Nominal-Rigidities | Macro Paper Warehouse

A Theory of Macroprudential Policies in the Presence of Nominal Rigidities

This is a theory paper, with no calibration or empirical estimates: its output is a set of analytical formulas rather than numbers. It asks what justifies macroprudential intervention in financial markets, and answers that nominal rigidities alone – without the incomplete markets or price-dependent borrowing constraints that earlier work relied on – are enough. In the baseline model financial markets are complete and frictionless; the only imperfections are sticky goods and labor prices and, in the cases the authors care most about, a constraint that stops monetary policy from undoing them, such as the zero lower bound or a fixed exchange rate. The mechanism is what the authors call an aggregate demand externality: once a state of the world is realised, who holds the wealth matters for how much the economy spends, because agents differ in their marginal propensities to spend, but no atomistic agent takes that macroeconomic consequence into account when choosing a portfolio ex ante. Two sets of results follow. First, using a perturbation argument in the spirit of Geanakoplos and Polemarchakis (1985), any equilibrium that is not first best can be improved by intervening in financial markets “except in non-generic knife-edged cases” – the constrained-inefficiency claim is a genericity claim, not a claim that intervention always helps. Second, optimal monetary and macroprudential policy are characterised jointly by explicit formulas in three sufficient statistics: elasticities of substitution, marginal propensities to spend, and good-specific wedges. The optimal financial tax on an agent’s claim in a given state is the marginal-propensity-weighted sum of that state’s wedges, so wealth should be tilted toward states where the goods an agent buys heavily are depressed. Monetary policy, in parallel, targets weighted averages of wedges, adapting the standard New Keynesian targeting rules. The framework is then extended to include pecuniary externalities as well, and – a result the authors call remarkable – the macroprudential formula is literally unchanged: market incompleteness and price-dependent constraints alter the wedges but not the mapping from wedges to taxes. Four applications illustrate the theory: household deleveraging into a liquidity trap, where the optimal policy mix restricts pre-crisis borrowing (in practice a loan-to-value or debt-to-income limit) while monetary policy still delivers perfect stabilisation during the boom; capital controls under a fixed exchange rate, read as a second-best way of regaining interest-rate autonomy; and two cases with a flexible exchange rate where capital controls are still warranted, one with terms-of-trade-dependent collateral constraints and one with non-contingent local- and foreign-currency debt.

An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area

The paper develops and estimates a stochastic dynamic general equilibrium model of the euro area in which prices and wages are both set in staggered Calvo contracts with partial indexation to past inflation, consumption is subject to external habit formation, capital utilisation is variable with a utilisation cost expressed in consumption goods, and capital adjustment costs are a function of the change in investment rather than its level – a structure assembled from Christiano, Eichenbaum and Evans (2001), Kollmann (1997), Erceg, Henderson and Levin (2000), Greenwood, Hercowitz and Huffman (1988) and King and Rebelo (2000). What distinguishes it from that lineage is the estimation: ten orthogonal structural shocks (two supply, three demand, three cost-push and two monetary policy) are introduced so that the model can be confronted with seven euro area macroeconomic series – real GDP, consumption, investment, the GDP deflator, real wages, employment and the nominal short-term interest rate – over 1970:1-1999:4, with the likelihood computed by the Kalman filter and the posterior explored by a Metropolis-Hastings algorithm. Because euro area hours worked are unavailable, employment enters instead, with only a fixed fraction of firms able to adjust employment each period and unobserved hours per employee absorbing the remainder. A small set of parameters is fixed rather than estimated – the discount factor at 0.99 (a 4 percent annual steady-state real rate), quarterly depreciation at 0.025, the capital share at 0.3, consumption and investment shares of output at 0.6 and 0.22, and the wage mark-up parameter at 0.5 because it is not identified – leaving 34 estimated parameters. On marginal likelihood the estimated model beats standard VARs of lag order one to three and is nearly matched by the best Bayesian VAR with a Minnesota prior, the BVAR(3), over 1980:2-1999:4. The parameter estimates imply considerable nominal stickiness, with average price contract duration of about two and a half years against about one year for wages – an ordering the authors call counterintuitive but robust, and attribute partly to their assumption of a flat marginal cost curve in the intermediate goods sector. Price indexation is estimated at 0.4, implying a weight on lagged inflation of only 0.28; external habit is about 55 percent of past consumption; the labour supply elasticity is estimated to be relatively high but imprecisely; and the estimated policy rule satisfies the Taylor principle with substantial interest rate smoothing. In the variance decomposition, three shocks – preference, labour supply and monetary policy – explain significant fractions of output, inflation and interest rates at medium to long horizons, with the price mark-up shock important for inflation but not output and productivity accounting for at most about 12 percent of output forecast error variance. Using the model to construct potential output, defined as the flexible-price-and-wage level in the absence of mark-up shocks, the authors obtain a path very different from a smoothed output trend, with a sharp fall in potential from 1973 to 1975; but they emphasise that the confidence bands are wide, and that the real interest rate gap “is hardly significant over the sample period,” suggesting it “may be a poor guide for monetary policy.”

