve provides the bridge between monetary shocks and real output fluctuations.

P = P_{-1} + \frac{(1-\theta)\, a}{\theta}(Y - \bar{Y})

Short-run aggregate supply: the price level exceeds last period's level when output is above potential, with slope determined by the share of flexible firms and the marginal cost sensitivity $a$ .

\pi = \frac{(1-\theta)\, a}{\theta}(Y - \bar{Y})

SRAS in inflation form: inflation is proportional to the output gap, with the proportionality factor decreasing in the degree of price stickiness $\theta$ .

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