The Basic New Keynesian Model

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1 The Basic New Keynesian Model January 11 th 2012 Lecture notes by Drago Bergholt, Norwegian Business School I

2 Contents 1. Introduction Prologue The New Keynesian model Key features Households Setup Optimal consumption vector and the aggregate price index Optimal allocation of consumption and labor Firms Aggregate inflation Optimal price setting Log-linearization Equilibrium Market clearing The New Keynesian Phillips curve and the Dynamic IS equation Equilibrium determinacy System representation Blanchard and Kahn conditions Shocks Effects of a monetary policy shock Effects of a technology shock Distortions to the efficient allocation The efficient steady state Distortions caused by market power Distortions caused by sticky prices Monetary policy solutions to equilibrium distortions The welfare loss function Introduction The simplest case A welfare loss function when real rigidities are absent Introduction of cost push shocks A welfare loss function when real rigidities are present Welfare based evaluation of monetary policy Introduction An efficient steady state under discretion An efficient steady state under commitment A distorted steady state under discretion II

3 9.5 A distorted steady state under commitment Wage rigidities Introduction Firms Households Inflation equations and the Dynamic IS equation System representation and equilibrium determinacy Shocks Monetary policy design with sticky wages A small, open economy model Introduction Households Terms of trade, domestic inflation and CPI inflation The real exchange rate International risk sharing Uncovered interest rate parity Firms and technologies Equilibrium Aggregate demand and output Equilibrium The trade balance Equilibrium The supply side: Marginal cost and inflation dynamics The New Keynesian Phillips curve and the Dynamic IS equation Equilibrium determinacy Equilibrium dynamics Optimal monetary policy in the small open economy Welfare losses References Appendix A. Dynare codes A monetary policy shock with sticky prices B. Dynare codes A technology shock with sticky prices C. Dynare codes A monetary policy shock with sticky prices and wages III

4 1. Introduction 1.1 Prologue These lecture notes take the reader through a basic New Keynesian model with utility maximizing households, profit maximizing firms and a welfare maximizing central bank. I follow Gali s (2008) book as closely as possible. The notes were born during my participation at a couple of PhD courses in monetary policy, taught by Antti Ripatti (Bank of Finland) and Krisztina Molnar (Bank of Norway), respectively. Both courses built on the excellent book by Gali. The aim of the notes is to provide the reader with all relevant calculations which are left out of the book. In addition, the notes also go through equilibrium determinacy conditions in more detail, following benchmark articles such as Blanchard and Kahn () and Bullard and Mitra (2002). Chapters 2, 3 and 4 characterize the basic New Keynesian model. I first analyze households, then firms. Results are combined to establish general equilibrium. I derive a dynamic IS equation and a New Keynesian Phillips curve. Determinacy and shocks are discussed in chapters 5 and 6. I perform some welfare analysis of monetary policy in chapters 7, 8 and 9. Chapter 10 augments the basic model with sticky wages in addition to sticky prices, following Erceg et al. (2000). Finally, the small open economy model established by Gali and Monacelli (2005) is derived in chapter 11. Dynare codes are provided in the appendix. A few words about notation: Variables in levels are denoted with capital letters, logged variables with small letters. Percentage deviations are denoted with small letters with a hat. Let us illustrate by an example: The percentage deviation in from is presented by a first-order Taylor expansion: 1.2 The New Keynesian model Key features So, what kind of features do the New Keynesian models possess? The most important are: Dynamic, stochastic, general equilibrium (DSGE) modeling: Agents behavior today affects future environments. Agents know this and behave accordingly. Still, uncertainty arises because at least some processes in the economy are exposed to exogenous shocks. General equilibrium, in the sense that it incorporates all markets in the economy, is provided. Monopolistic competition: Prices are set by private economic agents in order to maximize their objectives, as opposed to being determined by an anonymous Walrasian auctioneer seeking to clear all competitive markets at once. Nominal rigidities: At least some firms are subject to constraints on the frequency with which they can adjust prices of the goods and services they sell. Alternatively, firms may face some 1

5 costs of adjusting those prices. The same kind of friction applies to workers in the presence of sticky wages. Short run non-neutrality of monetary policy: As a consequence of nominal rigidities, changes in short term nominal interest rates are not matched by one-for-one changes in expected inflation, thus leading to variations in real interest rates. The latter brings about changes in real quantities. In the long run, however, all prices and wages adjust, and the economy reverts back to its natural equilibrium. While the first bullet point is a common feature in most modern macroeconomic models, including those in the RBC literature, the last three are special ingredients in New Keynesian models. Now it is time to present the basic model. 2

6 2. Households 2.1 Setup We will study households and the implications of market power first. Consider an economy consisting of many identically, infinitely-lived households, with measure normalized to one. The representative household has an instantaneous (and time separable) money-in-utility function of the form: (2.1) The consumption level is denoted, is labor, and is real money holdings. One can think of as a composite of many goods. We make the following assumptions about preferences: 1,,,,, To simplify the analysis, we also assume that the marginal utility of one specific element in the utility function is independent of the level of other elements, i.e. that. A representative household maximizes lifetime utility, and discounts the future proportionally by a factor : { } (2.2) The consumption index continuum of goods represented by the interval [ ]: is the sum of consumption of all goods, and there exists a ( ) (2.3) Note that utility is a nested function of, where is increasing in and is increasing in. Thus, utility is increasing in. The CES-aggregator given in (2.3) is an assumption about preferences. Given this assumption, goods become imperfect substitutes, a feature which equips firms with market power. 2 Households maximization problem is subject to a one-period budget constraint: In this setup, and is the number of bonds purchased last period, each yielding a payoff of one, is the price per bond bought today. (2.4) 1 The expression represents throughout the text. 2 Equation (2.3) also nests free competition as a special case. In particular, taking the limit as approaches infinity, (2.3) becomes. 3

