DYNAMIC NEW KEYNESIAN MODEL (Encyclopedia)

The Dynamic New Keynesian (DNK) model represents the current workhorse theoretical framework for the analysis of monetary policy. From a philosophical and methodological point of view, it achieves the synthesis between the New Classical Macroeconomics with the Keynesian school of thought. The New Classical Macroeconomics had brought about a revolution in modern macroeconomics, by proposing a new class of models, which are able to overcome, from a methodological perspective, the Lucas’ critique to the reduced-form models that were popular until the 1970s.1In order to do this, this new class of models introduces three main features. Firstly, rather than postulating behavioural equations describing the aggregate economic relations, it derives the latter from first principles, as the equilibrium outcome of intertemporal problems faced by optimizing agents, like households and firms. 2 Secondly, it substitutes the several, somewhat ad hoc, assumptions about the formation of expectations with the new, elegant and rigorous paradigm of rational expectations, according to which the agents form expectations by using efficiently all the available information. Thirdly, the equilibrium outcome of the model requires the simultaneous clearing of all active markets: the focus shifts from partial equilibrium models to general equilibrium ones. The baseline formulation of this new class of models is the Real Business Cycle (RBC) model. From a conceptual perspective, this new strand of literature builds on the Classical ideas that markets are frictionless and always clear, that all production inputs are completely and efficiently used at all times, and that money is neutral. As a consequence, the main claim of the RBC analysis is that business cycle fluctuations are mainly caused by shifts in productivity and reflect the efficient, dynamic response of a frictionless economy. Hence, there is no welfare-improving role for any stabilization policy.
The Keynesian school of thought, on the other hand, pushes the idea that the existence of nominal and real frictions may prevent the system from reaching a full-employment equilibrium in the short-run, and makes it converge to a sub-optimal and inefficient equilibrium. In such an economy, business cycle fluctuations may be affected and amplified by the real and nominal rigidities, and the latter then assign a central role to monetary policy for the equilibrium level of real variables (as in the basic IS-LM model). 
The DNK model blends these conceptual ideas with the methodological advances of the RBC framework. For this reason, the current literature also refers to it as the New Neoclassical Synthesis (as opposed to the "old" Neoclassical Synthesis of the 1950s, from which the IS-LM model had emerged).
With the RBC model, therefore, the DNK shares the methodological core: a Dynamic Stochastic General Equilibrium (DSGE) model of the economy, in which the demand side builds on the optimal intertemporal behaviour of infinitely-lived households which maximize the utility from consumption and leisure subject to an intertemporal budget constraint, and the supply side on the optimal behaviour of a continuum of firms using a common technology and interacting with the households in the markets for production inputs. Both households and firms have rational expectations. On top of this core structure, the DNK model adds a number of features that are typical of the Keynesian school: monopolistic competition and nominal rigidities. Therefore, differently from the RBC model, firms do not operate in perfect competition, as price-takers. On the contrary, in the DNK model, there is a large set of differentiated goods, so that each firm producing a given variety has some degree of market power, and sets its price optimally in order to maximize the discounted stream of future profits, over its planning horizon (which is infinite). Moreover, in resetting their prices, firms are subject to frictions of some kind (either in the form of constraints on the frequency of adjustment or of adjustment costs), which effectively prevent the aggregate price level from adjusting flexibly to the shocks that hit the economy. As a consequence, variations in the nominal interest rate do not affect one-for-one the expected inflation rate, implying variations also in the real interest rate and therefore non-neutrality of monetary policy.
These additional assumptions make the DNK model differ sharply from the RBC benchmark, in terms of positive and normative implications. From a positive perspective, nominal rigidities act as an amplification device, which make the dynamic response of the economy to structural shocks inefficient, as opposed to the efficient fluctuations implied by the RBC framework. From a normative perspective, these inefficiencies leave room for a welfare-improving role of economic (and foremost monetary) policy. The DNK model, therefore, restates the main results of the basic IS-LM-AS model within a fully-fledged structural dynamic framework, in which the aggregate demand and supply schedules are not simply postulated, but rather derived from first principles, as the result of the optimal behaviour of microeconomic agents.
 Structure of the model
The DNK model, in its baseline formulation assuming a fixed amount of physical capital and abstracting from public consumption, builds upon three main blocks, referring to the three different classes of agents that interact in the economy. The first block describes the demand side of the economy, and consists of a representative, infinitely-lived household. The household consumes a basket of all the differentiated goods produced in the economy
 

