A Model of the Twin Ds: Optimal Default and Devaluation

Transcription

1 A Model of the Twin Ds: Optimal Default and Devaluation by Na, Schmitt-Grohé, Uribe, and Yue June 20,

2 Motivation (I) There is a strong empirical link between sovereign default and large devaluations. Reinhart (2002) examines data for 58 countries over the period 1970 to 1999 and finds that: The unconditional probability of a large devaluation in any 24- month period is 17%. The probability of a large devaluation conditional on the 24- month period containing a default is 84%. Reinhart refers to this phenomenon as the Twin Ds. 2

3 Motivation (II) We extend the Twin Ds fact dynamically and find that devaluations around default resemble a step function: They are more akin to a change in the level of the nominal exchange rate than to a switch to a higher rate of depreciation. This fact suggests that typically devaluations around default are a one-time event and do not mark a switch to a regime of more future devaluations as in first-generation models of BOP crises. The fact that virtually of the public debt in the countries included in the panel is either dollarized or indexed (original sin), suggests that the observed one-time devaluations around default do not aim at inflating away the debt burden. 3

4 The Twin Ds: Six Recent Examples Argentina Uruguay Ukraine Exchange Rate Exchange Rate Exchange Rate Year Year Year Russia 2.2 Paraguay 10 Ecuador Exchange Rate Year Exchange Rate Year Nominal Exchange Rate Exchange Rate Year Default Date Note: Exchange rates are nominal dollar exchange rates, annual average, first observation normalized to unity. Data sources: Default dates, Uribe and Schmitt-Grohé (2015). Exchange rates, WDI. 4

5 Excess Devaluation, Evidence from 116 Default Episodes Excess devaluations Default date percent Years after default Note. Median of cumulative devaluations conditional on default in year 0 minus unconditional median. Sample contains 116 default episodes between 1975 and 2013 in 70 countries. Data sources: Default dates, Uribe and Schmitt-Grohé (2015). Exchange rates, WDI. 5

6 This Paper Develops a model that explains the Twin Ds phenomenon as an optimal policy outcome aimed at correcting relative-price misalignments. Main Elements of the Model Imperfect enforcement of debt contracts. (Eaton Gersovitz, RES 1981) Downward nominal wage rigidity. (Schmitt-Grohé and Uribe, JPE 2016) Intuition Optimal default occurs during large recessions. A contracting demand for labor puts downward pressure on real wages. A large devaluation reduces the real value of wages, thereby preventing unemployment. 6

7 Illustration of Narrative: Argentina Nominal Exchange Rate (E t ) Unemployment Rate + Underemployment Rate 40 Pesos per U.S. Dollar Percent Year Year Nominal Wage (W t ) 1.4 Real Wage (W t /E t ) 1.2 Pesos per Hour 12 Index 1996= Year Year Vertical Line 1998, beginning of recession. Vertical Line 2002, default and devaluation. 7

8 Related Literature Fiscal consequence of devaluations, flow effects: Balance of payment crisis literature. Devaluation rate as a way to create seignorage to finance fiscal deficits (Krugman, 1979). Fiscal consequence of devaluations, stock effects: Devaluation as indirect default on stock of local currency denominated debt (Calvo, 1988; Corsetti and Dedola, 2014; Aguiar et al., 2013; Da Rocha et al., 2013; Kriwoluzky et al., 2014; Moussa, 2013; Du and Schreger, 2015). Real models of default à la Eaton and Gersovitz. (Arellano, 2008; Aguiar and Gopinath, 2006; Hatchondo et al. 2010; Martinez and Sapriza, 2010; Chatterjee and Eyigungor, 2012; Mendoza and Yue, 2012.) 8

9 The Model 9

10 Households choose c T t, cn t, d t+1 to maximize subject to E 0 t=0 β t U(c t ) c t = A(c T t, cn t ), P T t ct t + P N t cn t + E t d t = P T t ỹt t + W th t + E t (1 τ d t )q td t+1 + E t φ t + E t f t, h t h. Households take h t as given. Debt is denominated in foreign currency (original sin). The Demand for Nontradables p t = A 2(c T t, cn t ) A 1 (c T t, cn t ) where p t P N t /E t is the relative price of nontradables in terms of tradables (the foreign price of tradables is normalized to 1). 10