7 2.2 Optimal consumption vector and the aggregate price index The household s decision problem can be dealt with in two stages. First, for any given level of consumption expenditures, it will be optimal to purchase the consumption vector that maximizes total consumption. 3 Second, given this optimal bundle of consumption goods, the household must choose the utility maximizing combination of consumption, labor and money. Let us find the optimal consumption vector first. For a given level of consumption expenditures, say, the consumption maximization problem is given by: ( ) s.t. (2.5) This problem can be used to derive an aggregate price index in addition to the optimal consumption vector. Let us solve the problem: FOC: ( ) ( ) : ( ) ( ) [( ) ] The equality must hold for all goods, so the relationship between two different goods must be: (2.6) Insert (2.6) into the constraint and solve for : 3 Alternatively, one can find the consumption vector that minimizes total consumption expenditures for a given level of consumption. The two problems are equivalent and give identical results. 4

8 Insert the result above into and evaluate the result for : ( ) [ ( ) ] [ ( ) ] [( ) ] ( ) Define as the expenditure needed to purchase a unit-level of, that is. Using this definition we can solve the above equation for : ( ) (2.7) Thus, equation (2.7) can conveniently be defined as an aggregate price index. We will use it throughout the notes. To find the optimal consumption vector, insert (2.6) into the expenditures level equation. Then, insert (2.7) and solve for consumption of good : [( ) ] (2.8) Insert (2.8) into (2.3) and rearrange: ( ) ( [ ] ) ( [ ] ) ( ) [( ) ] (2.9) Finally, we get the demand function for good by inserting (2.9) into (2.8): (2.10) 5

9 Equation (2.10) is the solution to (2.5), the first stage of a representative household s decision problem. Once the household knows prices and has decided on, it also knows how much to consume of each good. The next step is to decide. 2.3 Optimal allocation of consumption and labor The problem in the second stage is established by using (2.2), (2.4) and (2.9): { } s.t. (2.11) Problems such as the one above are most often solved by using either Kuhn-Tucker conditions or by dynamic programming. The results should be the same, of course. I will now show both of these methods. First, the Kuhn-Tucker approach starts by setting up the Lagrangian. Let us go through the steps: { } (2.12) FOC: : : : : From (2.16): (2.13) (2.14) (2.15) (2.16) (2.17) From (2.13): { } { } { } (2.18) From (2.14) and (2.13): (2.19) From (2.15) and (2.13): (2.20) 6

10 Equations (2.18), (2.19) and (2.20) determine the intertemporal consumption allocation (the Euler equation), the labor-leisure choice and the money demand, respectively. Together, those equations determine the rational, forward-looking household s allocation decisions. An alternative approach to derive (2.18)-(2.20) from (2.11) is to use dynamic programming. Point of departure is the observation that the structure of the household s optimization problem in period is identical to the one in period,, etc. To see this, we first define total financial wealth at the beginning of period as: Second, rewrite the budget constraint to: Third, assume that the budget constraint holds with equality and solve for : ( ) (2.21) Fourth, recast (2.11) into a Bellman equation where is treated as the state variable and as the control variable: { } (2.22) Equation (2.22) captures the core idea of dynamic programming, as it already defines a necessary condition any solution to (2.11) has to fulfill. The Bellman equation basically states that the highest obtainable value of the decision problem in period, ( ), is given by the control which maximizies the sum of current period utility and the discounted value of the decision problem next period. The Euler equation for this problem states that the marginal cost of allocating more wealth today is equal to the marginal benefit of allocating more wealth tomorrow. It is written as: When we plug (2.21) into (2.1), this optimality condition becomes: (2.23) The envelope theorem for the problem states that the marginal change in the value function today from a change in total wealth must be equal to the marginal change in today s utility. This optimality condition is written as: When we plug (2.21) into (2.1), the envelope theorem yields: 7

11 (2.24) Iterate (2.24) one period forward: (2.25) Insert (2.25) into (2.23) and we get the following consumption Euler equation: (2.26) Further, we characterize the remaining optimality conditions using (2.21) and (2.1): : (2.27) : (2.28) From (2.26): (2.29) From (2.27): (2.30) From (2.28): (2.31) Equations (2.29)-(2.31) determine the intertemporal consumption allocation (the Euler equation), the labor-leisure choice and the money demand, respectively. Notice that they are identical to (2.18)-(2.20), highlighting the fact that the household s optimization problem should have the same solutions regardless of solution method. To proceed we need to specify utility. As an example, consider the following per-period utility function: 4 (2.32) The marginal utilities of consumption, labor and money become: The Euler equation given by (2.18) or (2.29) writes: { } (2.33) 4 Gali (2008) excludes real money balances from the utility function, but instead imposes an ad-hoc log-linearized money demand given by, where is the interest rate elasticity in the money demand equation. We will see soon that this is equivalent to setting in (2.32). 8

