Deep Habits

Habits are formed over each individual variety of good rather than over a consumption aggregate, and the authors argue this is at least as well supported empirically as the standard assumption. They have in mind “an environment in which consumers can form habits separately over narrowly defined categories of goods, such as clothing, vacation destinations, music, cars and not just over consumption defined broadly. We believe that this description of preferences, to which we refer as ‘deep habits’, is more compelling than its standard, or, in our terminology, ‘superficial’, counterpart.” Two supports are cited: “the deep-habit formulation is embedded in Houthakker and Taylor’s (1970) classic work on consumption demand,” and “the empirical literature on consumer behaviour often finds that consumers’ choices over different brands of goods are affected by past brand choices.” In the conclusion the claim is stated as a reading rather than a demonstration: “this paper is motivated by our reading of the available empirical literature on consumption behaviour suggesting that the former alternative is at least equally compelling.”

The demand side of the macroeconomy is unchanged – the consumption Euler equation is identical to the superficial-habit case – which means existing estimates of habit strength can be reinterpreted, and also that the model brings extra identifying restrictions. “The demand side of the macroeconomy – in particular, the consumption Euler equation – is indistinguishable from that pertaining to an environment in which agents have superficial habits,” and so “existing Euler-equation-based empirical estimates of the degree of habit formation can be interpreted as uncovering the degree of deep-habit persistence.” The gain is that the habit parameter now also appears elsewhere: “because in our model the parameter [theta] that measures the strength of deep habits appears in equations other than the consumption Euler equation, our model provides additional identification restrictions,” which the estimation exploits “to obtain a more efficient estimate.”

Firms’ pricing becomes dynamic, because current sales build the future habit stock and therefore future demand. “When habits are formed at the level of individual goods, firms take into account that the demand they will face in the future depends on their current sales. This is because higher consumption of a particular good in the current period makes consumers, all other things equal, more willing to buy that good in the future through the force of habit. Thus, when habits are deeply rooted, the optimal pricing problem of the firm becomes dynamic.” Combined with imperfectly competitive product markets, this “results in a model of endogenous, time-varying mark-ups of prices over marginal cost.” Formally, the deep-habit model replaces the superficial model’s requirement that the mark-up be constant at the elasticity ratio with two extra equilibrium conditions, which “introduces a dynamic wedge between factor prices and their associated marginal products.”

Habitual consumption contributes a perfectly price-inelastic block to each firm’s demand, and that block shrinks in relative weight when aggregate demand rises, so the elasticity – and hence the mark-up – moves. Demand for firm i’s good has two terms: one with the elasticity of substitution across varieties, “and the second term is [habitual], which originates exclusively from habitual consumption of good i. Therefore, the second term is perfectly price inelastic. The price elasticity of the demand for good i is a weighted average of the elasticities of the two terms just described… When aggregate demand rises, the weight of the price-elastic term in total demand increases, and as a result the price elasticity increases.” In equilibrium the short-run elasticity is the substitution elasticity scaled by one minus the ratio of habitual to current consumption, which “is smaller than the price elasticity of demand in the absence of deep habits.” Under superficial or no habits the demand function collapses to the standard constant-elasticity form, so “the price elasticity of demand for good i is independent of the level of aggregate demand… implying a time-invariant mark-up.”

Firms cut mark-ups today to build customer base when future profits look high – so the mark-up falls with expected future per-unit profits and rises with the real interest rate. “When the present value of future per-unit profits are expected to be high, firms have an incentive to invest in customer base today. They do so by building up the current stock of habit. In turn, this increase in habits is brought about by inducing higher current sales via a decline in current mark-ups.” On interest rates: “the equilibrium mark-up is decreasing in the discount factor, which implies that, all else constant, a rise in the real interest rate should be associated with an increase in the current mark-up. This is because if the real interest rate is higher, then the firm discounts future profits more, and thus has less incentives to invest in market share today.” The authors note these two implications are what distinguishes deep habits from the implicit-collusion mechanism of Rotemberg-Saloner and Rotemberg-Woodford, where the maximum sustainable collusive mark-up is increasing in the long-run benefit of staying in the collusive relationship.

It provides their microfoundations, and in doing so reverses the prediction they were criticised for. “The dynamic pricing problem at the level of the individual firm that is induced by the introduction of deep external habits is akin to that studied in partial equilibrium models of customer-market pricing (Phelps and Winter, 1970) or brand-switching costs (Klemperer, 1995).” One difference is substantive: “in the deep-habit model there is gradual substitution between differentiated goods rather than discrete switches among suppliers,” which brings a tractability advantage – “under the deep-habit formulation one does not face an aggregation problem. In equilibrium, buyers can distribute their purchases identically, and still suppliers face a gradual loss of customers if they raise their relative prices.” On the critique: “existing general equilibrium versions of customer-market and switching-cost models have been criticized on the grounds that they predict procyclical mark-ups. This criticism, however, is based upon customer-market models in which the demand function faced by individual firms is specified ad hoc and not derived from the optimizing behaviour of households. Our results show that once the demand for individual goods is derived from first principles, a customer-market model is indeed capable of predicting an empirically relevant cyclical behaviour of mark-ups.” The mechanical difference is identified precisely: the Rotemberg-Woodford customer-market model has no price-insensitive term in demand, “as a result, the elasticity of demand for an individual variety is independent of current aggregate demand conditions.”

