As remarked, labour supply is relatively fixed. Utility maximisation and rational calculation will primairily be directed at finding a competitive wage (competition not necessarily meaning full competition - as we e.g. referred to a Nash equilibrium). An individual who sets his wages too high will become unemployed. Even the probability of becoming unemployed will have a sobering effect. Given this framework, the model must concern a dynamic process of unemployment (threats) and wage adjustment.
First consider a homogeneous market with price level P. Price adjustment towards the market clearing equilibrium price P° depends upon excess demand, and since excess demand is determined by the price level, we get a differential equation:
P’ = dP / dt = f[ D[P] - S[P] ] = f ° [ P° - P ]
Note that the choice of ‘excess demand’ as the explanatory variable is arbitrary. We might as well take excess supply, or allow demand and supply to react differently, or have a different sensitivity to prices and quantities. Similarly, we can also take the quantity as the explained variable. And we can also formulate the equation in expectational variables.
Some authors hold that above relationship for price dynamics is an hypothesis that needs further clarification. I think that this is too cautious. Admittedly, it might be too simple to only presume that agents know that they are involved in a market ‘tatonnement’ process, and further explanations can be helpful. Agents have various tools available, and the choice of offering and accepting prices and quantities can be described, using an optimising framework. The speed of adjustment in markets depends upon characteristics like the size of the market, the historical relationships between agents, ‘menu costs’, and the like. It is also useful to distinguish ‘normal’ periods and ‘shocks’. However, the level of detail depends upon the use of the model, and above relationship suffices our goal.
Inflation is the rate of growth of prices, i.e. p = dLog[P] / dt = P’ / P. The change in inflation is dp / dt = P”/ P - (P’)2 / P2 in terms of the original price level. Acceleration of inflation would be d2 p / dt2.
We need to clarify a term. The economic literature uses the term “Non-Accelerating-Inflation Rate of Unemployment” (NAIRU) for that rate of unemployment that causes dp / dt = 0.
This term thus should be “non-accelerating prices” or “non-changing, or constant, inflation”.
Secondly, it appears that the formulation in terms of differentials is less useful for practical economics than the formulation in differences. So we will use differences instead. Inflation then is p = (P /P[-1] - 1) (often expressed as a percentage).