12 The labor-leisure choice given by (2.19) or (2.30) writes: (2.34) The money demand equation given by (2.20) or (2.31) becomes: (2.35) Finally, it is convenient to log-linearize (2.33)-(2.35). We denote small letter variables as the log of large letter variables. With respect to the Euler equation, define the following: Using this, (2.33) can be rewritten to: [ ( ) ] It is clear from the equation above that in steady state where. Thus, a first-order Taylor expansion of the Euler equation around steady state yields: [ ] (2.36) The linearized version of the labor supply equation (2.34) is: Finally, let us linearize the money demand equation given by (2.35): (2.37) [ ] 9

13 : (2.38) The Basic New Keynesian Model [ ] [ ] [ ] If we discard the constant term and assume an income elasticity of one, where this assumption implies that, the money demand equation can be written as (2.38), where This ends the analysis of households in the New Keynesian model. We now turn to firms. 10

14 Assume Cobb-Douglas technology: 5 (3.1) The Basic New Keynesian Model 3. Firms 3.1 Aggregate inflation Here, is the output produced by firm in period, is the economy-wide technology level and is the labor force used by the firm. One key ingredient in the New Keynesian model is price rigidity. When firms set their prices, they can do so freely. However, they do not know a priori when the next opportunity to price change emerges. The probability of being unable to change the price in any given period is. Thus, this is the fraction of all firms that is stuck with the price they had last period while the remaining price dynamics (inflation) in period can be calculated as follows, where firms reset their prices. The aggregate is the aggregate price level, is the optimal price set by firms who are able to reoptimize in that period, and [ ] represent the set of firms not reoptimizing their posted price: [ ] [ ] ([ ] ) (3.2) The aggregate gross inflation is defined as. Steady state is defined by zero inflation, implying that and. Linearizing (3.2) around steady state yields: 6 [ ] (3.3) Equation (3.3) makes it clear that inflation results from the fact that firms reoptimizing in any given period choose a price that differs from the economy s average price in the previous period. 5 The capital stock is treated as fixed and investment is set to zero in the short run. These two specifications follow McCallum and Nelson (1999), who argued that capital do not play a major role in most monetary policy and business cycle analyses. 6 Remember the first-order Taylor expansion: where is the vector of variables one wants to linearize around. Using this as a point of departure, it is often convenient to define a new variable as the log deviation in from :. This implies that, and the Taylor expansion can be rewritten to a formula for log-linearization via Taylor series expansion:. 11

15 Hence, in order to understand inflation over time one needs to analyze the factors underlying firms price setting decisions. 3.2 Optimal price setting Basically, when firms are faced with the problem of setting optimal price today, they must take into consideration that this price often determine profit in the future as well, as the probability of being stuck with today s price periods ahead is. Thus, a firm who reoptimizes in period will choose the price that maximizes current market value of the profits generated while that price remains effective. The stochastic discount factor for nominal payoffs in period is, which is given by: 7 (3.4) The representative firm s maximization problem is thus given by: { [ ( ( ))]} s.t. (3.5) Let us spend a couple of seconds on the problem given in (3.5). is the output in period for a firm that last set its price in period. ( ) is the total cost in period as a function of this output. The nominal, undiscounted profit in period is thus ( ). The firm s problem is subject to a sequence of demand constraints as given by (2.10), and market clearing in period implies that the firm produces. The problem can be rewritten to an unconstrained one by inserting the constraint into the profit function. We also insert for the discount factor. This gives us: { [ ( ( ))]} (3.6) Let us find the optimal price : [ ( ( ))] FOC: 7 Go back to the Euler equation of the consumers to get the intuition. 12

16 [ ( )] [ ( )] [ ( )] [ ( )] ( ) ( ) Next, we insert for and and solve for the optimal price : ( ) ( ) ( ) ( ) (3.7) Divide both sides by to get the optimal real price as a weighted average of future real marginal costs: 8 (3.8) Notice that in the case with flexible prices, i.e. when, (3.6) collapses to a one period problem and (3.7) becomes: (3.9) Thus, (3.9) gives the desired or frictionless markup. 3.3 Log-linearization 8 Note that the real marginal cost in period is denoted. 13

17 The next step is to log-linearize (3.7) around the steady state. In a zero inflation steady state, we must have that: The last three identities follow from the zero inflation definition and from market clearing. Before log-linearizing it is convenient to divide both sides of (3.7) by : (3.10) A first-order Taylor expansion of the LHS of (3.10): 9 9 This is just a simple first-order Taylor expansion. The first term is the LHS of (3.10) in steady state. The four last terms contain the first derivatives with respect to,, and respectively, all evaluated in steady state. 14

18 [ ] [ ] A first-order Taylor expansion of the RHS of (3.10): The first term is the RHS of (3.10) in steady state. The four last terms contain the first derivatives with respect to,, and respectively, all evaluated in steady state. 15

19 ( ) ( ) [ ( )] [ ( )] Finally we equate LHS with RHS and solve for : [ ] [ ( )] 16

20 ( ) [( ) ] [( ) ] [ ] (3.11) We see from (3.11) that firms will set a price that corresponds to the desired markup, 11 given by, over a weighted average of their current and expected nominal marginal costs, with the weights being proportional to the probability of the price remaining effective at each horizon. 11 Because, we have that ( ). 17