Firm-specific capital, nominal rigidities and the business cycle

Macroeconomic data show inertial inflation, and the standard way of accounting for it inside New Keynesian models is to assume firms re-optimise prices only once every six quarters or, without indexation to lagged inflation, once every two years or more – an assumption that clashes directly with micro evidence that firms change prices more often than once every two quarters. This paper formulates and estimates a three-shock US business cycle model that reproduces inflation inertia while firms re-optimise prices on average once every 1.8 quarters, and traces the difference to a single modelling assumption: capital is firm-specific rather than homogeneous and traded in economy-wide rental markets. With a predetermined firm-level capital stock, a firm’s short-run marginal cost curve slopes up in its own output, so a contemplated price rise – which cuts the firm’s demand and output – also cuts its marginal cost, working against the price rise. The authors work with two versions of the Christiano-Eichenbaum-Evans (2005) model that differ only in this respect, and show that the log-linearised equilibrium equations differ only in the mapping from structural parameters to the reduced-form coefficient linking the change in inflation to average real marginal cost. Parameterised in terms of that coefficient the two models are observationally equivalent for aggregate data, which means macro evidence cannot adjudicate between them and the case must be made on micro implications. Estimation follows the CEE limited-information strategy, matching model impulse responses to those from a ten-variable identified VAR on quarterly US data for 1982:1-2008:3, with long-run restrictions identifying neutral and capital-embodied technology shocks and a recursive-timing restriction identifying the monetary policy shock; the three shocks together account for roughly 60 percent of the cyclical variance of aggregate output, with capital-embodied technology the largest single contributor and, notably, about 30 percent of the cyclical variation in the real wage. The point estimate of the inflation-marginal cost coefficient is 0.014, implying that a temporary one percent change in marginal cost moves the aggregate price level by only about 0.02 percent; under homogeneous capital this implies price re-optimisation once every 9.36 quarters, while under firm-specific capital it implies once every 1.8 quarters. Wage contracts are re-optimised on average once every 4.5 quarters, the habit parameter is 0.76, and the estimated cost of varying capital utilisation is higher than in CEE. The decisive micro comparison concerns the cross-firm distribution of production after a monetary policy shock: under homogeneous capital roughly 70 percent of firms produce essentially all of the economy’s output four periods after the shock while the rest effectively shut down, an implication the firm-specific capital model does not share. The authors conclude they “strongly prefer the firm-specific capital model,” while leaving open that other propagation mechanisms – firm-specific labour, sectoral heterogeneity in price-change frequency, intermediate inputs, rational inattention and sticky information – “may be at least as important.”

Habit Formation in Consumption and Its Implications for Monetary-Policy Models

The argument starts from a complaint about method: a model used to rank monetary policies must be trusted to represent how consumers and firms actually behave over the policy horizon, and the author contends that most optimisation-based sticky-price models of the late 1990s had not been validated in a way that would earn that trust. Matching first and second unconditional moments is not enough, and neither is matching a single impulse response – especially the response to a monetary policy shock, since the unanticipated component of policy accounts for only a small share of the variance of output, inflation or interest rates. Instead the paper advocates likelihood-based evaluation, comparing the full vector autocovariance function of a structural model against that of an unconstrained VAR in which the structural model is nested. Judged that way, the standard life-cycle consumption model fails in a specific and diagnosable manner: consumption behaves like a “jump variable,” front-loading its entire response to a shock, whereas identified VARs show a gradual hump-shaped response peaking around a year out. Adding Campbell-Mankiw rule-of-thumb consumers does not fix this. The paper’s proposed fix is habit formation in the Carroll-Overland-Weil form, in which utility depends on consumption relative to a reference level built from past consumption; the author’s own explanation of why it works is that this “mixes utility from the level of consumption with utility from the change in consumption,” so consumers acquire a motive to smooth changes as well as levels. He deliberately rules out the alternative repair – assuming serially correlated structural errors – on the grounds that a model with its dynamics hidden in the errors “becomes vulnerable to a Lucas critique of its errors.” Estimating the linearised consumption function by numerical maximum likelihood on U.S. quarterly data for 1966:1-1995:4, the habit parameter comes in at 0.80 with a standard error of 0.19, the rule-of-thumb income share at 0.26, the curvature parameter at 6.11 (implying a small intertemporal elasticity of substitution), and the forward-looking discount parameter at 0.99 per quarter. The restriction that habit formation is unimportant is rejected with a chi-squared statistic of 21.4 and a p-value of 4 times ten to the minus six; the restriction that rule-of-thumb behaviour is unimportant is rejected with a statistic of 12.6 and a p-value of 4 times ten to the minus four. Notably, the full set of cross-equation and zero restrictions implied by the structural model and rational expectations yields a test statistic of 32.8, “not significant at even the 10 percent level” – which the author calls “one of relatively few cases” where an optimisation-based rational-expectations model survives against the VAR that nests it. One estimate comes out lower than expected: the memory parameter in the habit reference level is essentially zero, implying the reference level is simply last quarter’s consumption, and the author defends this at length by showing that a single lag suffices to deliver the smoothing and that substituting a long memory changes the model’s disinflation dynamics only slightly. In a disinflation simulation that cuts the inflation target from about 5% to 2% unexpectedly, consumption in the habit model responds gradually with a peak at about a year and a full response over three to four years, and the author shows that the counterfactually fast real-side response in the no-habit model also significantly damps the persistence of inflation – so misspecifying the real side corrupts the nominal side. He declines to compute optimal policy in the model, giving three reasons: with empirically significant rule-of-thumb consumers it is unclear whose utility to maximise; the model contains no explicit cost of inflation, so the consumer-optimal policy degenerates to minimising consumption fluctuations; and the representative-agent structure is a poor vehicle for welfare costs that arguably fall on discrete employment shifts for a small fraction of the population. He is also explicit that the specification “might not be robust across shifts in monetary or other policy regimes,” and that only testing across regime shifts can settle that.

Higher-Order Perturbation in Sequence Space: the Certainty Correspondence

Sequence-space methods have made first-order solutions of heterogeneous-agent models with aggregate shocks fast, by exploiting “certainty equivalence”: to first order, aggregate risk is neutral, so the response to a one-time, perfect-foresight (“MIT”) shock is the same as the true stochastic impulse response. But that restriction to first order rules out any role for precautionary behavior, welfare effects of risk, or history- and size-dependent responses – exactly the questions a growing literature wants to ask of these models. This paper shows how to go beyond first order in the sequence space by establishing a “certainty correspondence”: nearly all the terms in the second- and third-order Taylor expansion of the model’s full nonlinear sequence-space solution – including the risky steady state and the interaction between shock size and shock history – can be computed purely from perfect-foresight (“MIT shock”) impulse responses, differentiated with respect to shock size and shock timing, without ever manipulating the derivatives of the underlying equilibrium system directly. Combined with a “one-shot principle” that lets each order of the general-equilibrium solution be obtained by evaluating equilibrium conditions on the previous order’s solution and applying a single already-computed inverse Jacobian, this makes third-order solutions of large heterogeneous-agent models computationally practical: an unreduced HANK model with roughly 5,000 idiosyncratic grid points needs only about 300,000 terms at third order in the sequence space, versus a state-space alternative that would require roughly 62 trillion terms and could not be stored on a computer. Applying the method to a quantitative HANK model, the paper finds the fiscal (transfer) multiplier is about four times larger when a sequence of shocks has pushed output several percent below steady state than when output is above steady state, consistent with empirical evidence on state-dependent fiscal multipliers. Applying it to a menu-cost model with strategic complementarity, the paper finds the responsiveness of inflation to nominal marginal cost is significantly steeper when trend inflation is already high, echoing recent findings on nonlinear Phillips curves. Extensive accuracy checks – against third-order state-space perturbation on a small model, and against a global Bellman solution in partial equilibrium – show the third-order sequence-space solution tracks both closely, and in fact tracks the global solution more closely than the perfect-foresight solution does, because it captures the effect of aggregate precautionary saving on marginal propensities to consume that a purely perfect-foresight calculation misses.

