New Keynesian Model (NK)

NK model takes the RBC model backbone and adds to that the price stickiness. To incorporate price stickiness we assume that firms are price-setters and thus move away from the perfect competition assumption. In the NK model we assume goods heterogeneity i.e. firms make a variety of goods which are not perfect substitutes. There are numerous or continuous of firms, each of which set their own price but does not impact the industry aggregates and thus takes all other prices as given. Further, they only take labour as the factor of production and have the same exogenously decided technology factor. On the other hand, households consume a variety of goods and supply labour to earn wages. 

The only difference in the monopolistic competition we get is the two-stage budgeting of households. They have to allocate spending across different varieties for a given amount of income and also have to choose between consumption and leisure. So, first, we solve for how much of each variety of goods should an individual consumer subjected to the constraint. The solution to this yields us the demand curve i.e. relationship between the quantity of specific good consumed and the price of that specific good. This demand curve is then used to find the aggregate consumption which is fed into the optimization problem at stage two.

What we observe from the solutions is that under this static monopolistic competitive setting, optimal pricing equals markup times the marginal costs. Also, markups create distortions. 

Now, when we shift from the static to the dynamic model, firms are allowed to adjust their prices in each period. Two standard ways have been discussed regarding the price rigidities - (a) Calvo price stickiness (b) Rotemberg price stickiness.

Under the Calvo setting, a given firm can update the price with certain probability or in other words only certain firms would be able to adjust their prices in each period. Also, all the firms which update their price in a particular period set the same optimal price because every firm face the identical problem. Under the static setting and dynamic setting without price rigidities, the firms objective will maximize the profit of that period itself, but when price rigidities come into play, the price chosen by the firm today will even affect future profits and therefore needs to be taken into account while deciding the optimal pricing. Thus we discount future profits and also take those into considerations.

In Rotemberg setting very firm can adjust their prices in each period but doing so would incur some adjustment cost. Further we see that both the settings lead to distortions, however, they only differ w.r.t. the source of distortions.

Therefore reiterating that the New Keynesian proposed series of models based on microeconomic foundations with price and/or wage rigidities and showed that macroeconomics can be understood with optimizing agents having market power. The New Keynesian model basically tries to explain the short run fluctuations through real and monetary shocks by incorporating rigidities in the basic RBC structure. The NK model basically boils down to 3 main equations:

1. The AS curve represented by the NK Phillips curve that related inflation to output gap and expected inflation over the next period.
2. The AD curve combines a dynamic IS curve that related evolution of output gap to the interest rate.
3. Monetary Policy schedule that describes how the nominal interest rate is set by the central bank depending on the fluctuations of inflation rate and output gap.

The model is micro founded where the households maximize their utilities subjected to the inter temporal budget constraint and firms maximize their profit subjected to nominal rigidities.

1. The NK Philips curve comes from the aggregation of the supply decision of the firms to optimize their profits. The curve has been derived from the Calvo model which assumes that only a certain proportion of firms can re-optimize their prices in each period and other firms have to stick with the prices that they had decided earlier. Therefore for determining the optimal re-setting price that can maximize their profits, the firms take 3 factors into account
a. Anticipated future inflation
b. Output gap
c. Cost push shock
Thus it was shown that the equation becomes

2. The dynamic IS curve is a log-linearization of the Euler bond equation that describes the intertemporal allocation of consumption of agents in the economy. It arises by maximization of welfare index of a representative household subjected to budget constraint. It links the current output gap to the difference between real interest rate, to the expected future output gap and to an exogenous demand shock.

3. Monetary Policy Schedule is based on the Taylor rule. It relates the nominal interest rate to inflation rate, the output gap and the monetary policy shock.

Now having a system of linear equations we can solve these to study the dynamic behaviour of the endogenous variables in response to the exogenous shocks. But before that, we need to express all the endogenous variables only in terms of the exogenous variables. We use the guess and verify approach, therefore. According to this, we guess a solution, substitute it back into the obtained equations and then solve the set of equations to get the restrictions on the parameter for which the original equations would hold true. For simpler equations and easy guess functions, we can solve analytically, otherwise, we need to use the numerical method using DYNARE.