Algebra of New Keynesian Models with Calvo price rigidities
Channel: Willi Mutschler
Date: 2022-05-03

This video is part of a series of videos on the baseline New Keynesian model with a linear production function and nominal price rigidities a la Calvo.
In this video we deep dive into the math and algebra, i.e. how to derive the nonlinear model equations with pen and paper.

Slides and notes: https://mutschler.eu/dynare/models/nk/
Course on Computational Macroeconomics: https://github.com/wmutschl/Computational-Macroeconomics

**Timestamps**

0:00:45 - Model structure
0:03:42 - Household sector
0:05:44 - Consumption index according to Dixit-Stiglitz
0:06:55 - Budget restriction
0:09:00 - Nominal vs real interest rate
0:11:03 - Debt, solvency constraint, transversality condition
0:15:36 - Consumption cost minimization
0:20:57 - Household: optimal labor, consumption and savings
0:29:25 - Firms: final vs intermediate goods
0:29:59 - Stochastic discount factor
0:31:40 - Final good sector: optimality conditions
0:34:49 - Intermediate good sector: objective function
0:36:03 - Intermediate good sector: optimal labor demand and marginal costs
0:39:10 - Intermediate good sector: Nominal price rigidities a la Calvo
0:41:21 - Intermediate good sector: optimal reset price
0:44:06 - Intermediate good sector: rewriting infinite sums recursively
0:50:07 - Intermediate good sector: law of motion for optimal reset price
0:53:13 - Market clearing: overview
0:54:09 - Market clearing: bond and labor market
0:54:37 - Market clearing: aggregate demand
0:57:23 - Market clearing: aggregate supply and price inefficiency distortion
1:02:14 - Monetary policy
1:03:40 - Stochastic processes
1:04:41 - Overview of all nonlinear model equations
1:06:34 - If you find mistakes and errors, please let me know

**References**
- Galí (2015, Ch.3)
- Heijdra (2017, Ch.19) • Romer (2019, Ch.7)
- Walsh (2017, Ch.8)
- Woodford (2003, Ch.3)

**Corrections**
- 49:50 The last simplified s_{2,t} formula on my notes misses the expectation operator, the lambdas and pi. The formulas should be:
S_{1,t} = y_t + \theta \beta E_t \frac{\lambda_{t+1}}{\lambda_t} \Pi^{\epsilon-1} S_{1,t+1}
S_{2,t} = y_t mc_t + \theta \beta E_t \frac{\lambda_{t+1}}{\lambda_t} \Pi^{\epsilon} S_{2,t+1}
Now, to get the expressions on the slides (which are still correct), you need to multiply both equations by \lambda_t and re-define lower-case s_{1,t} = \lambda_t S_{1,t} and s_{2,t} = \lambda_t S_{2,t}. Sorry that I did not show this step. Thanks to Camilo Granados for pointing this out in the comments! It doesn't really matter though as this is just a scaling of the auxiliary variables.

Checkout https://mutschler.eu/dynare for more stuff on DSGE models and Dynare.