Labor Markets and Monetary Policy: A New Keynesian Model with Unemployment

Standard New Keynesian models generate no unemployment, only voluntary movements in hours or employment, which the authors call a surprising basis for the workhorse models used by central banks. They extend the framework with a labour market in which hiring is costly and the cost per hire rises with labour market tightness – defined as the ratio of aggregate hires to the pool of jobless individuals available at the start of the period, which is also the job-finding rate facing an unemployed worker – and they proceed in two steps. With flexible prices and their utility specification (log consumption, power disutility of employment), the constrained-efficient allocation has a constant job-finding rate and hence a constant unemployment rate, invariant to productivity shocks, because income and substitution effects on labour supply exactly offset; the same invariance survives under Nash bargaining, though the bargained unemployment rate generally differs from the efficient one unless a Hosios-like condition holds (no effective market power by final goods firms, and worker bargaining power equal to the elasticity of hiring costs with respect to tightness). The authors are careful to distinguish this invariance from the Shimer puzzle: Shimer derived small unemployment responses assuming a constant marginal rate of substitution, whereas here “our neutrality result follows entirely from movements in the marginal rate of substitution,” which moves one-for-one with productivity so that labour market frictions play no role – and they note that under more general assumptions “the Shimer puzzle will be even stronger than in the original Shimer set-up.” Because the one-for-one wage response looks counterfactual, they impose real wage rigidity through a schedule indexed by a parameter running from Nash bargaining at zero to Hall’s fully rigid wage at one, and add Calvo price staggering. Real marginal cost then depends on tightness and on the rigidity index, which yields a Phillips curve linking inflation to expected inflation and to the current, lagged and expected unemployment rate – with the weights on the level versus the change in unemployment determined by how fluid the labour market is. Since constrained-efficient unemployment is constant, both stabilisation goals are desirable, but with partial wage adjustment neither can be achieved alone: there is no divine coincidence. Calibrating quarterly (discount factor 0.99, unit Frisch elasticity, elasticity of substitution 6 implying a gross markup of 1.2, Calvo slope 1/12, rigidity index 0.5, hiring cost elasticity 1) to a fluid U.S. market (5 percent unemployment, job-finding rate 0.7, separation rate 0.12) and a sclerotic European one (10 percent unemployment, job-finding rate 0.25, separation rate 0.04), with hiring costs set at 1 percent of GDP in the U.S. case, they find that after a persistent (AR(1) coefficient 0.9) one percent fall in productivity, stabilising unemployment requires about a 150 basis point rise in inflation on impact under both calibrations, while strict inflation targeting raises unemployment by about 3 percentage points on impact under both and, in Europe, produces a hump-shaped path peaking near 8 percentage points – a response the authors themselves label possibly unrealistic while noting the policy assumed is also unrealistically extreme. Optimal policy sits between: unemployment rises 50 basis points in the U.S. calibration and about half that in Europe, at the price of persistently higher inflation of roughly 1 and 1.4 percentage points, and the welfare losses under strict inflation targeting are 25 times those under optimal policy in the European calibration.

Large Devaluations and the Real Exchange Rate

This paper asks why large devaluations are typically followed by large, persistent declines in the real exchange rate (RER) rather than being quickly offset by domestic inflation. The authors argue that the primary force behind this pattern is slow adjustment in the price of nontradable goods and services, not slow adjustment in the price of goods that are actually imported or exported (Section 1, Introduction). Using data from five large devaluation episodes – Argentina (2001), Brazil (1999), Korea (1997), Mexico (1994), and Thailand (1997) – they show that at-the-dock import and export prices move roughly in line with the nominal exchange rate, so relative purchasing power parity (PPP) is a reasonable description of “pure-traded-goods” prices even during large devaluations; it is the retail price of nominally tradable goods, and the price of nontradable goods and services, that adjust slowly. The key insight the authors offer for why the traditional tradable/nontradable CPI split is misleading is that the retail price of a tradable good embeds two nontradable components – distribution costs (wholesaling, retailing, marketing, and local transport, estimated at roughly 50 percent of the retail price) and “local goods” that are conventionally classified as tradable but are in practice produced and consumed only domestically. Once the CPI basket is re-decomposed to account for these components, a series of “price-accounting” exercises show that the resulting predicted inflation rates are close to actual post-devaluation inflation in each country, and an Engel (1999) decomposition of the RER confirms that the great majority of the RER’s post-devaluation movement reflects the price of nontradables relative to pure-traded goods, not movements in the relative price of pure-traded goods themselves. A detailed case study of Argentina’s 2001 devaluation, using disaggregated CPI data, a purpose-built survey of prices in Buenos Aires, and supermarket scanner data, corroborates these findings and documents that retail-goods prices adjust far more frequently than services prices, and that there was substantial “flight from quality” toward cheaper goods after the devaluation. The authors test the robustness of their findings against two large real-appreciation episodes (Argentina 1991-95, Mexico 1988-94), four medium-size devaluations (Finland, Italy, Sweden, and the UK in 1992), and small business-cycle-frequency exchange-rate fluctuations in ten OECD countries (1971-2001), finding that the nontradables-relative-to-pure-traded-goods channel remains important in the medium and small cases too, though relatively less dominant than in the large-devaluation episodes, where fluctuations in pure-traded-goods prices instead account for most of the movement.

Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy

This 2005 Journal of Political Economy paper by Christiano, Eichenbaum, and Evans (CEE) builds and estimates a general-equilibrium model to answer a specific question: what combination of frictions lets a DSGE model reproduce two features economists had already documented in the data - an inertial response of inflation and a persistent, hump-shaped response of output - after a shock to monetary policy? Rather than picking parameters to match a few moments, CEE identify the monetary policy shock as the seventh element (ordered after prices, output, consumption, investment, the real wage, and labor productivity, but before profits and M2 growth) of the Cholesky-orthogonalized innovations to a nine-variable recursive VAR estimated on quarterly U.S. data from 1965Q3 to 1995Q3, then estimate a subset of the model’s structural parameters by minimum-distance matching of the model’s impulse responses to the first 25 periods of the VAR-implied impulse responses. The model combines Calvo price contracts with lagged-inflation indexation for firms that cannot reoptimize, Calvo wage contracts with analogous lagged-inflation indexation for households, habit formation in consumption, convex costs of adjusting the flow of investment, variable capital utilization, and a working-capital channel through which firms borrow to finance their wage bill in advance, so that the nominal interest rate enters marginal cost directly. The estimated benchmark model puts the average price contract at about 2.5 quarters (Calvo parameter 0.60) and the average wage contract at about 2.8 quarters (0.64), with habit parameter 0.65 and an investment-adjustment-cost parameter implying a temporary 1 percent increase in the price of capital raises investment by 0.40 percent; the capital-utilization curvature parameter is driven to its lower bound of 0.01, indicating a highly elastic supply of capital services. With these parameters, the model’s impulse responses lie within the two-standard-deviation confidence bands of the VAR-estimated responses for most variables: inflation shows no noticeable rise until roughly three years after an expansionary shock, output rises for nine quarters with a cumulative response of 3.14 percent, of which more than 78 percent occurs after the typical wage and price contract in effect at the time of the shock has been reoptimized (a “contract multiplier” of 3.7). Counterfactual re-estimations show sticky wages, not sticky prices, are the crucial nominal friction - setting price stickiness to zero barely affects the estimated wage-contract length or the model’s qualitative fit, while setting wage stickiness to zero forces price stickiness to an extreme (contracts averaging over three years, which the authors call inconsistent with microeconomic evidence) and destroys the hump-shaped output response - and that variable capital utilization is the crucial real friction, since removing it roughly halves the output response and again forces implausibly long price contracts when re-estimated. The paper is explicit that its results are estimated on a single U.S. sample ending in 1995Q3, that Calvo pricing is treated as a reduced-form device for nominal sluggishness rather than a literal description of contracting, and that the model is disciplined only against monetary-policy-shock responses, leaving its performance against other shocks as a separate, only preliminarily addressed question.

Optimal fiscal and monetary policy under sticky prices

This paper resolves a contradiction between two branches of optimal monetary policy theory. Ramsey models with flexible prices (Calvo and Guidotti; Chari, Christiano, and Kehoe) find that an optimizing government, restricted to distortionary income taxation and nominal non-state-contingent debt, should make inflation highly volatile and serially uncorrelated, using unanticipated price-level changes as a non-distorting, state-contingent tax on nominal wealth so that regular tax rates can stay smooth; New Keynesian models with sticky prices, by contrast, typically find optimal inflation should be zero or near-zero at all times – but usually by assuming the government can also rely on lump-sum taxes, eliminating any need for inflation to double as a fiscal instrument. Schmitt-Grohé and Uribe build a single model that combines the empirically relevant assumptions of both literatures – only distortionary income taxation and nominal non-state-contingent debt available to the fiscal authority, plus monopolistic competition and Rotemberg-style costly price adjustment on the supply side – and solve for the Ramsey-optimal fiscal and monetary policy under full commitment. Their central finding is that the tradeoff between using inflation as a shock absorber and avoiding the real costs of price adjustment is overwhelmingly resolved in favor of price stability: calibrating price stickiness to even one-tenth of available U.S. estimates already reduces the optimal standard deviation of inflation from about 7% per year under full price flexibility to well under 1%, and at their full baseline calibration it falls to just 0.17% per year. They trace this fragility to Aiyagari et al.’s result that the welfare gain from being able to issue real state-contingent debt (which flexible-price surprise inflation effectively replicates) is itself small, so even minor price-adjustment costs are enough to make the Ramsey planner abandon front-loading altogether. In its place, the government relies on ordinary tax-rate and debt adjustments, smoothed over time to minimize distortion – which induces near-random-walk behavior in both taxes and public debt, reproducing the Barro (1979)/Aiyagari et al. finding usually derived by assuming the government can issue only real (not nominal) non-state-contingent debt, but here obtained instead from a purely nominal, non-state-contingent debt structure plus even minimal price rigidity. The paper further shows that price stickiness induces a systematic, quantitatively significant deviation from the Friedman rule (roughly half of it attributable to stickiness itself, the rest to an existing monopoly-profit-taxation channel), and that a regression of the Ramsey-optimal nominal interest rate on inflation and output, estimated on simulated data, produces an inflation coefficient statistically indistinguishable from zero (and negative in point estimate) – the opposite of what an actual Taylor rule requires – a result the authors present as a cautionary finding about inferring policy rules from optimal-policy time series, not as a claim that a passive rule can implement the Ramsey outcome.