(1)

,

wherecaptures the elasticity of substitution between any two given types, and the latter are indexed . Such differentiation among consumption goods is the first departure from the RBC framework, implying a monopoly market power for the firms producing them. At the limit, when goes to infinity, such differentiation fades away, each brand becomes a perfect substitute for the others, and the market structure converges to perfect competition.
The household chooses consumption, saving and hours worked in order to maximize the expected discounted stream of utility flows, over an infinite planning horizon, subject to a sequence of budget constraints. The problem can be formalized as

such that

,

 (2)

for all , aggregate consumption obeys equation (1) and asymptotic solvency is granted. In the formulation above, denotes the rational-expectation operator conditional on information available at time , is the time-discount factor, the level of real consumption at time t, the amount of hours worked, the price of the type-i consumption good, the aggregate price level, a riskless nominal one-period bond in which the household allocates its savings, the nominal return on such a bond between period t-1 and t, the nominal hourly wage and the nominal profits produced and distributed by the monopolistic firms.

The formulation above nests two distinct optimization problems faced by the household: one in the intra-temporal dimension, and the other in the inter-temporal one.
The intra-temporal problem requires choosing the optimal combination of differentiated goods for a given level of aggregate consumption, in each period. The solution to this intra-temporal problem yields as equilibrium conditions the demand for each type i, as a decreasing function of its relative price and given aggregate consumption

(3)

 

which highlights the interpretation of as the price-elasticity of demand for brands; and the consumption-based aggregate price index

(4)

The inter-temporal problem, instead, is related to the optimal consumption-saving decision (how much to consume today as opposed to saving for future consumption), and the simultaneous optimal consumption-leisure decision (how much to work to finance consumption as opposed to enjoying leisure). The solution to such a problem provides two additional equilibrium conditions, implying the equilibrium path for aggregate consumption and hours worked. The optimal consumption-saving decision requires the equalization of the marginal-utility costs of giving up consumption today, to the expected discounted marginal-utility benefits of consuming tomorrow the payoff from current savings:

(5)

where denotes the marginal utility of consumption at time t. In turn, the optimal consumption-leisure decision requires the equalization of the Marginal Rate of Substitution (MRS) between consumption and leisure and their relative price (where captures the opportunity cost of choosing an additional hour of leisure, in terms of foregone earnings):

(6)

The equation above implies the labour supply schedule, which determines the optimal amount of hours worked at time t, given the real hourly wage in the same period and the level of desired consumption.

The functional form of the utility U is usually assumed "isoelastic"

(7)

in that the elasticity of intertemporal substitution in consumption and the Frisch elasticity of labour supply are constant, primitive parameters.
The second block describes the supply side of the economy, and consists of a continuum of firms, indexed , each producing a differentiated good out of labour services hired from the representative household, according to a Constant Returns to Scale (CRS) technology of the form

(8)

where is an aggregate productivity index, which follows a log-stationary process:

(9)

Each firm chooses the optimal amount of labour services to hire in each period, which, under the technology specification of equation (8), requires that the real marginal costs are equal to the real wage per efficiency unit (and therefore common across firms):

(10)

The problem of each firm i is choosing the selling price for the differentiated goods it produces, subject to two constraints. The first one is given by the demand for its specific brand, coming from the household, as described by equation (3). The second constraint is on the frequency of price adjustment. Each firm is able to set its price optimally following a time-dependent, stochastic rule: in each period, each firm will get the chance to re-set its price optimally with probability , while with probability it will have to keep the price unchanged.3 These probabilities are history-independent, meaning that every firm in each period faces the same probability of having to keep the price unchanged, regardless of what happened in the previous periods. For the law of large numbers, therefore, in each period t a set of mass of firms will charge the last period’s price, while a set of mass re-optimizes. As a consequence, the aggregate price index (4) can be cast in the simplified form