11 The Demand For Nontradables price, p A 2(c T 0,cN ) A 1(c T 0,cN ) A 2(c T 1,cN ) A 1(c T 1,cN ) quantity, c N c T 1 < ct 0 A Contraction in Traded Absorption, c T t, Shifts the Demand for Nontradables Down and to the Left 11

12 Firms Choose h t to max { P N t F(h t) W t h t } The Supply of Nontradables p t = W t/e t F (ht) 12

13 The Supply of Nontradabels p W 0 /E 0 F (h) W 0 /E 1 F (h) (E 1 > E 0 ) h A Devaluation (E t ) Shifts The Supply Schedule Down 13

14 Nominal Wages are Downwardly Rigid W t γw t 1 W t = nominal wage in period t. γ = degree of downward wage rigidity. γ = 0 fully flexible wages. Think of γ as being around 1. Schmitt-Grohé and Uribe (2013) estimate γ = 0.99 at quarterly frequency. 14

15 Closing of the Labor Market The following slackness condition is assumed to hold at all times: ( h h t ) ( W t γw t 1 ) = 0. Express in real terms where ( h h t ) ( w t γ w t 1 ɛ t ) = 0, w t W t E t = real wage in units of tradables. ɛ t E t E t 1 =the gross devaluation rate in period t 15

16 Optimal Exchange-Rate Policy p A 2(c T 0,F(h) ) A 1(c T 0,F(h) ) p 0 p PEG A 2(c T 1,F(h) ) A 1(c T 1,F(h) ) B A W 0 /E 0 F (h) p OPT C W 0 /E 1 F (h) h PEG h = h OPT h c T 1 < ct 0 (negative endowment shock, yt t ) E 1 > E 0 (optimal devaluation) 16

17 The Government Each period t, the government can be either in good financial standing or in bad financial standing. If the government is in good financial standing, it can choose to either honor its debt (indicated by I t = 1) or default. If it defaults, it immediately acquires bad financial standing (indicated by I t = 0). If the government is in bad financial standing in t, then it regains good financial standing in t + 1 with exogenous probability θ, and maintains bad standing with probability 1 θ. 17

18 Two Exogenous Costs of Default (1) Financial Exclusion: While the country is in bad financial standing (I t = 0), it cannot participate in international credit markets, (1 I t )d t+1 = 0. (2) Output Loss: The endowment received by households is given by ỹ T t = { y T t if I t = 1 (good standing) y T t L(yT t ) if I t = 0 (bad standing) where y T t is an exogenous stochastic process, and L( ) is an increasing function. 18

19 Risk-Neutral Foreign Lenders The price of debt, q t, is given by q t = Prob{I t+1 = 1 I t = 1} 1 + r where r is the risk-free interest rate. 19

20 Competitive Equilibrium c T t = yt t (1 I t)l(y T t ) + I t[q t d t+1 d t ] (1 I t )d t+1 = 0 λ t = U (A(c T t, F(h t)))a 1 (c T t, F(h t)) I t [ (1 τ d t )q t λ t βe t λ t+1 ] = 0 w t γ w t 1 ɛ t given policies {I t, ɛ t, τ d t }. A 2 (c T t, F(h t)) A 1 (c T t, F(h t)) F (h t ) = w t ; (h t h) ( w t γ w ) t 1 ɛ = 0; and ht h t I t [q t E ti t+1 1+r ] = 0 20

21 Competitive equilibrium with unrestricted τ d t & ɛ t c T t = yt t (1 I t)l(y T t ) + I t[q t d t+1 d t ] (1 I t )d t+1 = 0 λ t = U (A(c T t, F(h t)))a 1 (c T t, F(h t)) I t [ (1 τ d t )q t λ t βe t λ t+1 ] = 0 A 2 (c T t, F(h t)) A 1 (c T t, F(h t)) F (h t ) = w t w t γ w ( t 1 ; (h t h) w t γ w ) t 1 ɛ ɛ = 0; and h t t t h I t [q t E ti t+1 1+r ] = 0 given I t. Policies {ɛ t, τ d t } picked to satisfy nonboxed conditions 21

22 Optimal Policy Problem max E 0 β t U(A(c T {c T t,h t, F(h t ))) s.t. t,d t+1,i t,q t } t=0 c T t = yt t (1 I t)l(y T t ) + I t[q t d t+1 d t ] (1 I t )d t+1 = 0 I t [q t E ti t+1 1+r ] = 0 h t h States are: y T t, d t (note w t 1 is not relevant under optimal exchangerate policy). Full employment is optimal (h t = h for all t). After setting h t = h, the above problem is exactly Arellano (2008). 22