Optimal monetary policy with staggered wage and price contracts

In an optimizing-agent model in which both product and labour markets are monopolistically competitive and both prices and wages are set in Calvo-style staggered nominal contracts, the authors show that monetary policy cannot reach the allocation a frictionless economy would reach. Their argument runs through welfare: approximating average household utility gives an objective that depends on just three unconditional variances – the output gap, price inflation, and wage inflation – and each enters with a strictly negative weight. Price inflation stays constant only if firms are continuously on their labour demand schedules; wage inflation stays constant only if households are continuously on their labour supply schedules; and those two conditions together imply a zero output gap. But holding nominal wages and prices both fixed would pin the real wage at its steady-state value, whereas the Pareto-optimal real wage moves in response to productivity and labour-supply shocks. The contradiction means no more than one of the three variables can have zero variance, so a three-way stabilisation tradeoff is unavoidable, and the Pareto optimum is attainable only in the two special cases where either wages or prices are completely flexible. Solving numerically under a quarterly calibration – discount factor 0.99, Cobb-Douglas capital share 0.3 (labour elasticity of output 0.7), wage and price markup rates of 1/6, and Calvo parameters of 0.75 for both contracts (a mean duration of four quarters) – the authors compute the optimal interest rate rule and find the expected welfare loss relative to the Pareto optimum to be about 0.0024 percent of steady-state consumption, roughly a quarter of Lucas’s (1987) baseline estimate of the gains from eliminating aggregate consumption fluctuations. Two regularities emerge from grids over contract durations: it is optimal, other things equal, for the more flexible nominal variable to absorb a larger share of the required real-wage adjustment, and output-gap volatility is low under the optimal rule for essentially every combination of contract durations. Comparing rules against the optimal benchmark, strict price inflation targeting is clearly suboptimal – at the baseline it produces a welfare loss roughly eight times the optimal rule’s, and when the labour elasticity of output goes to zero the ratio explodes – while strict output-gap targeting does nearly as well as the optimal rule except when the price markup rate is much smaller than the wage markup rate, and two hybrid rules (price inflation plus the output gap, or price inflation plus wage inflation) perform nearly as well as the optimal rule in every case considered.

Real Wage Rigidities and the New Keynesian Model

Most central banks behave as though stabilising inflation and stabilising the gap between output and its desired level are competing goals, yet the standard New Keynesian framework implies no such conflict: because the New Keynesian Phillips curve makes inflation a function of expected inflation and the output gap alone, holding inflation constant delivers a zero output gap, and because the log distance between the efficient (first-best) and natural (second-best) levels of output is a constant in that model, a zero output gap is also a zero welfare-relevant gap. The authors name this property the “divine coincidence” and argue it is an artefact of the absence of non-trivial real imperfections rather than a robust feature. Introducing one such imperfection – real wages that adjust only partially toward the marginal rate of substitution, with the adjustment weight serving as an index of real rigidity – makes the distance between first- and second-best output fluctuate with both supply and preference shocks, so that stabilising inflation, while still equivalent to stabilising the output gap, is no longer equivalent to stabilising the welfare-relevant gap, and the central bank faces a genuine tradeoff. The authors show the tradeoff is quantitatively non-trivial: with a real-rigidity index of 0.9 (a six-quarter half-life for real wage adjustment), an oil share in production of 0.025, an average price duration of six months and the discount factor taken to one, a 10 percent rise in the price of oil requires annualised inflation slightly above 4 percent on impact if the welfare-relevant gap is fully stabilised, or a 1.1 percent first-quarter fall in the welfare-relevant output gap if inflation is fully stabilised – with both magnitudes falling sharply, to roughly 2 percent and 0.5 percent at an index of 0.8 and to below 0.5 percent and 0.1 percent at 0.5, since the expressions are highly non-linear in the rigidity index. The same friction multiplies the short-run output cost of a disinflation by a factor of ten at an index of 0.9, turning a move from 5 percent inflation to zero from a 0.25 percentage point permanent output loss into a 2.5 percentage point short-run loss. On the positive side, real wage rigidities generate inflation inertia – persistence in inflation beyond that inherited from the output gap – and yield an inflation equation in lagged and expected inflation, unemployment and the change in the real price of the non-produced input that is close to traditional Phillips curve specifications; estimated by instrumental variables on annual U.S. data for 1960-2004 (GDP deflator inflation, the civilian unemployment rate, and the PPI raw materials index relative to the GDP deflator, instrumented with four lags of each), all coefficients carry the predicted sign and are statistically significant, and the restriction that the coefficients on lagged and expected inflation sum to one cannot be rejected at the 5 percent level, though the authors note it is not rejected by much.

Rule-of-thumb behaviour and monetary policy

Standard optimisation-based sticky-price models have no lagged variables in their structural equations, which makes them hard to square with the high serial correlation actually observed in output and inflation; this paper asks what happens to optimal monetary policy once a fraction of agents is allowed to skip the optimisation and follow a simple backward-looking rule instead. The model is otherwise identical to Woodford’s – a closed economy with no capital accumulation, a continuum of monopolistically competitive household-producers, and Calvo price setting – and the two departures are deliberately symmetric. Each period a household draws an independent optimisation cost; a fraction of households with costs above a threshold sets consumption equal to last period’s aggregate per-capita consumption rather than solving its Euler equation, and among firms offered a Calvo price-reset opportunity a fraction follows Gali and Gertler’s rule of setting its price to last period’s average newly chosen price scaled up by last period’s inflation. Both departures put a lagged endogenous variable into the structural equation – lagged output into the IS curve, lagged inflation into the Phillips curve – and both, the paper shows, also change the welfare criterion that policy should be maximising, a point not previously noted in the literature: rule-of-thumb price setting adds a penalty on the squared change in inflation, and rule-of-thumb consumption adds a penalty on the squared change in output. Rule-of-thumb behaviour works in two opposing directions. It raises endogenous persistence, which on its own would make inflation and the output gap more variable; but it also weakens transmission, reducing the sensitivity of inflation to the output gap and of output to expected real interest rates. In the paper’s calibration – Woodford’s parameter values, based on Rotemberg and Woodford’s estimates on U.S. data for 1980-95, with Calvo parameter 0.66 per quarter, discount factor 0.99, and a natural-rate-of-interest shock with standard deviation 0.93 percent per quarter – the weakening of transmission dominates, so inflation variability falls as rule-of-thumb price setting becomes more prevalent and output gap variability falls sharply as rule-of-thumb consumption becomes more prevalent. The central policy result is that highly inertial, indeed “superinertial,” interest rate policy – a sum of coefficients on lagged interest rates exceeding one – remains optimal at every fraction of rule-of-thumb behaviour examined (the paper reports results for optimising fractions of 1, 0.6 and 0.2), and survives every robustness check it runs: a lower weight on interest rate variability, logarithmic preferences, serially correlated shocks, and the introduction of inefficient supply shocks that create a genuine inflation/output-gap trade-off. Two rules stand out as robust: the four-argument rule that implements the optimal plan when all agents optimise (current inflation, the change in the output gap, and two lags of the interest rate), and a first-difference version of Taylor’s 1993 rule. By contrast, rules feeding back only from inflation and the lagged interest rate, and price-level rules, have optimal coefficients that shift dramatically with the rule-of-thumb fraction – an unattractive property given how hard that fraction is to measure. Throughout, the policymaker is assumed able to commit; the authors are explicit that further work is needed to show these particular rules of thumb are good approximations to actual decision making.

Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach

The paper estimates an extended New Neoclassical Synthesis model on US data covering 1966:1-2004:4 using Bayesian likelihood methods, with seven observable series – real GDP, hours worked, consumption, investment, real wages, prices and the short-term nominal interest rate – and seven orthogonal structural shocks: total factor productivity, risk premium, investment-specific technology, wage mark-up, price mark-up, exogenous spending and monetary policy. The model carries sticky Calvo price and wage setting with backward indexation, habit formation in consumption, investment adjustment costs, variable capital utilisation and fixed costs in production. Three questions organise the paper. First, does the model describe the data? Compared against unrestricted VARs and a Sims-Zha Bayesian VAR over the full sample (with 1956:1-1965:4 as a training sample to standardise priors), the tightly parameterised model beats the unrestricted VARs decisively and is matched by the best BVAR(4) on marginal likelihood; in a rolling out-of-sample RMSE exercise over 1990:1-2004:4 the two are comparable one quarter ahead but the structural model “does considerably better than both the VAR(1) and BVAR(4) model” over horizons up to three years, with the improvement “quite uniform across the seven macro variables.” Second, which frictions earn their place? Cutting the Calvo probability for prices or for wages to 0.10 each costs about 50 in log marginal likelihood; removing investment adjustment costs costs about 160; reducing habit formation is costly but much less so; shutting off variable capital utilisation “comes at no cost”; and restricting price indexation to 0.01 actually improves the marginal likelihood. Third, what drives the US business cycle? Within a year, output is dominated by the exogenous spending, risk premium and investment-specific shocks, which together account for more than 50 percent of forecast error variance; beyond two years the productivity and wage mark-up shocks account for more than half, with the wage mark-up dominant in the long run, while monetary policy shocks “contribute only a small fraction of the forecast variance of output at all horizons.” Inflation is driven by price mark-ups in the short run and wage mark-ups in the medium to long run, which the authors attribute to a very small estimated slope of the New Keynesian Phillips curve and to an aggressive estimated policy response. A positive productivity shock reduces hours worked immediately and significantly, turning positive only after two years, and does so even under flexible prices and wages, which the authors trace to habit persistence and capital adjustment costs. Finally, sub-sample estimates for the “Great Inflation” (1966:2-1979:2) and the “Great Moderation” (1984:1-2004:4) show most structural parameters stable, the standard deviations of productivity, monetary policy and price mark-up shocks lower in the second period, the policy response to the output gap level halved and no longer significant, and a counterfactual exercise attributing the fall in volatility mainly to milder shocks rather than to policy or structural change.

Staggered prices in a utility-maximizing framework

This 1983 Journal of Monetary Economics paper by Guillermo Calvo builds a model of staggered price-setting that is more analytically tractable than the earlier discrete-contract-length models of Phelps (1978) and Taylor (1979, 1980), while grounding the demand side in fully optimizing, infinitely-lived Sidrauski-Brock households. Each firm can revise its price only when a random signal arrives, with the probability that a firm has not yet received a signal after h periods falling exponentially at a constant hazard rate; because signals arrive independently across a continuum of firms, at any instant the economy contains a smooth, non-degenerate distribution of outstanding price vintages, so the aggregate (log) price level becomes a predetermined variable that cannot jump, even though individual firms set prices under perfect foresight over the entire future path of the average price and excess demand. Calvo shows the resulting dynamics can be characterized with largely graphical, phase-diagram techniques, and derives the notable implication that it is the rate of change of inflation, not the level of inflation itself, that is a decreasing function of excess demand – a higher-order inverse Phillips relationship – even though the more familiar positive association between the inflation level and excess demand can still emerge along the equilibrium path. On the household side, families maximize a discounted stream of utility from consumption and real money balances subject to a flow budget constraint, and Calvo introduces a “Price Regulating Mechanism” – a stylized tax-and-subsidy scheme ensuring every consumer effectively pays the same average price – to sidestep the problem of how demand is allocated across differently priced firms. Using this framework, he shows that a one-time unanticipated increase in the money supply can move the economy from excess supply to full employment and, chosen optimally, can attain the first-best outcome; that this monetary policy is welfare-superior to an equivalent fiscal expansion through government spending, because the latter permanently lowers steady-state private consumption; and that pegging the nominal interest rate at a fixed level produces a continuum of equilibrium inflation paths, demonstrating that the indeterminacy problem identified by Sargent and Wallace (1975) under interest-rate pegs is not an artifact of assuming fully flexible prices.

Sticky Price Models of the Business Cycle: Can the Contract Multiplier Solve the Persistence Problem?