Because not every component of demand is habit-forming, so shocks that shift the composition change how strong the habit mechanisms are. “Because not all components of aggregate demand may be subject to deep-habit formation, it follows that changes in the composition of aggregate demand will, in general, affect the strength of the aforementioned price-elasticity and intertemporal effects of deep habits on mark-ups. For instance, if investment spending is not subject to habit-forming behaviour, a shock that increases the share of investment in aggregate spending, such as an aggregate productivity shock, would reduce the overall importance of habits and, as a result, alter the pricing behaviour of firms.” In the full model the authors also state their maintained assumption on the public side plainly: the government-spending results “hinge on our maintained assumption that government consumption is subject to good-specific habit formation.”

Nonlinear GMM on U.S. quarterly data for 1967:Q1-2003:Q1, exploiting both the Euler equation and the supply-side mark-up conditions, with remaining parameters calibrated to Rotemberg and Woodford’s targets. “The approach that we take is to exploit the fact that the deep-habit parameters enter both the intertemporal consumption Euler equation – as in superficial-habit-formation models – and the equilibrium conditions determining the dynamics of the mark-up, which originate on the supply side of the economy. This characteristic of the deep-habit model is particularly useful, because it allows for a more efficient estimation of the habit parameters than the standard estimates of habit parameters that are derived solely from the consumption Euler equation.” Utility is assumed separable in consumption and leisure to facilitate estimation. The resulting settings are a habit strength of 0.86, a habit-stock persistence of 0.85, an elasticity of substitution across varieties of 5.3, and an inverse intertemporal elasticity of 2. The calibration targets – “taken from Rotemberg and Woodford (1992)… to facilitate the comparison of our model of endogenous mark-ups due to deep habits to their ad hoc version of the customer-market model” – are a labour share of 75%, consumption share 70%, government consumption share 12%, a 4% annual real interest rate and a Frisch labour supply elasticity of 1.3, which imply a capital elasticity of output of 0.25, a quarterly depreciation rate of 0.025, a discount factor of 0.99 and a leisure curvature parameter of 3.08. Following Prescott, households devote 20% of their time to market activities in steady state. All three shocks are given serial correlation 0.9. Fixed costs are included “to ensure that profits are relatively small on average, as is the case of the U.S. economy, in spite of equilibrium mark-ups of prices over marginal cost significantly above” zero.

It raises it: the calibration implies 1.32, which the authors call “somewhat high,” against 1.23 without deep habits. The steady-state mark-up depends on the elasticity of substitution scaled by a factor involving the shares of consumption, government purchases and investment in output; with no deep habits that factor is one and the mark-up reduces to the usual elasticity ratio, 1.23. “Because under deep habits, the parameter m is less than unity, firms have more market power under deep habits than under superficial habits. This is because, in the former formulation, firms take advantage of the fact that when agents form habits on a variety-by-variety basis, the short-run price elasticity of demand for each variety is less than [the substitution elasticity].” In the conclusion the authors turn this into a claimed advantage relative to rival theories: most theories of countercyclical mark-ups “face a trade-off between the elasticity of the mark-up with respect to aggregate demand and the level of the mark-up,” predicting too low an elasticity once the average level is restricted to realistic values, whereas “our theory can predict high mark-up elasticities without requiring empirically unrealistic levels of average mark-ups.”

Mark-ups fall and wages rise, reversing the standard model’s prediction. For a preference shock worth 1% of steady-state habit-adjusted consumption, consumption and output rise in all three model variants, and labour supply expands, which “puts downward pressure on wages. In the economies with superficial habit or no habit, the labour demand schedule is unaffected by the preference shock on impact. In these two economies, the combination of an unchanged labour demand schedule and an increase in the labour supply causes the equilibrium wage to fall. By contrast, in the deep-habit model the mark-up falls significantly on impact by about 0.4%. This decline in the mark-up leads to an increase in the demand for labour at any given wage rate. This expansion in labour demand more than compensates the increase in labour supply, resulting in an equilibrium increase in wages of about 0.3%.”

Mark-ups fall by about half a per cent, real wages rise, and private consumption rises – qualitatively matching the VAR evidence but smaller in magnitude. Government spending, like a preference shock, raises absorption and – through the negative income effect of unproductive public spending – labour supply. “In the economy with deep habits, firms reduce mark-ups by about half a per cent. The resulting expansion in labour demand is strong enough to offset the income effect on labour supply. As a result, real wages rise in equilibrium.” The consumption result is the striking one: “in the model without deep-habit formation (with or without superficial-habit formation), private consumption spending declines as government spending rises… Under deep habits, the negative income effect is offset by a strong substitution effect away from leisure and into consumption induced by the increase in wages associated with the fall in mark-ups.” Against the data, Galí, López-Salido and Vallés “document that for the U.S. economy, a 1% increase in government spending is followed by a persistent and significant increase in private consumption that peaks at over 0.25% after 10 quarters. Although the deep-habit model underpredicts the magnitude of the consumption increase, it is remarkable that it can overturn the prediction of standard neoclassical models of a negative relationship between public and private consumption.”