(11)

This price-setting mechanism implies a degree of stickiness in the general price level proportional to , in that any unexpected disturbance that would require a price adjustment as an optimal response from the firms will induce an actual adjustment only from a subset of mass of them, while the others will respond by adjusting the amount of production. This asymmetry in the response of the firms implies a misallocation of resources between sectors that are able to re-set their prices and those that instead will have to adjust production, which is the origin of the welfare cost of unstable inflation in this model.4

When a given firm does get the chance to re-optimize, it will take into account that the chosen price will have to be charged for k more periods with probability . This feature of the price-setting mechanism effectively makes the problem of the firm dynamic, since the optimal price will have to be set in order to maximize not only the current profits, but rather the entire expected discounted flow of future profits. Formally, the problem of a generic firm that gets the chance to re-optimize in period t, is:

(12)

such that


(13)

the real marginal costs obey equation (10), and

                
(14)

denotes the Inter-temporal Marginal Rate of Substitution (IMRS) in consumption, at which the household (which is the ultimate owner of the firms) discounts future cash-flows. The solution to such a problem implies that all firms that are able to re-set their price at time t, will choose the common level

 
(15)

where  is the net mark-up that each firm can charge over marginal costs as a consequence of its market power, and are period weights (decreasing with k) related to the way in which the firms discount future cash-flows. Equation (15), therefore, implies that the optimal price for a firm that is able to re-optimize should be set as a constant mark-up over a weighted average of current and expected future nominal marginal costs. In the limiting case of full price flexibility, arising when , all firms can re-set their price in every period, and therefore only the current marginal costs are relevant:


(16)

.

The third and last block accounts for the behaviour of the monetary policy maker. In the cash-less specification adopted here, monetary policy is usually defined as the direct control, by the Central Bank, of the short-term nominal interest rate , according to some instrument feedback rule of the form


(17)

The above specification of the monetary policy rule, originally proposed by Taylor (1993), implies that the nominal interest rate at time t is raised above the long-run level rr whenever the actual inflation rate or real output are higher than a given target (respectively 0 and ), or as a result of some other, un-modelled, objective of the Central Bank ,and , are the response coefficients, measuring the degree of aggressiveness towards the two targets.

Equilibrium and Implications

The model outlined above, in the absence of exogenous shocks hitting the level of labour productivity, converges to a non-stochastic long-run equilibrium in which the level of the interest rate is tied to the time-discount factor  

(18)

    
The first implication of the DNK model is in the relation above. 
The presence of the static, real distortion related to monopolistic competition implies an inefficiently low level of equilibrium output. The (simplified) RBC version of the specification adopted here, arising when , indeed, would imply a higher level of output, , as a direct consequence of perfect competition.  
From a dynamic perspective, the implications of the model above will be affected by the second dynamic distortion: nominal rigidities. Such implications can be studied by resorting to first-order approximation of the equilibrium conditions outlined above, which allows to reduce the model economy to a system of five linear stochastic equations, in which lower-case variables denote log-deviations from long-run equilibrium values. The demand side of the economy is described by the Euler Equation (5) and the aggregate resource constraint, requiring that aggregate private consumption equal aggregate output, given the absence of investment and public consumption:

(19)

The equation above is sometime referred to as a dynamic stochastic IS curve, in that it implies a negative relation between output and the interest rate, just like the static IS schedule of the Neoclassical Synthesis of the 1950s, discussed in the undergraduate macroeconomics courses. An important difference, however, is worth mentioning. While in the static IS curve, indeed, the negative impact of the interest rate on output is triggered by the contractionary effect on investment, here the negative effect works through the intertemporal substitution in consumption: a higher interest rate makes it more convenient to consume less today, save more and postpone consumption to the future.
The supply side is described by the aggregate production function, obtained by aggregating across firms, equation (8); the equilibrium in the labour market, equations (6) and (10); the pricing behaviour of the firms, described by equations (11) and (15). All together, these equilibrium conditions imply the short-run New Keynesian Phillips Curve (NKPC) describing the equilibrium dynamics of the inflation rate:


