Sticky Prices, Sticky Information or Rational Inattention? Evaluating Microfoundations.

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1 Sticky Prices, Sticky Information or Rational Inattention? Evaluating Microfoundations. Jorge Diego Solorzano University of Warwick. September 2017 Abstract This paper evaluates competing sets of microfoundations on firms price-setting behaviour widely used in monetary economics. The contribution of this project is twofold. The first contribution is to use as inflation driver a relevant measure of marginal cost from wage microdata. To that end, I calculate the User Cost of Labor as in Basu and House (2016) for the first time in a different dataset, and confirm their findings. The second contribution is the micro-evidence on the three leading price-setting theories used in macro models. I show compelling evidence that Calvo s (1983) sticky price model fits well for good prices; service prices favour Mankiw and Reis (2002) sticky information model; and that there is little empirical support for Mackowiak and Wiederholt s (2009) rational inattention model. These results suggest that models with a single price-setting decision process may be insufficient to adequately capture the monetary transmission mechanism. I would like to thank Huw Dixon for his useful comments. This paper has greatly benefited from comments received at the International-Macro Workshop at Warwick. Errors are all mine. Department of Economics, University of Warwick, Coventry CV4 7AL, UK. j.d.solorzano-rueda@warwick.ac.uk

2 1 Introduction Microfoundations are key elements in nowadays mainstream macroeconomic models. Micro-founded frictions are found crucial for the degree of monetary non-neutrality, inflation persistence and optimal monetary policy rule in DSGE modelling. Perhaps the most important difference between microfoundations is how firms set prices and process information. Three widely known models are: sticky prices, as proposed by Calvo (1983); sticky information, as developed by Mankiw and Reis (2002); and rational inattention, as introduced by Mackowiak and Wiederholt (2009). Yet, there is only a handful of papers on the empirical credibility of these three models. In this paper we use three large price and wage micro-datasets to shed light on the validity of these frameworks. The contribution of this paper is twofold. First, we calculate the User Cost of Labor (UCL) as in Basu and House (2016) and Kudlyak (2014) using a different dataset. The UCL provides the cost of expanding labor input with an unchanged composition of worker type, which is one of the main criticisms faced by other measures of labor cost over the business cycle. Our data confirms results from Basu and House (2016) and Kudlyak (2014) that UCL is cyclical. In addition, we show that the UCL does not vary substantially between formal and informal workers, though it does vary by industry. Second, we evaluate the three competing frameworks in price formation. Since marginal cost is potentially an endogenous variable, we use Generalised Methods of Moments (GMM) with the UCL, as well as lag values, instrumenting for marginal cots. Carlsson and Skans (2012) follow the same econometric strategy. It is worth noticing that our analysis relies on actual cost proxies driving price-setting, as instead of most aggregate data studies using output as an indirect marginal cost measure. Our results can be summarised as follow. Firstly, regarding the sticky price framework, we find that current and future marginal costs explain firms price-setting practices, as suggested in the Calvo model. However, the original model predicts that the role of future marginal cost monotonically decays depending how far ahead its forecasted. We do not find supporting evidence of this prediction for the full dataset. For prices in the service sector, current and future marginal cost have similar effects on pricing decisions. In contrast, in line with the Calvo framework, the role of future marginal costs diminish for goods prices. Secondly, with respect to the sticky information model, our estimates show that firms do 1