23 Summary of Analytical Results Decentralization of the EG Model: Real models of sovereign default in the tradition of Eaton and Gersovitz (1981) can be interpreted as the centralized version of economies with default risk and downward nominal wage rigidity. Implied Use of Policy Instruments: Default and devaluation policies are determined jointly, with the latter serving as a means to bring about the full-employment real wage. 23

24 Quantitative Analysis 24

25 Functional Forms and Calibration U(c) = c1 σ 1 1 σ ; σ = 2 A(c T, c N ) = [ ] 1 a(c T ) 1 1 ξ + (1 a)(c N ) ξ 1 ξ ; ξ = 1/2,a = 0.26 y N t = h α t ; α = 0.75 L(y T t ) = max {0, δ 1y T t + δ 2(y T t )2 } Set β = 0.85, δ 1 = 0.35, and δ 2 = 0.44 to ensure: (a) E(d t /y T t ) = 60%, (b) Prob of default equal to 2.6 per century, and (c) Average output loss in autarky of 7%. Set γ = 0.99 wages can fall by up to 4% per year. y T t = 0.93y T t µ t, µ N(0,1) (Argentina, ) 25

26 Asymmetric Output Cost of Default y t y t L(y t ) (flat) y t L(y t ) (quadratic) y t 26

27 A Typical Default Episode Under Optimal Policy Tradable Output 0.85 yt T L(yt T ) y T t Real Wage, w t Consumption of Tradables, c T t Relative Price of Nontradables, p t Nominal Exchange Rate, E t Country Interest-Rate Premium %/yr

28 Observations On The Dynamics Around A Typical Default Episode Under Optimal Policy Default takes place after a short but sharp contraction in tradable output. Default coincides with end of contraction and beginning of recovery. Contrary to what the intertemporal approach to the BOP would suggest (but consistent with empirical evidence), the contraction of consumption of tradables is more severe than that of output, TB surplus. Reason: the interest rate premium doubles. In spite of the severe external crisis, the economy displays full employment at all times. Full employment is achieved through large devaluations, > 35 percent, ( Twin Ds) that cause a large reduction in the real wage and a significant real depreciation. Real depreciation mainly due to increase in nominal price of tradables, as in actual data. 28

29 Empirical Evidence on the Behavior of Capital Controls and Reserve Requirements Around Default The model predicts that debt taxes increase as the economy approaches default. Is this prediction of the model supported in the data? Take a look at the next two plots. The first shows that capital control measures increase and the second shows that reserve requirements increase Capital Controls 4 Reserve Requirements Index % Years after default Years after default Source. Own calculations based on data on capital controls from Fernández et al. (2015) and on reserve requirements from Federico et al. (2014). 29

30 Optimal Default Under Currency Pegs Suppose the central bank picks ɛ t = 1 t 30

31 In the data, currency pegs rarely survive default. Thus there are no stylized facts that we aim to explain here. Instead our motivation is to ask what would happen if, contrary to what is typically observed, a currency peg survives a default. The case is potentially of interest because of the recent experience in the periphery of Europe where countries chose to stick to currency pegs through severe external crises involving large increases in country spreads and even sovereign defaults (Greece and Cyprus). However, those defaults came with bailouts and thus do not conform directly to the type of defaults analyzed here. 31

32 The Peg-Constrained Optimal Policy Problem: Pick polices {I t, τt d } and all endogenous variables of the private-sector equilibrium to maximize subject to E 0 t=0 β t U(A(c T t, F(h t ))) c T t = yt t (1 I t)l(y T t ) + I t[q t d t+1 d t ] (1 I t )d t+1 = 0 A 2 (c T t, F(h t)) A 1 (c T t, F(h t)) F (h t ) = w t ; w t γw t 1 ; h t h I t [q t E ti t+1 1+r ] = 0 Note: (1) Impossible to get rid of wage-rigidity constraints (except for slackness); (2) States are now: y T t, d t, w t 1 32

33 Typical Default Episodes With Fixed Exchange Rates Tradable Output, y T t Real Wage, w t Consumption of Tradables, c T t Relative Price of Nontradables, p t 3 % Unemployment Rate Country Interest-Rate Premium %/yr peg optimal devaluation 33