(20)

in which the composite parameter k  is defined as and the additional stochastic shock captures inflationary cost-push shocks.5 The third equation describes the behaviour of the Central Bank

(21)

the fourth equation defines the target, frictionless level of output, which is derived in the limiting case ,

(22)

and the last one describes the dynamics of the stochastic driving force, the productivity shock:

(23)



The main implications of the DNK model, even in the small-scale version discussed here, are sharply different from the corresponding RBC version. In the latter, fluctuations in the real output would only follow the dynamics of productivity, as described by equation (22), and would therefore be efficient. Monetary policy, moreover, would be completely ineffective and the Phillips Curve would turn vertical. The system in that case would reduce to equation (19), implying the dynamics of the interest rate, (22) and (23) only. In the general DNK version, however, the presence of monopolistic competition and nominal rigidities implies a series of departures from the implications of the RBC. First of all, the dynamics of actual output are now the result of the amplification mechanism triggered by nominal rigidities, through the interaction between NKPC, the dynamic stochastic IS curve and the monetary policy rule. Secondly, a positively-sloped NKPC implies a short-run trade-off between inflation and output stabilization. Thirdly, the dynamics of the price level are sticky, and they depend mainly on the wedge between actual output and its frictionless counterpart. Fourthly, monetary policy, through changes in the interest rate and their feedback effect on output, is able to have real effects on real activity, the more so the stickier consumer prices are.

References
Calvo, G. A., (1983). "Staggered prices in a utility-maximizing frame work". Journal of Monetary Economics, 12 (3).
Clarida, R., J. Galí and M. Gertler, (1999). "The science of monetary policy: a new Keynesian perspective". Journal of Economic Literature, 37 (4)
Galí, J., (2003). "New Perspectives on Monetary Policy, Inflation and the Business Cycle". In: Dewatripont, M., Hansen, L., Turnovsky, S. (Eds.), Advances in Economic Theory, vol. III. Cambridge University Press.
Galí, J., (2008). Monetary Policy, Inflation and the Business Cycle, Princeton University Press, Princeton
Goodfriend, M. and R. King, (1997) "The New Neoclassical Synthesis and the Role of Monetary Policy". In: Bernanke, B. and J. Rotemberg (Eds.), NBER Macroeconomics Annual 1997, MIT Press
Long, J. B. and C. Plosser. (1983) "Real Business Cycles." The Journal of Political Economy, 91 (1)
Lucas, Robert (1976), "Econometric Policy Evaluation: A Critique", in Brunner, K.; Meltzer, A., The Phillips Curve and Labor Markets, Carnegie-Rochester Conference Series on Public Policy, 1
Nisticò, S., (2007). "The Welfare Loss from Unstable Inflation". Economics Letters, 96
Rotemberg, J.J., (1982). "Sticky Prices in the United States". The Journal of Political Economy, 90 (6)
Woodford, M., (2003). Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press, Princeton
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1"Given that the structure of an econometric model consists of optimal decision rules of economic agents, and that optimal decision rules vary systematically with changes in the structure of series relevant to the decision maker, it follows that any change in policy will systematically alter the structure of econometric models" (Lucas, 1976, p. 41): a reduced-form econometric model is useless for predicting the effects of a change in policy, because the model parameters, conditional upon which the prediction is made, are themselves dependent on policy.
2This is the sense in which such a model overcomes the Lucas’ critique: since the model parameters are now related to first principles like the structure of preferences or technology, they are independent of the economic policy, and can be reliably used to predict the effects of changes in the latter.
3This assumption is due to Calvo (1983), hence the denomination "Calvo rule".
4An alternative way to introduce nominal rigidities has been proposed by Rotemberg (1982), and it is based on the assumption that all firms can re-optimize in each period, but in doing so they have to pay a real menu cost proportional to the price adjustment. Notwithstanding the lack of asymmetry implied by this price-setting mechanism, however, it can be shown that its positive and normative implications in this baseline specification of the model are equivalent to those of the "Calvo rule" (see, for example, Nisticò, 2007).
5Although here this term has been added ad hoc, it can be easily derived from the underlying structure of the economy, for example by assuming a time-varying price-elasticity of the demand for brands .

Editor: Salvatore NISTICO'
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