21 4. Equilibrium 4.1 Market clearing Market clearing in the goods market implies: Let aggregate output be defined as: (4.1) ( ) (4.2) Insert (4.1) into (4.2), and then (2.10) into (4.2), to get the aggregate market clearing condition: ( ) { [ ] } ( ) ( ) ( ) [( ) ] Finally, taking logs on both sides yields: Equation (4.3) is the aggregate market clearing condition. Insert (4.3) into (2.36) and get: (4.3) (4.4) Market clearing in the labor market: (4.5) From (3.1) we see that. Insert this into (4.5) as well as the goods market clearing condition (4.3) and the consumption demand (2.10): ( ) (4.6) Next we take the log of (4.6): [ ] 18

22 [ ] (4.7) I will now show that because up to a first-order approximation around, but first I must show that. Recall the consumer price index ( ). Rearranging gives: ( ) ( ) A second order approximation of (4.8) gives us: (4.8) [ ] (4.9) From equation (4.9) it is also clear that up to a first-order approximation. Next, let us do a second order approximation of : [ ] [ ] Now, insert (4.9) and get: 19

23 ( ) [ ] (4.10) From (4.10) we conclude that up to a first-order approximation,. This implies that: [ ] Thus, (4.7) can be rewritten to: (4.11) 4.2 The New Keynesian Phillips curve and the Dynamic IS equation Next, an expression for individual firms marginal cost as a function of the economy s average real marginal cost is derived. The latter is derived in (4.12), where we insert from (4.11): The nominal marginal cost by using labor is the wage. The nominal marginal gain is the income increase, that is the price times the marginal increase in production by adding a little more labor. Thus, the real marginal cost is the nominal cost relative to the nominal gain, i.e.. Linearizing gives. It follows from the average production function that marginal productivity is. Thus,. 20

24 (4.12) Similarly, a firm s real marginal cost in period is: (4.13) Now, the market clearing condition and the demand schedule (2.10) imply that firm output is ( ), which in linearized terms gives ( ). 13 Use this as well as (4.13) and (4.12) to get: [ ] [ ] ( ) [ ] (4.14) Notice that the last term in (4.14) disappears if there is constant returns to scale, i.e. if. Then, which implies that the marginal real cost is independent of the production level; it is common across all firms. We shall now derive an expression for inflation. The point of departure is (3.11), which we rewrite to: [ ] Insert (4.14): [ ] ( ) ( ) ( ) 13 Note that. 21

25 Define and subtract on both sides.: ( ) ( ) [ ] [ ] [ ] [ ] [ ] If we take out the terms of each summation operator, the equation can be written more compactly as a difference equation: [ ] Next we insert (3.3) and solve for inflation : 22

26 (4.15) Equation (4.15) expresses inflation as the sum of (discounted) expected inflation and real marginal costs, and we have defined to ease the notation. It is clear from (4.15) that inflation is strictly decreasing in price stickiness, in the measure of decreasing returns, and in the demand elasticity. An alternative presentation of inflation is found by solving (4.15) forward: { [ ( ) ] } Equivalently, and defining the average markup in the economy as, we see that inflation will be high when firms expect average markups to be below their steady state or desired level. In that case firms that have the opportunity to reset prices will choose a price above the economy s average price level in order to realign their markup closer to its desired level. Thus, in the present model, inflation results from the aggregate consequences of purposeful price-setting decisions by firms, which adjust their prices in light of current and anticipated cost conditions. Next, a relation is derived between the economy s real marginal cost and a measure of aggregate economic activity. We have derived earlier that. Insert this into, and use that : [ ] [ ] Insert for and get to: (4.16) In the case with flexible prices we know from before that. Define natural output level as the equilibrium level under full price flexibility. In this case (4.16) can be rewritten to: (4.17) Solve (4.17) for natural output: [ ] [ ] 23

27 If we subtract (4.17) from (4.16) we get a measure of the real marginal cost gap as a function of the output gap from natural output, denoted : (4.18) [ ] [ ] (4.19) Finally, the New Keynesian Phillips curve is established by inserting (4.19) into (4.15): (4.20) The New Keynesian Phillips curve (NKPC) is one of the key building blocks of the New Keynesian model, and the parameter is defined by. The second key equation is the dynamic IS equation. If we use the definition of the real interest rate,, equation (4.4) becomes. In a similar vein, the natural output is given as a function of the natural interest rate: (4.21) Subtracting (4.21) from (4.4) gives the output gap from the natural output, i.e. the dynamic IS equation (DIS): [ ] [ ] (4.22) Equations (4.20) and (4.22) together with an equilibrium process for the natural rate, which in general will depend on all exogenous forces in the model, constitute the non-policy block of the basic New Keynesian model. That block has a simple recursive structure: The NKPC determines inflation given a path for the output gap, whereas the DIS equation determines the output gap given a path for the exogenous natural rate and the actual real rate. To see the latter, assume the transversality condition. Then one can solve (4.22) forward to yield: (4.23) 24