34 Observations On Typical Defaults With Fixed Exchange Rates Default occurs after protracted contractions in tradable output. Defaults occur in the context of massive involuntary unemployment (over 20%) and highly depressed levels of consumption. The lack of nominal depreciation causes the real wage to stay above the full employment real wage before, during, and after default. Firms do not lower prices because costs (wages) remain high, leading to much less real depreciation relative to optimal exchange rate policy. 34

35 Exchange Rate, Pesos per U.S. Dollar Default, Devaluation, and Unemployment: Argentina Greece Unemployment Rate, % Exchange Rate, Index Unemployment Rate, % Exchange Rate, Krona per Euro Iceland Unemployment Rate Unemployment Rate, % Exchange Rate, Index Cyprus Unemployment Rate, % Nominal Exchange Rate Note. Vertical line indicates the year of default. Own calculations based on data from INDEC (Argentina), EuroStat, and the Central Bank of Iceland. 35

36 Conclusions Paper shows that real economies with default risk à la Eaton- Gersovitz can be interpreted as the centralized version of economies with default risk, downward nominal wage rigidity, optimal debt taxes, and optimal devaluation policy. Paper presents a model that predicts that under the optimal policy defaults are accompanied by large devaluations, that is, the model can explain the Twin Ds phenomenon. Under optimal policy, the central role of devaluations around default episodes is to fend off unemployment. In fixed-exchange rate economies defaults are predicted to be accompanied by massive unemployment. 36

37 Extras 37

38 Sensitivity Analysis 1. Long-Maturity Debt 2. Incomplete Exchange Rate Pass Through 3. Patience and the Twin Ds (higher β) 38

39 Optimal Devaluation Around the Typical Default Episode Under Long-Maturity Debt 1.25 Nominal Exchange Rate, E t

40 Optimal Devaluation Around the Typical Default Episode Under Imperfect Pass-Through 1.6 Nominal Exchange Rate, E t Perfect Pass Through (η=1) Imperfect Pass Through (η=0.5)

41 3.) Predicted Twin Ds And Patience β=0.85 β=0.95 β=0.98 Nominal Exchange Rate, E t

42 Calibration Parameter Value Description γ 0.99 Degree of downward nominal wage rigidity σ 2 Inverse of intertemporal elasticity of consumption y T 1 Steady-state tradable output h 1 Labor endowment a 0.26 Share of tradables ξ 0.5 Elasticity of substitution between tradables and nontradables α 0.75 Labor share in nontraded sector β 0.85 Quarterly subjective discount factor r 0.01 World interest rate (quarterly) θ Probability of reentry δ parameter of output loss function δ parameter of output loss function ρ serial correlation of ln yt T σ µ std. dev. of innovation µ t Discretization of State Space n y 200 Number of output grid points (equally spaced in logs) n d 200 Number of debt grid points (equally spaced) n w 125 Number of wage grid points (equally spaced in logs) [y T, y T ] [0.6523,1.5330] traded output range [d, d] float [0,1.5] debt range under optimal float [d, d] peg [-1,1.25] debt range under peg [w, w] peg [1.25,4.25] wage range under peg 42

43 Do Countries Default in Bad Times? Output Around Default Episodes Percent deviation from trend Years after default Source: Uribe and Schmitt-Grohé, Output is measured as real per capita GDP, log quadratically detrended at annual frequency. Median across 105 default episodes over the period

44 Consumption, Investment, The Trade Balance, and The Real Exchange Rate Around Default Episodes 6 Consumption 20 Investment Percent deviation from trend Percent deviation from trend Years after default Years after default 1.5 Trade Balance To Output Ratio 4 Real Exchange Rate Percent deviation from trend Percent deviation from trend Years after default Years after default Source: Uribe and Schmitt-Grohé, An increase in the real exchange rate indicates a real appreciation of the domestic currency. 44

45 Long-Run Debt Sustainability under a Currency Peg 45

46 Density Function of External Debt 2.5 Currency Peg Optimal Devaluation Policy density external debt (d t ) Note. Debt distributions are conditional on being in good financial standing. 46

