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If you had to summarize New-Keynsian thought into one paragraph, or a set of equations, what would it be?

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Gali's Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework, provides an advanced undergraduate / first-year graduate student introduction to these models, and I'd recommend it for self studies. According to chapter 1, available online, which offers an overview of the New Keynesian Model, the key elements are:

  1. Structural characteristics of Real Business Cycle (RBC) Dynamic Stochastic General Equilibrium (DSGE) models
  2. Monopolistic competition
  3. Nominal rigidities
  4. Short run non-neutrality of monetary policy
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I will try to answer this question although it is rather hard using only one paragraph, if not impossible. A paragraph explaining what New-Keynesian Economics is:

Models explaining price and wage stickiness using rational expectations and utility maximizing behaviour.

See Gordon, R.J, 1990 for more details.

For a defining equation the New-Keynesian Phillips curve is one of the cornerstones of New Keynesian economics:

$\pi_{t}=\beta E_{t}\left\{ \pi_{t+1}\right\} +\kappa x_{t}$

Where $x_{t} $ is the output gap, $\pi_{t} $ is inflation at time $t $. $\kappa=\frac{\gamma\left(1-\omega\right)\left(1-\beta\omega\right)}{\omega} $ which is an increasing function of firms being able to change their price that period. Note that this curve is based upon Calvo pricing but one can also use Rotemberg pricing etc.

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