28 The Basic New Keynesian Model Equation (4.23) emphasizes the fact that the output gap is proportional to the sum of current and anticipated deviations between the real interest rate and its natural counterpart. To gain further insight into the natural interest rate, note first that (4.4) implies. Second, note that the first difference of (4.18) gives. Now, solve (4.22) for and use these two observations to yield an expression for the natural real rate. From (4.22): ( ) [ ] [ ] [ ] (4.24) Thus, the natural real rate is a function of households discount rate and expected technological progress. In some cases it is convenient to work with deviations in the natural real rate from the discount rate, which we define as: (4.25) Note that if one turns off technology shocks, the real rate becomes the discount rate. Once a process for the technological progress is specified, one can identify the real interest rate path in (4.24). In order to close the model, we supplement (4.20) and (4.22) with one or more equations determining how the nominal interest rate evolves over time, i.e. with a description of how monetary policy is conducted. Observe from (4.23) that the equilibrium path of real variables cannot be determined independently of monetary policy when prices are sticky. The output gap is directly determined by the real interest rate gap, which is directly determined by the nominal interest rate set by central banks. This important feature of the New Keynesian Model is in contrast to classical models where monetary policy is neutral. 25

29 The Basic New Keynesian Model 5. Equilibrium determinacy 5.1 System representation Throughout this and the next sections I will look at a specific monetary policy rule, more specifically an interest rate rule. I assume the central bank follows a rule of the form: (5.1) Standard reasoning implies that and are non-negative, which we assume from now on. The first task when analyzing monetary rules is to check whether the specified policy yields a unique and stable equilibrium. While doing this, it is convenient to work with a reduced form representation of (4.20) and (4.22) who takes into account the policy rule under consideration. We first derive a forward looking version of the dynamic IS equation. Insert (5.1) into (4.22), and then (4.20) into (4.22). Solve the resulting equation for : ( ) [ ( ) ] [ ( )] (5.2) Equation (5.2) shows the current output gap as a function of expected output gap, expected inflation, and shocks. We next achieve a similar representation of current inflation. Insert (5.2) into (4.20) and get: { [ ( )]} ( ) ( ) { [ ] ( )} (5.3) 26

30 Finally, the two equations (5.2) and (5.3) can be written as a system of forward looking difference equations: [ ] [ ] [ ] [ ] ( ) [ [ ] [ ] ( ) ] [ ] ( ) where (5.4) [ ] [ ] The system given in (5.4) is a reduced form representation of the dynamic IS curve and the New Keynesian Phillips curve, which takes into account effects from the policy defined in equation (5.1). The coefficient matrix represents effects from expectations on current output gap and inflation while the coefficient vector represents effects from technology shocks in and monetary policy shocks in. We have defined to ease the notation. 5.2 Blanchard and Kahn conditions In the case we consider here we have two non-predetermined variables, and. Following Blanchard and Kahn (1980), the system (5.4) has a locally unique equilibrium if and only if both eigenvalues of the 2x2-matrix are inside the unit circle. 14 Let us characterize necessary and sufficient conditions for this property to hold. The two eigenvalues, denoted and, are generally solutions to the following system written in matrix form, where is an identity matrix: In our case the system becomes: [ When this is written out: ] [ ] 14 Consider a recursive system of the form { }, where is a vector of predetermined and nonpredetermined variables and is a vector of exogenous variables. Blanchard and Kahn (1980) proved that there exists a locally unique equilibrium if and only if the number of eigenvalues of inside the unit circle is equal to the number of non-predetermined variables. If the number of eigenvalues inside the unit circle is less than the number of non-predetermined variables, then an infinite number of equilibria exist. If the number of eigenvalues inside the unit circle exceeds the number of non-predetermined variables, then no equilibrium exists. 27

31 [ ] ( ) Following LaSalle (1986) 15 we know that the two eigenvalues of the circle if and only if the following two inequalities are met: matrix are inside the unit and Let us derive conditions the policy parameters and must meet for these two inequalities to hold. From the first inequality: (5.5) It is clear that condition (5.5), and consequently the first inequality, are fulfilled as long as, which we assume. Thus, the only relevant inequality is the second one, which we rewrite to: (5.6) We see from condition (5.6) that the equilibrium is unique as long as the policy parameters and have sufficiently high values, i.e. as long as monetary authorities respond to deviations of 15 LaSalle (1986) showed that both solutions to are smaller than one if and only if at the same time as. 28

32 inflation and output with adequate strength. Note also that our assumptions about the other parameters imply that is sufficient for (5.6) to hold. This is referred to as the Taylor principle. Let us give (5.6) some intuition. Suppose the economy is exposed to a permanent change in inflation;. From (4.20) we see that without any policy this leads to a permanent change in the output gap equal to. However, with the policy rule described here, we can find the nominal interest rate response by inserting into the differentiated version of (5.1): Furthermore, by rearranging (5.6) we get that. This implies that the change in inflation should be met by a larger change in the nominal interest rate. Eventually this will drive the real rate upwards and act as a stabilizing force. Thus, from (5.6) we see that when the central bank responds aggressively enough to changes in output gap and inflation, i.e. when and are large enough, output is forced back to natural output and inflation back to zero. 29

33 6. Shocks 6.1 Effects of a monetary policy shock Assume that the exogenous component of (5.1) follows an AR(1) process, where [ ): (6.1) Notice that a positive (negative) realization of is interpreted as a contractionary (expansionary) monetary policy shock, leading to a rise (decline) in the nominal interest rate for given levels of inflation and output gap. We want to find the contemporaneous effects of a monetary policy shock on the output gap and inflation. One way to identify these effects is by using the method of undetermined coefficients. Let us start by making the following guess: (6.2) (6.3) The coefficients and are yet to be determined. First, insert (6.1)-(6.3) into the New Keynesian Phillips curve given by (4.20). Find an expression for : (6.4) Then, insert the monetary policy rule (5.1) into the dynamic IS equation (4.22): 16 [( ) ] ( ) Insert (6.1)-(6.4) and solve for the coefficient : 16 To make the analysis as transparent as possible we set. That is, we turn off the technology shocks. 30