3 not fully react to changes in marginal cost that could have been forecasted by their information set. Recall that in this model information is sticky, not prices. Hence, this result is at odds with the benchmark model proposed by Mankiw and Reis (2002). Our calculations are obtained by using instruments lagged back sufficiently far to ensure that firms have updated their information set and in order to deal with expectations. However, if one allows for strategic complementarities in the model, firms would not fully pass-through their projected marginal cost. 1 Yet, our estimates with the complete dataset are significantly lower than what we would expect and therefore reject the sticky information model. Nevertheless, service prices display support to Mankiw and Reis (2002) model allowing for strategic complementarities passing nearly one-third of changes in forecasted marginal cost; whereas goods prices reject the model. Thirdly, we find little evidence in favour of the rational inattention model. Mackowiak and Wiederholt (2009) propose that firms react strongly to and immediately to idiosyncratic shocks. Instead, we find a price-cost elasticity of about one-fifth. This incomplete adjustment forcefully reject the rational inattention model, regardless of focusing on services or goods prices. Closest to this paper is research by Carlsson and Skans (2012). We deviate from this study in three main directions. First, Carlsson and Skans (2012) use annual data, while our microdata sets report monthly price and wage records. Annual data is not optimal for analysing intermittent price adjustments as these price-setting theories suggest. Second, their study centres its attention on goods prices, while our research provide evidence on goods and services. Third, we use a different instrument in our econometric strategy. We use the UCL, as first proposed by Kudlyak (2014), as the main price inflation driver over the business cycle. Carlsson and Skans (2012) find evidence supporting the Calvo (1983) framework and the Mankiw and Reis (2002) model. The latter only after allowing for strategic complementarities (real rigidities). Their findings speak against the Mackowiak and Wiederholt (2009) framework. Our analysis is based on three main sources reporting 5.5 million earnings and 150 million wage observations, as well as 4 million price quotas. These micro-datasets are merged by (4-digit) industry codes. Our first microdata comes from the largest labor survey in Mexico, Encuesta Nacional de Ocupacion y Empleo (ENOE). It contains self-reported earnings from 1 Carlsson and Skans (2012) uses simulations for Swedish data in order to determine what estimates are to be expected and finds a coefficient of

4 individual workers. All in all, the data consists of over 5.5 million earnings observations. Importantly for the calculation of UCL, we observe the starting date of the worker s current position, among other job and demographic characteristics. The second dataset comes from administrative wage records from the Mexican Institute of Social Security (IMSS). These administrative records constitute a census of all formal workers employed in the private sector. Our data is a panel of employee clusters reporting the wage bill and number of workers in the cluster on a monthly basis. 2 We combine our wage dataset with disaggregated measures of output at industry level (4-digit) in order to construct a measure of unit labor cost (consistent with the vast majority of DSGE models in the literature). Finally, the third microdata, reports monthly price dynamics of individual goods and services collected for CPI calculations. This paper is organised as follows. Section 2 outlines the three models we consider: sticky prices, sticky information and rational inattention. Section 3 describes our empirical strategy and presents preliminary results. Section 4 concludes. 2 Models If in sector k at time t firms have market power, the optimal frictionless price P k,t is set as a markup µ k,t over marginal cost MC k,t. That is, lnp k,t = µ k,t MC k,t From the cost minimisation problem, the cost associated with each possible margin of adjustment should be the same at the optimum. Hence, it is sufficient to look at one of them, in this case we focus on labour-input margin. MC k,t = Cost k,t L k,t If we have a production function of the form L k,t Y f,t Y k,t = (L k,t ) α g(others k,t ) The marginal cost is equal to MC k,t = 1 W agebill k,t α Y k,t 2 Clusters are defined by state, district, county, firm s size, age, gender, industry and income level. Given the granular characteristics defining each cluster, nearly 80% of clusters have only one or two workers. 3

5 Thus, the frictionless model implies that lnp k,t = β 0 + lnmc k,t Rational inattention Mackowiak and Wiederholt (2009) propose that prices adjust freely, however, firms face constraints on the amount of information to be processed each period. In their original calibration based on micro-evidence, firms allocate 96% of their attention to idiosyncratic conditions. Hence, firms react strongly and quickly to idiosyncratic conditions, whereas reaction is dampened and delayed to aggregate conditions. lnp k,t = β 0 + β MW lnmc k,t H 0 : β MW 1 (1) Sticky information Mankiw and Reis (2002) suggest firms update their information set with a fixed probability. After updating their information set, firms decide upon a price path that will remain in place until the firm is drawn to update the next time. The firm s optimal price is period t + s is given by lnp k,t+s = β 0 + β MR E t r lnmc k,t+s where t + s denotes the firm s price plan, t r is the time period when the information set was last updated. Thus, our hyphotesis is H 0 : β MR < 1 (2) Sticky prices Calvo (1983) propose that only a fraction of firms can reset their prices every period. Thus, the optimal price under Calvo-style nominal rigidities is characterised as the discounted value of current and expected future marginal cost. Therefore, we have lnp k,t = β 0 + (1 θβ) E t (θβ) s lnmc k,t+s lnp k,t = β 0 + s=0 β C,t+s lnmc k,t+s s=0 4

