Formula

Derivation of the Inflation Formula with Adaptive Expectations

The inflation rate under adaptive expectations can be determined through a derivation that incorporates the assumption that expected inflation equals the previous period's inflation (πtE=πt1\pi_t^E = \pi_{t-1}). The derivation proceeds in several steps: inflation is defined as the percentage increase in prices, which is equal to the increase in costs per unit of output. If wages are the only cost, this is equivalent to the percentage increase in wages. This wage increase is composed of expected inflation plus the bargaining gap. Finally, by substituting the previous period's inflation for expected inflation, we arrive at the formula πt=πt1+gapt\pi_t = \pi_{t-1} + \text{gap}_t. The full derivation is as follows:

\begin{align*} \text{inflation (%)} &\equiv \text{increase in prices (%)} \\ &= \text{increase in costs per unit of output (%)} \\ &= \text{increase in wages (%)} \; (\text{if wages are the only costs}) \\ &= \text{expected inflation (%)} + \text{bargaining gap (%)} \\ &= \text{last period's inflation} + \text{bargaining gap (%)} \\ \pi_t &= \pi_{t-1} + \text{gap}_t \end{align*}

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Updated 2026-01-15

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