34 [ ] [ ] [ ] [ ] (6.5) Insert (6.5) into (6.4) to obtain: (6.6) Finally, this means that the solutions to (6.2) and (6.3) are: (6.7) (6.8) To ease the notation,. It is also straight forward to show that [ ] as long as (5.6) is satisfied. Note that if we insert (6.1) into (6.7) and (6.8), we get: Hence, an exogenous increase in the interest rate leads to a persistent decline in both output gap and inflation. Because the natural level of output is unaffected by the monetary policy shock, the response of output matches that of the output gap. Furthermore, (4.22) and (6.2) can be used to obtain an expression for the real interest rate deviation from its steady state counterpart, the natural real rate: ( ) (6.9) The response on nominal interest rate combines both the direct effect of and the indirect effect induced by reduced output gap and inflation. From (6.8) and (6.9): [ ] (6.10) Note that if the persistence of the monetary policy shock,, is sufficiently high, the nominal interest rate will decline in a response to a rise in. In that case, and despite the lower nominal rate, the policy shock still has a contractionary effect on output, because the latter is inversely related to the real rate, which goes up unambiguously. Finally, one can use (2.38) and (4.3) to determine the change in the money supply required to bring about the desired change in the interest rate. From, where we insert (6.8), (6.7) and (6.10): 31

35 [ ] [[ ] ] [[ ] ] (6.11) The sign of the change in money supply that supports the exogenous policy intervention is, in principle, ambiguous. Note however, that is a sufficient condition for a contraction in the money supply. Let us simulate the effects of a monetary policy shock. Parameters are calibrated as follows: Table / / / Setting = 0.99 implies a steady state real return on financial assets of about = 4%. Log utility is implied by = 1 and = 1. = 2/3 implies an average price duration of = 3 quarters. = 1.5 and = 0.5/4 is roughly consistent with observed variations in the federal funds rate over the Greenspan era. Finally, = corresponds to a monetary policy shock of 25 basis points. Simulated impulse responses are shown in the figure Consistent with the analytical results, it is seen that the policy shock leads to an increase in the real interest rate, and a decrease in inflation and output. The latter two effects correspond to that of the output gap because the natural level of output is not affected by the monetary policy shock. Under the calibration given in table 1 the nominal interest rate goes up, though by less than its exogenous component, as a result of the downward adjustment induced by the decline in inflation and output gap. In order to bring about the observed interest rate response, the central bank must engineer a reduction in the money supply. The calibrated model thus displays a liquidity effect. Note also that the response of the real rate is higher than that of the nominal rate as a result of the decrease in expected inflation. Overall, figure 1 shows dynamic responses which are qualitatively similar to those estimated using structural vector auto regressive methods. Figure 1: A monetary policy shock 17 Simulations are done with Dynare and Matlab. See Appendix A for the Dynare codes. 32

36 6.2 Effects of a technology shock Assume that the technology parameter follows an AR(1) process, where [ ]: (6.12) Here we want to find contemporaneous effects of the technology shock on the output gap and inflation. Given (4.25), the implied natural rate expressed in terms of deviations from steady state, is given by: [ ] (6.13) Again we use the method of undetermined coefficients. Guess the following: (6.14) (6.15) Insert (6.12) and (6.14)-(6.15) into the New Keynesian Phillips curve (4.20). Find an expression for : (6.16) 33

37 Then, insert the monetary policy rule (5.1) into the dynamic IS equation (4.22): 18 [( ) ] ( ) Insert (6.12)-(6.15) and solve for the coefficient : ( ) ( ) [ ] [ ] [ ] [ ] (6.17) Insert (6.17) into (6.16) to obtain: (6.18) Finally, this means that the solutions to (6.14) and (6.15) are: (6.19) To ease the notation, (6.19) and (6.20), we get: [ ] (6.20). Note that if we insert (6.12) into Hence, a positive technology shock leads to a persistent decline in both output gap and inflation. To find the implied equilibrium response of output, decompose output into. Then insert for from (4.18) and from (6.19): 18 To make the analysis as parsimonious as possible we now set. That is, we turn off the monetary policy shocks. 34

38 [ ] (6.21) To find the equilibrium response of employment, insert for (6.21) into (4.11): [ ] (6.22) Hence, the sign of the response of output and employment to a positive technology shock is in general ambiguous, depending on the configuration of parameter values, including the policy parameters and. Parameters are calibrated as follows in the simulation exercise: Table / / / We assume that = 0.9, i.e. that it takes some time for a technology shock to die out. Simulated impulse responses are shown in the figure Notice that the improvement in technology is partly accommodated by the central bank, which lowers nominal and real rates while increasing the money in circulation. That however is not enough to close the negative output gap, which is responsible for a decline in inflation. Output increases, but less than its natural counterpart. Figure 2: A technology shock 19 Simulations are done with Dynare and Matlab. See Appendix B for the Dynare codes. 35