47 Observations on the figure In the long run, economies undergoing currency pegs can sustain less debt than economies with optimal floats (20 vs. 60 percent of tradable output, respectively). Reason: Ex-ante, stronger incentive to default under a peg, since default, by accelerating the recovery of consumption, helps reduce unemployment. Under optimal devaluation, this channel is not there, because the central bank guarantees full employment at all times, through appropriate movements in the nominal exchange rate. However, ex-post, the peg economy does not predict a higher default rate than the optimal float economy. The reason, is that the lower level of debt reduces incentives to default. 47

48 External Debt to GDP Ratios: GIPS versus Ecuador, 1990 to Ecuador GIPS % of GDP year Source. Own calculations based on the updated and extended version of the dataset constructed by Philip R. Lane and Gian Maria Milesi-Ferretti, The external wealth of nations mark II: Revised and extended estimates of foreign assets and liabilities, , Journal of International Economics 73, November 2007,

49 Average Debt And Default Probability Across Devaluation Policies Optimal Devaluation Currency Policy Peg Debt-to-traded-output ratio (qtr) 60% 20% Number of Defaults per century Country Premium 3.5% 2.5% 49

50 Data and Model Predictions: Optimal Devaluation Policy E(r r ) σ(r r ) corr(r r, y) corr(r r, tb/y) Data Model Note. Data moments are from Argentina over the inter-default period 1994:1 to 2001:3, except for the default frequency, which is calculated over the period 1824 to In the theoretical model, all moments are conditional on the country being in good financial standing. 50

51 Business-Cycle Statistics: Data and Model Predictions Under Optimal Exchange Rate Policy σ(c)/σ(y) σ(tb/y)/σ(y) corr(c, y) corr(tb/y, y) Data Emerging Countries Argentina Model

52 Is there wage restraint during booms? Example: Periphery of Europe during the Boom 52

53 Boom-Bust Cycle in Peripherical Europe: Current Account / GDP 110 Labor Cost Index, Nominal 14 Unemployment Rate Percent Index, 2008 = Percent Date Date Date Data Source: Eurostat. Data represents arithmetic mean of Bulgaria, Cyprus, Estonia, Greece, Ireland, Lithuania, Latvia, Portugal, Spain, Slovenia, and Slovakia Wages grew by 70 percent between 2000 and 2008! 53

54 Let s take a closer look at Spain and Ireland... 54

55 Nominal hourly wages in Spain increase by 44 percent during the boom 110 Nominal Hourly Wages: Spain Index, 2008 =

56 Nominal hourly wages in Ireland increase by 57 percent during the boom 110 Nominal Hourly Wages: Ireland Index, 2008 =

57 ... Despite No Growth in Total Factor Productivity 115 Spain Ireland 110 TFP, Index 1995=

58 Total Factor Productivity: (value added based), Index (1995=100) Spain Ireland Source: EU KLEMS Growth and Productivity Accounts. This database includes measures of output and input growth, and derived variables such as multifactor productivity at the industry level. The input measures include various categories of capital (K), labour (L), energy (E), material (M) and service inputs (S). The measures are developed for 25 individual EU member states, the US and Japan and cover the period from 1970 to The variables are organized around the growth accounting methodology, a major advantage of which is that it is rooted in neo-classical production theory. It provides a clear conceptual framework within which the interaction between variables can be analyzed in an internally consistent way. The data series are publicly available on November 2009 release. 58

59 Evidence On Downward Nominal Wage Rigidity Downward nominal wage rigidity is the central friction in the present model natural to ask if it is empirically relevant. Downward wage rigidity is a widespread phenomenon: Evident in micro and macro data. Rich, emerging, and poor countries. Developed and underdeveloped regions of the world. Byproduct: Will obtain an estimate of the parameter γ governing wage stickiness in the model (useful for quantitative analysis). 59

60 Downward Nominal Wage Rigidity: Evidence From Micro Data 60

61 Probability of Decline, Increase, or No Change in Wages U.S. data, SIPP panel , between interviews one year apart. Interviews One Year apart Males Females Decline 5.1% 4.3% Constant 53.7% 49.2% Increase 41.2% 46.5% Source: Gottschalk (2005) Large mass at Constant suggests nominal wage rigidity. Small mass at Decline suggests downward nominal wage rigidity. 61

62 Distribution of Non-Zero Nominal Wage Changes United States Source: Barattieri, Basu, and Gottschalk (2012) 62

63 Evidence From The Great Contraction Of 2007 Distribution of Nominal Wage Changes, U.S Source: Daly, Hobijn, and Lucking (2012). 63