Since the early 1970s macroeconomists have known how to build general equilibrium models in which monetary shocks move output contemporaneously; the harder problem, as the authors frame it, is generating the defining feature of business cycles – persistent output movements – without simply assuming prices are fixed for long stretches. Staggered price-setting has long been the promising candidate, following Taylor’s argument that because contracts are written relative to one another, shocks are “passed on from one contract to another – a sort of ‘contract multiplier’.” This paper asks quantitatively whether that mechanism delivers, in a general equilibrium model with a continuum of monopolistically competitive firms producing differentiated goods from capital and labour, real balances in the utility function, and prices set for a fixed number of periods in staggered cohorts. The authors define the contract multiplier as the ratio of the half-life of output after a monetary shock under staggering to one-half the length of exogenous price stickiness (the half-life under synchronized setting, since shocks arrive randomly between adjustments), and note it is approximately invariant to that length in their models. Fitting an ARMA to quadratically detrended log real GDP gives an output half-life of 10 quarters, so with one quarter of exogenous stickiness the required multiplier is 20 (60 with one month, 5 with one year). Under a benchmark calibration – money demand parameters estimated from a regression of log consumption velocity on the interest rate using 1960:1-1995:4 Citibase data, giving an interest elasticity of 0.39; an 11 percent markup and demand elasticity of 10 following Basu and co-authors; a capital-output ratio of 2.65, investment-output ratio of 0.23, one-third of time in market work, and a capital share of one-third; money growth serial correlation of 0.57 from M1 over 1959:3-1995:2 – the multiplier is roughly 1, implying exogenous stickiness would have to last 5 years to match the data. The reason is that with constant-elasticity demand prices move one-for-one with costs, and with unit elasticity of substitution between consumption and leisure costs are extremely sensitive to output, so the elasticity of the equilibrium real wage with respect to consumption exceeds one and output is not persistent. The authors then test three escapes and find each fails once intertemporal links are restored: near-perfect substitutes preferences give a multiplier of 21.97 without capital and interest-sensitive money demand but 0.50 with them (and imply labour input would rise 57 percent per day under 2 percent growth in wages and consumption); Kimball-style convex demand yields 3.79 without links and 1.55 with them, at a parameterization under which a 2.3 percent rise in relative price drives demand to zero; and specific factors give 1.82 without links and 1.33 with them, needing a demand elasticity of about 6500 to reach 20. Combining all three raises the multiplier only to 1.81, and simultaneous parameter searches over wide ranges top out at 3.05 for the benchmark, 3.20 for convex demand and 4.17 for specific factors and the combined model. Their conclusion is that “the staggered price-setting mechanism is not the long-sought solution” and that mechanisms to solve the persistence problem must be found elsewhere.

Structural Reinforcement Learning for Heterogeneous Agent Macroeconomics

Standard recursive formulations of heterogeneous-agent models with aggregate risk force the entire cross-sectional distribution of agents into the Bellman equation characterizing individual decisions – the “Master equation” – purely because low-dimensional equilibrium prices, unlike the distribution itself, do not follow a Markov process, so rational agents forecasting prices end up needing to forecast the whole distribution. This extreme curse of dimensionality remains the central computational bottleneck for global solutions of heterogeneous-agent models, so severe that even a Huggett (1993) model with aggregate risk – despite looking simple – proved impossible for any team to solve in an influential benchmarking exercise and was dropped from the project altogether. This paper sidesteps the Master equation entirely using ideas from reinforcement learning (RL): agents learn equilibrium price dynamics directly from simulated paths, as standard RL would, but the paper’s “structural reinforcement learning” (SRL) approach departs from standard RL by assuming agents have structural knowledge of their own individual-state dynamics (their budget constraint and idiosyncratic income process), letting the authors compute exact policy gradients by differentiating through these known dynamics rather than relying on the noisy, approximate policy gradients standard RL methods estimate; only the equilibrium price process itself is treated as unknown and learned from simulation. By further restricting agents to condition their policies only on current (or briefly lagged) prices rather than the full price history or the distribution, the paper solves for a low-dimensional “restricted perceptions equilibrium” in the sense of Sargent (1991) rather than the full rational-expectations equilibrium – expectations are restricted in functional form but remain statistically consistent with actual outcomes. Because policy functions depend only on prices, they double as individual supply/demand schedules that can be integrated across the distribution and market-cleared period-by-period along a simulation, treating market clearing as part of the “environment” (in RL parlance) rather than something solved inside an optimization loop – which is what lets the method efficiently handle nontrivial market-clearing conditions that have historically been very hard. Implemented in JAX on a single GPU, the resulting structural policy gradient (SPG) algorithm solves the Krusell and Smith (1998) model in about 55 seconds, the previously-unsolved Huggett (1993) model with aggregate risk in around one minute, and a one-asset HANK model with a forward-looking New Keynesian Phillips curve in around three minutes – with the Krusell-Smith solution closely matching alternative global solutions of the rational-expectations equilibrium, and allowing agents a longer history of lagged prices barely moving the solution, indicating most of the information relevant for forecasting prices is already contained in current prices. The paper is explicit that its algorithm, as presented, is not itself intended as an empirically realistic theory of how real economic agents form expectations, though it suggests the “sampling”-based logic behind SRL could in principle be developed into one.

The New IS-LM Model: Language, Logic, and Limits

This article gives a simple, self-contained exposition of what King calls the “New IS-LM model” – a small, three-equation macroeconomic system, built from optimizing microfoundations and analyzed under rational expectations, consisting of a forward-looking IS equation (current output depends on expected future output and the real interest rate), a Fisher equation (the nominal rate equals the real rate plus expected inflation), and an expectational (“New Keynesian”) Phillips curve (current inflation depends on expected future inflation and the current output gap). King situates the model historically as the outgrowth of a “New Neoclassical Synthesis” that answers the rational-expectations-era critique of the original Hicksian IS-LM framework and its 1970s descendants, while explicitly noting the model is not itself derived from first principles in this article but is instead a distillation used to communicate results from more fully articulated, microfounded models. Working through the model’s implications, King shows that a “neutral” monetary policy – one that always keeps output at its capacity level – implies a specific, and in general history-dependent, inflation target: inflation should be exactly zero if there are no exogenous “inflation shocks,” and otherwise the target should absorb the persistence properties of those shocks while never responding to shocks to aggregate demand, capacity growth, or money demand. He derives the forward-looking New Keynesian Phillips curve explicitly from Calvo-style staggered, forward-looking price-setting by monopolistically competitive firms, under an admittedly “heroic” assumption linking real marginal cost to the output gap, and shows the resulting curve implies essentially no long-run trade-off between inflation and output, even though nominal disturbances can still generate output effects that persist for many periods – resolving an apparent tension between the model’s long-run classical neutrality and the empirically persistent business cycles that motivated earlier “old Keynesian” IS-LM analysis. Turning to policy-rule design, King shows that interest rate rules of the Taylor type must satisfy restrictive parameter conditions – broadly, an “aggressive” response of more than one-for-one to inflation – to guarantee a unique, stable rational-expectations equilibrium rather than a continuum of self-fulfilling “sunspot” equilibria, and demonstrates the specific and somewhat counterintuitive result that this zone of admissible, determinacy-preserving rules is actually smaller (the zone of indeterminacy larger) once prices are sticky than in the flexible-price benchmark, with the exact boundary conditions differing sharply depending on whether the rule responds to current or expected future inflation. He closes by explicitly flagging the model’s own limits: it cannot, from within itself, justify why output stabilization at capacity is welfare-improving, define what an “inflation shock” actually is at a structural level, or evaluate the consequences of omitting investment and capital altogether – questions that, King stresses, can only be answered by stepping outside the New IS-LM model into the fully articulated, microfounded models it is meant to summarize.