Good under demand shocks, where the superficial-habit model gets the sign wrong. The authors set up the test by appeal to the literature arguing that technology shocks explain little of the cycle – “Galí and Rabanal (2004) estimate that technology is responsible for about 15% of the variations in output and hours at business-cycle frequency,” and they point to demand factors as the main source of the output-labour comovement – so “a measure of the ability of any model to fit the data is its predictions regarding the comovement between labour and output, conditional on demand shocks being the main source of fluctuations.” Against an unconditional correlation between labour productivity and output of 0.34 reported by Cooley and Prescott, “our deep-habit-formation model predicts a correlation of 0.33 conditional on government-purchases shocks being the sole source of uncertainty and of 0.72 when only preference shocks are present. By contrast, the respective predicted correlations under the superficial-habit model are -0.1 and -0.85.” The conclusion drawn is appropriately hedged: the finding “suggests that the deep-habit model has the potential to offer a better explanation of salient business-cycle regularities.”

It isolates the price-elasticity effect – and shows that effect alone is not quantitatively enough. With subsistence points instead of habits, the price-inelastic term in demand “is exogenous to the firm. Consequently, the pricing problem of the firm is static and the intertemporal effect present in the deep-habit model ceases to exist,” making it “an ideal environment to study the price-elasticity effect in isolation.” Calibrated so that the steady-state mark-up is 32% as in the baseline, “for all three shocks considered, the mark-up behaves countercyclically in the good-specific subsistence-point model. However, in comparison to the deep-habit model, the mark-up movements are small. Indeed, the decline in mark-ups is insufficient to induce procyclical wage movements in response to demand shocks… Similarly, the good-specific subsistence-point model fails to deliver a procyclical consumption response, following an increase in government spending.” The explanation ties back to the level/elasticity trade-off: for a given steady-state mark-up and substitution elasticity, “the deep-habit model features a much larger price-inelastic component of demand,” and it can do so while keeping a realistic average mark-up “because the intertemporal effect pushes the long-run mark-up down.”

The intertemporal effect, whose sign then depends entirely on which component of demand a shock hits. Under relative rather than additive deep habits, all terms in the demand function are price elastic, so “the price-elasticity effect of deep habits, stemming from the presence of purely inelastic terms, is also absent.” What remains is the intertemporal effect, and its behaviour “is dominated by changes in the composition of aggregate demand,” because investment demand is not subject to habit formation. Ranking the relevant steady-state mark-ups – the one where all demand components are habit-forming, the actual one, and the one with no habits, in increasing order – the authors conclude that “because both the preference shock and the government-spending shock amplify the importance of the habitual demand for goods, firms have an incentive to lower the mark-up,” while “in response to a positive productivity shock that enlarges the component of aggregate demand not subject to habit formation, firms will have a tendency to raise the mark-up.”

External habits are adopted for analytical convenience; making them internal destroys the time consistency of the firm’s pricing problem. The external assumption “preserves the separation of the problem of choosing total consumption expenditures over time from that of choosing expenditures on individual varieties of goods at a given point in time. Under deep internal habits, such separation is broken.” More importantly for the supply side, under internal deep habits “current demand for a particular variety depends not only on its current relative price but also on all future expected relative prices. This difference dramatically changes the nature of the firm’s problem. In particular, under internal deep habits the firm’s problem is no longer time consistent, as in each period the monopolist has the incentive to renege from price promises made in the past.” The authors note the price-elasticity intuition still suggests a procyclical elasticity in that case, but state that “a full analysis of the optimal pricing behaviour under alternative economic environments (e.g. commitment or discretion in its many different forms such as Markov-perfect equilibria, or reputational equilibria) is beyond the scope of this paper and is left for future research,” and in the conclusion call it “perhaps, the most relevant next step in this research programme” – noting also that “there is vast empirical support for the hypothesis of rational addictive behaviour at the level of individual consumption goods.”

The two make distinguishable predictions, and the sticky-price model’s mark-up movements are far smaller. Comparing against Schmitt-Grohé and Uribe’s calibrated Calvo-Yun model with superficial habits and a Taylor rule: “in response to a positive productivity shock the deep-habit model predicts a decline in mark-ups, whereas the sticky-price model predicts an increase in mark-ups. Further, in response to a positive government-spending shock, consumption and real wages fall in the sticky-price model, whereas in the deep-habit model both variables increase. Like the deep-habit model, the sticky-price model predicts a decline in mark-ups in response to an increase in public spending. However, the movements in mark-ups implied by the sticky-price model are quantitatively negligible (of the order of 0.01% in response to a 1% increase in public spending).” The authors are careful to add that “doing justice to a comparison of the predictions of the deep-habit and sticky-price models lies outside the scope of the present study,” and point to a companion paper for the combined sticky-price-plus-deep-habits Phillips curve.