64 Micro Evidence On Downward Nominal Wage Rigidity From Other Developed Countries Canada: Fortin (1996). Japan: Kuroda and Yamamoto (2003). Switzerland: Fehr and Goette (2005). Industry-Level Data: Holden and Wulfsberg (2008), 19 OECD countries from 1973 to

65 Evidence From Informal Labor Markets Kaur (2012) examines the behavior of nominal wages, employment, and rainfall in casual daily agricultural labor markets in rural India (500 districts from 1956 to 2008). Finds asymmetric nominal wage adjustment: W t increases in response to positive rainfall shocks W t fails to fall, labor rationing, and unemployment are observed in response to negative rain shocks. Inflation (uncorrelated with local rain shocks) tends to moderate rationing and unemployment during negative rain shocks, suggesting downward rigidity in nominal rather than real wages. 65

66 Evidence From the Great Depression, Enormous contraction in employment: 31% between 1929 and Nonetheless, during this period nominal wages fell by 0.6% per year, while consumer prices fell by 6.6% per year. See the figure on the next slide. A similar pattern is observed during the second half of the Depression. By 1933, real wages were 26% higher than in 1929, in spite of a highly distressed labor market. 66

67 Nominal Wage Rate and Consumer Prices, United States 1923:1-1935: :8= log(nominal Wage Index) log(cpi Index) Year Solid line: natural logarithm of an index of manufacturing money wage rates. Broken line: logarithm of the consumer price index. 67

68 Evidence From the Great Depression In Europe Countries that left the gold standard earlier recovered faster than countries that remained on gold. Left Gold Early (sterlingbloc): United Kingdom, Sweden, Finland, Norway, and Denmark. Countries That Stuck To Gold (gold bloc): France, Belgium, the Netherlands, and Italy. Think of the gold standard as a currency peg (a peg not to a currency, but to gold). When sterling-bloc left gold, they effectively devalued, as their currencies lost value against gold. Look at the figure on the next slide. Between 1929 and 1935, sterling-bloc countries experienced less real wage growth and larger increases in industrial production than gold-bloc countries. 68

69 Changes In Real Wages and Industrial Production, Source. Eichengreen and Sachs (1985). 69

70 Evidence From Emerging Countries Argentina: pegged the peso at a 1-to-1 rate to the dollar between 1991 and Starting in 1998, the economy was buffeted by a number of large negative shocks (weak commodity prices, large devaluation in Brazil, large increase in country premium, etc.). Not surprisingly, between 1998 and 2001, unemployment rose sharply. Nonetheless, nominal wages remained remarkably flat. This evidence suggests that nominal wages are downwardly rigid, and that γ is about 1. Why γ 1? The slackness condition ( h h t )(W t γw t 1 ) (recall ɛ t = 1 during this period), implies that if unemployment is growing, wages must grow at the gross rate γ. 70

71 4 Argentina Nominal Exchange Rate (E t ) Unemployment Rate + Underemployment Rate 40 Pesos per U.S. Dollar Percent Year Year Nominal Wage (W t ) 1.4 Real Wage (W t /E t ) 1.2 Pesos per Hour 12 Index 1996= Year Year Implied Value of γ: Around unity. 71

72 Evidence From Peripheral Europe ( ) Look at the table on the next slide. Between 2008 and 2011, all countries in the periphery of Europe experienced increases in unemployment. Some very large increases. In spite of this context of extreme duress, nominal hourly wages experienced significant increases in most countries and modest declines in very few. The slide following the table explains how to use the information in the table to infer a range for γ. 72

73 Source: EuroStat. Unemployment, Nominal Wages, and γ Evidence from the Eurozone Unemployment Rate Wage Growth Implied W 2008Q1 2011Q2 2011Q2 W 2008Q1 Value of Country (in percent) (in percent) (in percent) γ Bulgaria Cyprus Estonia Greece Ireland Italy Lithuania Latvia Portugal Spain Slovenia Slovakia

74 How To Infer γ The model implies that if unemployment increases form one period to the next, then nominal wages must be growing at the rate γ. How to calculate γ: γ = ( W2011:Q2 W 2008:Q1 Subtract 0.6% per quarter to adjust for foreign inflation and long-run growth (because they are not explicitly incorporated in the model) to obtain the estimate: ) 1 13 γ [0.99, 1.022] 74