6 H 0 : 1 > β C,t > β C,t+1 >... > 0 (3) H 0 : s β C,t+s < (4) 3 Econometric framework We use Arellano-Bond estimator for a number of reasons. First, our panel is small-t with large-n. We observe around 100,000 individual price quotas on a quarterly basis for 10 years. Second, our independent variable (marginal cost) is not strictly exogenous, meaning it is correlated with past and possible current realisations of the error term. Third, individual fixed effects are likely to be present. Forth, hetoroskedasticity and autocorrelation within individual prices quotas but not across them. Our instruments must identify causal effects of marginal cost changes on price-setting behaviour. Similarly, in order to handle expectations when taking the Mankiw and Reis and the Calvo models to the data, we also rely on instruments. 3.1 Instrumentation Instruments must be correlated with the items marginal costs, but independent to the pricing decision. Beside lagged marginal cost, we construct an external instrument. We follow Basu and House (2016) to construct the User Cost of Labor (UCL) which is our external instrument. 3 The User Cost of Labor is defined as the (expected) difference between the present discounted value of wages paid to a worker hired in the current quarter and that paid to a worker hired in the next quarter. Kudlyak (2014) shows that the UCL is the relevant price for a firm considering to add a worker. Moreover, the microdata application of the marginal cost concept in macro theory is the cost of expanding labor input without changing the composition of worker types. Therefore, the UCL, as cost associated with expanding the number-of-employees margin while controlling for variations in the labor force composition, is an automatic guess as a relevant instrument. 3 See Kudlyak (2014) and Haefke et al. (2013) for more. 5

7 We start our calculation of the UCL by fitting one linear model ln w i,k τ,t = c + γt + ΨX i,k + T T do=1 d=do χ do,d D i,k do,d + εi,k t (5) Here w i,k τ,t is the real wage at time t of individual i employed in industry k hired at time-τ at her current position. Covariates in matrix X i are expedience (and experience squared), tenure (and tenure squared), schooling years, sex, industry (4-digit) and regional fixed effects. 4 The dummy variables D i,k do,d take the value 1 if do = τ and d = t and 0 otherwise. At time t, all workers who began their current job at date-τ get an additional adjustment to their predicted wage given by the coefficient χ τ,t. These adjustments imply that individuals who started working at date-τ experience an expected strip of (log) wage realisations given by {χ τ,τ, χ τ,τ+1, χ τ,τ+2...}. Notice that χ τ,τ indicates the wage of a newly hired worker, controlling for differences in human capital over the business cycle. We construct the projected wage payments as ln w k τ,t = ĉ + ˆγt + ˆΨX k + χ τ,t (6) The forecast of the present value of wage payments for workers hired at time-τ and still employed (with probability ψ) is calculated as P DV t=τ k = (βψ) j exp{ ln wτ,τ+j k } (7) j=0 Hence, UCL is uncovered by Kudlyak (2014) as UCL k t = P DV k t 4 Future analysis will include covariates of job position and type of job. βψp DVt+1 k (8) 6

8 Figure 1: User Cost of Labor 2005q1 2007q3 2010q1 2012q3 2015q1 date UCL New hires Unemployment 3.2 Taking models to the data Calvo (1983) The Calvo model only defines the optimal reset price. Hence, only prices which actually change can be used in our analysis. A further complication with this model is the infinite sum of current and future marginal cost. Using a quarterly estimation of θ = 38.59% and assuming β = 0.99, we expect the elasticity for E t ln MC t+4 is Since the terms in the sum fades fairly rapidly towards zero, we truncate the sum to 5 terms in our empirical application. In other to handle expectations we define Z Calvo,t n as a set of variables observed in time t-n. We need to use instruments lagged sufficiently far backwards. We rely on a GMM estimator with a dynamic instrument matrix, which allows us to use lags as they become available. Thus, our moment conditions are ) } 4 E t {(lnp i,t β 0 β k lnmc i,t+k Z Calvo,t 2 k= Mankiw and Reis (2002) Mankiw and Reis (2002) suggest that firms update their information set with a fixed probability. After updating, firms decide upon a price path that will remain in place until 5 Defined as (1 α(1 θ))(α(1 θ)) where 1 θ is the prob of not changing. 7