39 7. Distortions to the efficient allocation 7.1 The efficient steady state I will now consider monetary policy design in the New Keynesian framework. This section investigates distortions to the efficient allocation and how monetary authorities can cope with these distortions. First we need to determine the efficient allocation. A natural benchmark is the problem faced by a benevolent social planner seeking to maximize the representative households social welfare, given preferences and technology. Using (2.3) and (4.5), this problem reads as: 20 { } { [( ) ]} s.t. (7.1) The constraint is the resource constraint coming from all the firms. Notice how all goods enter the utility function symmetrically, at the same time as utility is concave in each good. Also, all goods are produced with identical technology. Thus, by symmetry, can never be optimal. This gives the following efficiency conditions: (7.2) (7.3) The problem therefore simplifies to: { } (7.4) Let us solve the social planner s problem. FOC: : (7.5) From the firm s problem in free competition we also have the following: { } { } (7.6) FOC: : (7.7) From (7.5) and (7.7): (7.8) 20 Here we depart from the MIU-specification used previously in order to keep the analysis as simple as possible. 36

40 Thus, (7.8) is the relevant efficient benchmark monetary authorities should opt for. However, there are two sources of inefficiencies built into the New Keynesian model setup. The first one is firm s market power, which allows firms to set prices individually instead of being price takers. The second is staggered price setting, which prevent firms from adjusting optimally to shocks in the economy in the short run. I will now study these two inefficiencies in turn. 7.2 Distortions caused by market power Market power, which yields monopolistic competition, stems from the construction that each firm perceives an imperfectly elastic demand for its differentiated product. This gives firms the opportunity to set prices above marginal costs. Market power is unrelated to the presence of sticky prices. To illustrate this, suppose for the moment that prices are fully flexible so that. Firm s problem then becomes: { } { } [ ] { } { [ ] } (7.9) FOC: [ ] [ ] [ ] [ ] Because all firms behave in the same way, : 37

41 (7.10) As before, is the gross optimal markup chosen by firms and is the marginal cost. If we insert (7.10) into the efficient allocation (7.8), where the first equality follows from the optimality conditions of the household, we immediately see that: (7.11) Thus, the presence of market power not only leads to higher prices than optimal, but also an inefficiently low level of employment, and therefore also of output. This kind of distortion to the efficient equilibrium can be dealt with in a simple way by means of an employment subsidy. Let denote the rate at which the cost of employment is subsidized, and let outlays associated with the subsidy be financed by a lump-sum tax. If the subsidy is set to, then, by construction, the equilibrium under flexible prices yields efficiency. 21 Equation (7.10) becomes: (7.12) 7.3 Distortions caused by sticky prices The assumed constraints on the frequency of price adjustment constitute a source of inefficiency on two grounds. First, the fact that firms do not adjust their prices continuously implies that the economy s average markup will vary over time in response to shocks, and will generally differ from the constant frictionless markup. Denote the economy s average markup, i.e. the ratio of average price to average marginal cost, as. Then, from (7.12): (7.13) The last equality follows from the assumption that the subsidy in place exactly offsets the monopolistic competition distortion, which allows us to isolate the role of sticky prices. Insert (7.13) into the efficient benchmark allocation (7.8): (7.14) Thus, (7.8) is violated whenever. Efficiency can only be restored if policy manages to stabilize the economy s average markup to its frictionless level. In addition to the inefficiency described above, staggered price setting is a source of a second type of inefficiency. The latter has 21 In much of the analysis below it is assumed that such an optimal subsidy is in place. 38

42 to do with the fact that relative prices of different goods will vary in a way unwarranted by changes in preferences or technologies, as a result of the lack of synchronization in price adjustments. Whenever we also get, and consequently, which violates (7.2) and (7.3). To cope with distortions caused by staggered price setting, one should therefore opt for markups that are equal across all firms at all times. I will now analyze how this goal can be achieved by monetary authorities. 7.4 Monetary policy solutions to equilibrium distortions To keep the analysis simple, assume that in the last period,, implying that we had an efficient allocation at. Then this efficient allocation can be attained by a policy that stabilizes marginal costs at a level consistent with firms desired markup, i.e., given the prices in place. If that policy is expected to be in place indefinitely, no firm has an incentive to adjust its price because it is currently charging its optimal markup and expects to keep doing so in the future. As a result, and, hence,. In other words, the aggregate price level is fully stabilized and no relative price distortions emerge. In addition,, and output and employment matches their counterparts in the flexible price equilibrium allocation with a subsidy in place. From (4.19) and (4.20) we immediately see that implies the following, where inflation is given by (3.3): (7.15) (7.16) From the dynamic IS equation (4.22) we see that once (7.15) and (7.16) are expected to take place indefinitely, the nominal interest rate becomes the natural real rate: (7.17) Two features of the optimal policy are worth emphasizing. First, stabilizing output is not desirable in and of itself. Instead, output should vary one for one with the natural level of output. Whenever real shocks cause natural output to fluctuate a lot, so should also output. Second, price stability emerges as a feature of the optimal policy even though, a priori, the policy maker does not attach any weight on such objective. The next step is to analyze how to implement (7.15) and (7.16) in practice. Because (7.15) and (7.16) imply (7.17), one could think of (7.17) as a natural candidate for monetary policy. Although one obvious equilibrium is, we need to whether this equilibrium is unique. Treat (7.17) as an exogenous interest rate rule and insert it into (4.22). Combine with (4.20) to yield a system of difference equations: 39