The new open economy macroeconomics: a survey

Since Obstfeld and Rogoff’s 1995 Redux model, open-economy macroeconomics has been rebuilt around dynamic general equilibrium models with explicit microfoundations, imperfect competition and nominal rigidities, with the declared aim of replacing the Mundell-Fleming model “still widely employed in policy circles as a theoretical reference point.” This survey describes that literature, focusing “almost exclusively on the analysis of monetary shocks” because nominal rigidities matter most starkly there, and organises it around a single diagnostic question: which of the benchmark’s conclusions are results about the world and which are artefacts of its assumptions? The answer is that very few survive. In the Redux model itself – two countries of yeoman-farmers, identical preferences, the law of one price, prices preset one period ahead, a single riskless real bond, labour the only factor – a surprise permanent home monetary expansion raises home output and consumption, lowers the world real interest rate, puts the home current account into surplus, makes money non-neutral in the long run through the resulting permanent net-foreign-asset position, rules out exchange rate overshooting, and raises home and foreign welfare by exactly the same amount. Each of those results is then shown to be contingent. Allow a fraction of firms to price to market with prices sticky in local currency and the expenditure-switching effect disappears, overshooting becomes possible, home and foreign consumption growth delink while output correlations rise (matching the international business-cycle evidence better), and under full pricing to market the current account stays in balance and a depreciation improves rather than worsens the depreciating country’s terms of trade – turning the Redux equal-gains result into a beggar-thy-neighbour effect. Replace symmetric CES preferences with a unitary home-foreign substitution elasticity (Corsetti and Pesenti) and the model becomes solvable in closed form with a permanently zero current account, but the terms of trade re-enter as a welfare channel, so the optimal monetary surprise becomes finite and it is no longer optimal to expand output to its competitive level. Add capital and a monetary shock may produce a current account deficit rather than a surplus; add non-traded goods or home bias and overshooting appears and the welfare gains stop being equally shared. Persistence beyond the imposed rigidity requires specific ingredients – convex demand, rigid real wages, or translog preferences with intermediate inputs – since with constant markups and rising marginal costs “a firm will raise its price as soon as it is given the opportunity.” Financial structure turns out to matter less than expected for monetary transmission, because equilibrium current account movements are quantitatively small, though it matters qualitatively and matters a great deal for fiscal shocks. The policy-interdependence results invert with the pricing assumption: under the law of one price spillovers are positive and coordination means faster joint monetary expansion, whereas under full pricing to market spillovers are negative and coordination means a slower common inflation rate. Stochastic versions permit the first utility-based welfare comparison of exchange rate regimes, with flexible rates dominating pegs under pricing to market for any risk aversion at least logarithmic, but fixed rates preferred under producer-currency pricing if risk aversion is high enough. The empirical section is short by the survey’s own account – “[t]hus far, the literature has been primarily theoretical in focus” – and its verdict is explicitly a warning: because “many welfare results are highly sensitive to the precise denomination of price stickiness, the specification of preferences and financial market structure … any policy recommendations emanating from this literature must be highly qualified.”

The Science of Monetary Policy: A New Keynesian Perspective

This widely cited survey derives monetary policy design from a simple forward-looking New Keynesian model built from first principles – a forward-looking IS-type output-gap equation and a forward-looking Phillips curve – and states its conclusions as a numbered sequence of general “Results” meant to be robust across a wide variety of macroeconomic frameworks. Under discretion (no commitment), optimal policy embeds implicit inflation targeting: the central bank should adjust the nominal rate more than one-for-one with expected future inflation (Result 3, later known as the Taylor principle), should perfectly offset demand shocks but let the nominal rate stay put in the face of shocks to potential output (Result 4), and, more generally, should raise real rates whenever inflation is forecast above target and let it return only gradually. Under commitment, the paper derives a genuinely new result: even when the central bank has no temptation to push output above its natural level (ruling out the traditional Kydland-Prescott/Barro-Gordon inflationary-bias motive for commitment), a credible commitment to a rule still improves the current output-inflation trade-off, because current inflation depends on expectations of future policy, and a rational private sector will discount an un-committed promise of future toughness (Result 7). The paper also formalizes Alan Blinder’s “opportunistic” approach to disinflation as optimal when policy-makers weight small output deviations more heavily than small inflation deviations, showing it is equivalent to targeting inflation within a zone rather than at a point (Result 12), and works through practical complications including imperfect information, interest-rate smoothing, and model uncertainty. Turning from theory to practice, the paper applies its “Taylor principle” criterion to U.S. monetary history, arguing pre-Volcker policy “tended to accommodate rather than fight increases in expected inflation” while Volcker-Greenspan policy adopted the kind of implicit inflation targeting the theory recommends, and closes with simple rules (including Taylor’s and the authors’ own forward-looking variant) and open questions for future research, including endogenous inflation persistence, open-economy extensions, and the zero lower bound.