9 the firm is drawn to update the next time. The firm s optimal price is period t+k is given by lnp k,t+k = γ k + E t r lnmc f,t+k (9) where t + k denotes the firm s price plan, t r is the time period when the information set was last updated Mackowiak and Wiederholt (2009) Prices can be changed freely in any period, but the firm faces a constraint on the amount of information that can be processed at each period. They calibrate their model to match micro-evidence on price. Firms allocate 96% of their attention to idiosyncratic conditions. Hence, firms reacting as strongly and quickly to idiosyncratic conditions whereas reaction is dampened and delayed to aggregate conditions. 8

10 Table 1: Calvo Model GMM GMM GMM Sample All Goods Services ln MC t *** *** (0.0336) (0.0234) (0.216) E t ln MC t *** * 0.695*** (0.0406) (0.0445) (0.220) E t ln MC t *** * 0.533*** (0.0228) (0.0405) (0.156) E t ln MC t *** *** (0.0275) (0.0250) (0.138) E t ln MC t *** *** (0.0468) (0.0348) (0.184) E t ln MC t *** (0.0461) (0.0368) (0.147) Sum 0.733*** 0.207* 2.652*** (0.087) (0.125) (0.563) Observations 599, ,273 18,676 Number of id 118,886 28,808 7,610 IV MC Lags IV UCL Lags Hansen J Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 Table 2: Mankiw and Reis model GMM GMM GMM Sample All Goods Services ln MC t *** *** 0.280*** (0.016) (0.022) (0.092) Observations 697, ,069 21,535 Number of id 125,499 29,841 8,082 IV MC Lags IV UCL Lags Hansen J Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 9

11 Table 3: Mackowiak and Wiederholt model GMM GMM GMM Sample All Goods Services ln MC t 0.153*** *** ( ) (0.0154) (0.0307) Observations 585, ,142 19,252 R-squared Number of id 99,515 25,055 5,543 IV MC Lags IV UCL Lags Hansen J Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 4 Conclusions In this paper we use three large price and wage micro-datasets to shed light on the validity of three widely known price-setting models: sticky prices, sticky information, and rational inattention. We evaluate these three competing frameworks in price formation by using Generalised Methods of Moments (GMM) with the UCL, as well as lag values, instrumenting for marginal cots. It is worth noticing that our analysis relies on actual cost proxies driving price-setting, as instead of most aggregate data studies using output as an indirect marginal cost measure. Our results are summarised as follows. Firstly, regarding the sticky price framework, we find that current and future marginal costs explain firms price-setting practices, as suggested in the Calvo model. However, the original model predicts that the role of future marginal cost monotonically decays depending how far ahead its forecasted. We find that for service prices current and future marginal cost have similar effects on pricing decisions. In contrast, in line with the Calvo framework, the role of future marginal costs diminish for goods prices. Secondly, with respect to the sticky information model, our estimates show that firms do not fully react to changes in marginal cost that could have been forecasted by their information set. Hence, this result is at odds with the benchmark model proposed by Mankiw and Reis (2002). However, if one allows for strategic complementarities in the model, firms 10

12 would not fully pass-through their projected marginal cost. Yet, our estimates with the complete dataset are significantly lower than what we would expect and therefore reject the sticky information model. Nevertheless, if one allows for strategic complementarities, then service prices, by passing nearly one-third of changes in forecasted marginal cost, support the Mankiw and Reis (2002) model. Thirdly, we find little evidence in favour of the rational inattention model. Mackowiak and Wiederholt (2009) propose that firms react strongly to and immediately to idiosyncratic shocks. Instead, we find a price-cost elasticity of about one-fifth. This incomplete adjustment forcefully reject the rational inattention model for both service or good sectors. These results indicate that models with a single price-setting decision process may be insufficient to adequately capture the monetary transmission mechanism. 11