43 ( ) [ ] [ ] [ ] [ ] where (7.18) [ ] Let us calculate the characteristic equation: [ ] With respect to the two conditions necessary for smaller-than-unity eigenvalues derived by LaSalle (1986), we get: and Clearly, the last condition does not hold, so both eigenvalues of cannot lie inside the unit circle. Thus, by the Blanchard and Kahn (1980) conditions, there exists a multiplicity of equilibria because the number of eigenvalues inside the unit circle is smaller than the number of nonpredetermined variables. The zero output gap and zero inflation target is only one of them, and there is nothing in the policy (7.17) that drives the economy back to the desired equilibrium given by (7.15) and (7.16). The second policy rule we consider is an interest rate rule with an endogenous component: (7.19) We first derive a forward looking version of the dynamic IS equation. Insert (7.19) into (4.22) and solve for : ( ) [ ( ) ] 40

44 [ ] (7.20) Next, insert (7.20) into (4.20) and solve for : { [ ]} { [ ] } (7.21) The two equations (7.20) and (7.21) can be written as a system of difference equations: [ ] [ ] [ ] [ ] [ ] [ ] where (7.22) [ ] Note that the transition matrix in (7.22) is identical to the one in (6.4). Thus, whenever condition (6.6) holds, which it does as long as one follows the Taylor principle, the policy rule given by (7.19) yields a unique and stable equilibrium with. A last monetary policy rule worth considering is a forward-looking interest rate rule: (7.23) Now, the monetary authorities adjust the nominal interest rate to variations in expected inflation and output gap, as opposed to their current values. Insert (7.23) into (4.22): ( ) [ ] (7.24) Insert (7.24) into (4.20): [ ] (7.25) 41

45 Write as a system of forward looking difference equations: [ ] [ ] [ ] [ ] where (7.26) [ ] The characteristic equation: In our case: [ ] [ ] Written out: [ ] [ ] [ ] The inequalities from LaSalle (1986) which should be met by the two eigenvalues of : and 42

46 From the first inequality: If : (7.27) Condition (7.27) cannot be binding when is non-negative because. If : (7.28) From the second equality: [ ] If [ ]: [ ] (7.29) If [ ]: [ ] (7.30) Condition (7.30) turns out to be identical to (6.6). However, in this case the two conditions (7.28) and (7.29) must hold in addition. We see from (7.28)-(7.30) that the policy responses should be neither too weak nor too strong. In particular, from (7.28) and (7.30) we see that a very high value on leads to indeterminacy, quite independently of. If the response to output is modest, then rules with can lead to determinate equilibria. However, from (7.29) we see that too large a value of again leads to indeterminacy. Finally, note that if, then could be set to zero. 43

47 8. The welfare loss function 8.1 Introduction I will now derive measures of the society s welfare losses caused by deviations in output and inflation from their steady state targets. The result will be a quadratic loss function that represents a quadratic second-order Taylor series approximation to the level of expected utility of the representative household in equilibrium with a given monetary policy. First I look at the simplest case where the only distortions in the economy are the presence of monopolistic competition and sticky prices. Then I look at an empirically more appealing case where what one refers to as cost push shocks exist. 8.2 The simplest case A welfare loss function when real rigidities are absent The first case we consider is the one analyzed previously, where the government implements an employment subsidy that removes the distortions caused by monopolistic competition. Thus, we assume that the subsidy given by (7.12) is in place. This case will also serve as a methodological framework for the welfare analysis conducted later. In order to lighten the notation, denote the period utility as and the steady state utility as. We will use the following second order approximation of relative deviation in consumption from its steady state counterpart, where logged consumption is approximated around logged steady state consumption: The same kind of second order approximation is performed on labor, so that: We need some more results as well. From (2.32) we have that and. From the market clearing condition we have that. Using all these results, a second-order Taylor approximation of around steady state leads us to the following criterion for welfare losses: From (19) we see that utility is separable in consumption and labor, i.e.. Also, we make the analysis simple and assume away money in the utility function. 44

48 The Basic New Keynesian Model ( ) ( ) ( ) ( ) ( ) ( ) (8.1) Our goal is to find a way to express (8.1) in terms of steady state deviations only, that is with the gap in output from natural output and the gap in inflation from zero inflation. The way to such a representation contains several steps. First, note from (4.7) that: { [ ]} [ ] ( ) (8.2) As before: [ ] (8.3) The next step is to get an alternative expression for. In the welfare analysis we do a secondorder approximation. Thus, while we earlier found that up to a first-order, this result can no longer be used. The following second-order approximation of will be useful, where is approximated around zero: From ( ), we have that ( ). Thus, when taking expectations on both sides of the above, where to good, we get: denotes the expectations operator with respect [ ] 45

49 (8.4) The price dispersion is denoted. Next, let us do a second-order approximation of in : (8.5) Finally, insert (8.4) and (8.5) into (8.3) to get the following second-order approximation of : [ ] { [ ] } { } [ ] [ ] { [ ] } [ [ [ ] ] ] [ ] [ ] [ ] (8.6) As before, we have defined. The next step is to insert (8.2) and (8.6) into (8.1), rearrange and also get rid of non-policy terms whenever possible: 46

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