13 5 Appendix Table 4: Calvo model - All broad sectors GMM GMM GMM GMM GMM Sample All Fresh Food Goods Services Other Servs ln MC t *** *** *** (0.0336) (0.0387) (0.0234) (0.216) (0.293) E t ln MC t *** 0.305*** * 0.695*** (0.0406) (0.0413) (0.0445) (0.220) (0.251) E t ln MC t *** *** * 0.533*** (0.0228) (0.0256) (0.0405) (0.156) (0.463) E t ln MC t *** 0.324*** *** 0.678** (0.0275) (0.0440) (0.0250) (0.138) (0.330) E t ln MC t *** 0.490*** *** 1.109*** (0.0468) (0.0425) (0.0348) (0.184) (0.327) E t ln MC t *** *** (0.0461) (0.0485) (0.0368) (0.147) (0.239) Sum 0.733*** 0.46*** 0.207* 2.652*** 1.456*** (0.087) (0.078) (0.125) (0.563) (1.2401) Observations 599, , ,273 18,676 8,616 Number of id 118,886 75,298 28,808 7,610 4,620 IV MC Lags IV UCL Lags Hansen J Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 12

14 Table 5: Calvo model - IV composition GMM GMM GMM GMM GMM GMM GMM GMM GMM GMM GMM GMM GMM GMM GMM Sample All All All Fresh Food Fresh Food Fresh Food Fresh Food Fresh Food Fresh Food Services Services Services Other Servs Other Servs Other Servs 13 E t ln MC t *** *** *** 0.128** *** *** 0.200*** (0.0336) (0.131) (0.0656) (0.0387) (0.0556) (0.0731) (0.0234) (0.0458) (0.0514) (0.216) (1.532) (1.060) (0.293) (0.850) (0.791) E t ln MC t *** 1.961*** 0.413*** 0.305*** ** * 0.245** 0.189* 0.695*** 4.392* *** (0.0406) (0.433) (0.0526) (0.0413) (0.116) (0.0777) (0.0445) (0.0961) (0.103) (0.220) (2.357) (0.857) (0.251) (0.550) (0.414) E t ln MC t *** *** *** *** *** * ** 0.533*** (0.0228) (0.0854) (0.0284) (0.0256) (0.0583) (0.0387) (0.0405) (0.125) (0.110) (0.156) (1.539) (0.251) (0.463) (0.596) (1.589) E t ln MC t *** 1.388*** 0.273*** 0.324*** *** *** 3.507* ** 0.926** (0.0275) (0.300) (0.0365) (0.0440) (0.137) (0.0814) (0.0250) (0.0951) (0.0476) (0.138) (1.811) (0.243) (0.330) (0.369) (0.918) E t ln MC t *** 0.600** 0.617*** 0.490*** *** *** 3.419* *** (0.0468) (0.263) (0.0763) (0.0425) (0.0785) (0.0559) (0.0348) (0.0826) (0.141) (0.184) (1.881) (0.973) (0.327) (0.972) (1.501) E t ln MC t *** *** *** *** 0.194* *** * (0.0461) (0.526) (0.0772) (0.0485) (0.101) (0.0815) (0.0368) (0.0905) (0.156) (0.147) (1.060) (0.718) (0.239) (0.879) (0.859) Sum 0.733*** 1.271*** 0.509*** 0.46*** * ** 2.652*** * 2.046** (0.083) (0.326) (0.112) (0.078) (0.109) (0.138) (0.125) (0.448) (0.248) (0.563) (7.5) (0.897) (1.24) (1.582) (4.093) Observations 599, , , , , , , , ,273 18,676 18,676 18,676 8,616 8,616 8,616 Number of id 118, , ,886 75,298 75,298 75,298 28,808 28,808 28,808 7,610 7,610 7,610 4,620 4,620 4,620 IV MC Lags IV UCL Lags Hansen J Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1

15 Table 6: Calvo model - More lags GMM GMM GMM GMM GMM Sample All Fresh Food Goods Services Other Servs ln MC t 0.238*** 0.334*** *** *** (0.0191) (0.0266) (0.0175) (0.181) (0.301) E t ln MC t *** *** *** 0.587*** *** (0.0188) (0.0231) (0.0239) (0.195) (0.230) E t ln MC t *** *** *** 1.631*** (0.0187) (0.0196) (0.0348) (0.156) (0.601) E t ln MC t *** *** 1.506*** (0.0175) (0.0245) (0.0232) (0.128) (0.393) E t ln MC t *** *** *** 0.615*** 2.250*** (0.0229) (0.0304) (0.0228) (0.175) (0.513) E t ln MC t *** *** *** (0.0303) (0.0364) (0.0230) (0.133) (0.308) Sum *** *** *** 4.469*** (0.062) (0.059) (0.101) (0.542) (1.610) Observations 599, , ,273 18,676 8,616 Number of id 118,886 75,298 28,808 7,610 4,620 IV MC Lags IV UCL Lags Hansen J Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 14

16 Table 7: Mankiw and Reis model - Broad sectors GMM GMM GMM GMM GMM Sample All Fresh Food Goods Services Other Servs ln mc *** *** 0.280*** (0.016) (0.020) (0.022) (0.092) (0.131) Observations 697, , ,069 21,535 9,948 Number of id 125,499 79,715 29,841 8,082 4,932 IV MC Lags IV UCL Lags Hansen J Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 15

17 Table 8: Mankiw and Reis model - IV composition GMM GMM GMM GMM GMM GMM GMM GMM GMM GMM Sample All All Fresh Food Fresh Food Goods Goods Services Services Soc Serv Other Services ln mc *** *** *** *** *** 0.458*** (0.026) (0.016) (0.031) (0.012) (0.036) (0.022) (0.115) (0.092) (0.132) (0.131) Observations 697, , , , , ,069 21,535 21,535 9,948 9,948 Number of id 125, ,499 79,715 79,715 29,841 29,841 8,082 8,082 4,932 4,932 IV MC Lags IV UCL Lags Hansen J Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 16

18 Table 9: Mackowiak and Wiederholt - Broad sectors GMM GMM GMM GMM GMM Sample All Fresh Food Goods Services Other Services ln MC t 0.153*** 0.201*** *** *** (0.007) (0.006) (0.015) (0.031) (0.096) Observations 585, , ,142 19,252 8,898 R-squared Number of id 99,515 63,485 25,055 5,543 3,109 IV MC Lags IV UCL Lags Hansen J Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 17

19 Table 10: Mackowiak and Wiederholt model - IV composition GMM GMM GMM GMM GMM GMM GMM GMM GMM GMM GMM GMM GMM GMM GMM Sample All All All Fresh Food Fresh Food Fresh Food Goods Goods Goods Services Services Services Soc Serv Soc Serv Soc Serv ln MC t 0.153*** 0.132*** *** 0.201*** 0.186*** *** *** * *** *** *** 0.987** (0.007) (0.005) (0.030) (0.007) (0.005) (0.028) (0.015) (0.010) (0.074) (0.031) (0.024) (0.182) (0.096) (0.072) (0.449) Observations 585, , , , , , , , ,142 19,252 22,962 19,252 8,898 10,828 8,898 R-squared Number of id 99, ,043 99,515 63,485 66,899 63,485 25,055 26,681 25,055 5,543 6,266 5,543 3,109 3,679 3,109 IV MC Lags IV UCL Lags Hansen J Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 18

20 References Basu, S. and C. L. House (2016): Allocative and Remitted Wages: New Facts and Challenges for Keynesian Models, Tech. rep., National Bureau of Economic Research. Calvo, G. A. (1983): Staggered prices in a utility-maximizing framework, Journal of monetary Economics, 12, Carlsson, M. and O. N. Skans (2012): Evaluating microfoundations for aggregate price rigidities: evidence from matched firm-level data on product prices and unit labor cost, The American Economic Review, 102, Haefke, C., M. Sonntag, and T. Van Rens (2013): Wage rigidity and job creation, Journal of monetary economics, 60, Kudlyak, M. (2014): The cyclicality of the user cost of labor, Journal of Monetary Economics, 68, Mackowiak, B. and M. Wiederholt (2009): Optimal Sticky Prices under Rational Inattention, American Economic Review, 99, Mankiw, N. G. and R. Reis (2002): Sticky Information versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve, Quarterly Journal of Economics,

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