Asset Bubbles and the Cost of Economic Fluctuations

: Lucas (1987, 2003) estimates that the cost of (cid:135)uctuations is less than 0.1% of consumption. In other words, a social planner would pay no more than 0.1% of (permanent) consumption to eliminate all future business cycle (cid:135)uctuations. The current paper extends Lucas(cid:146)calculations by studying the costs of (cid:135)uctuations arising from asset bubbles. We estimate two classes of costs: consumption volatility due only to asset price volatility, and consumption volatility due to asset trading interacted with price volatility. We show that the magnitude of these welfare costs is driven by heterogeneous household portfolios. If assets are held proportionately across the population, the welfare costs fall by an order of magnitude. Our benchmark calibration, which assumes a coe¢ cient of relative risk aversion of 3, implies that the asset bubbles of the last decade generated a social welfare cost equal to a permanent 3 percent reduction in the level of national consumption. Our calculations are sensitive to the details of the calibration, including the degree of balance sheet and trading heterogeneity, the coe¢ cient of relative risk aversion, and the magnitude of the asset bubble. Our speci(cid:133)cations with reasonable parameter values generate welfare costs ranging from 1 to 10 percent of (permanent) national consumption.


Introduction
Consumption is approximately the annuity-value of wealth, so volatile wealth dynamics can generate volatile consumption dynamics. 1The current paper models and calibrates the linkages from asset price movements to consumption and estimates the consequences of asset bubbles for social welfare.
Because of active market timing, lifecycle portfolio adjustment, or other trading motives, some households are net buyers of assets and some households are net sellers.Such transactions produce inter-household transfers when asset prices deviate from fundamentals.We refer to such inter-household transfers as an asset trading e¤ect.
Households that own bubble-priced assets perceive that they are wealthier than they actually are.Consequently, they raise consumption during a bubble period: i.e., borrow more and lower active savings.When asset prices eventually return to their fundamental values, these households need to reduce their consumption to re ‡ect their new net worth.This consumption reduction necessarily overshoots the initial consumption increase, since the households need to implicitly "pay back"the fraction of consumption during the bubble years that is discovered (ex-post) to be above the annuity value of their assets.In other words, during the bubble the agent overconsumes relative to the true annuity value of wealth.After the bubble, the agent correspondingly underconsumes relative to the pre-bubble annuity value of wealth.We refer to this dynamic pattern as a consumption boom/bust e¤ect.
Asset trading e¤ects and consumption boom/bust e¤ects jointly generate excess consumption volatility.In the current paper we model these e¤ects and calculate the resulting welfare costs.We provide exact calculations and we provide Taylor approximations that express these e¤ects with simple closed form equations.Four key expressions emerge from the Taylor approximations.
First, there is an asset trading welfare cost.The welfare cost arising from this channel -expressed as a fraction of permanent consumption -is given 1 An important exception is the case in which wealth ‡uctuations are driven by interest rate ‡uctuations.For evidence on the propensity to consume out of variation in housing wealth, see Greenspan andKennedy (2007, 2008) and Bosworth and Smart (2009).Carroll et al. (2006) and Campbell and Coco (2007) estimate a housing-wealth MPC of nine percent.by In this expression, is the coe¢ cient or relative risk aversion, is the magnitude of the asset bubble as a fraction of the fundamental value of the capital stock, is the annual discount rate, and N is the duration of the bubble (years).Finally, k i is the net value of capital (using pre-bubble prices) purchased by household i during the bubble; and W i is the total net worth of household i (including human capital).The asset-trading e¤ect is increasing in risk aversion, in the square of the magnitude of the asset bubble, in the cross-sectional variance of normed asset trading, and in the growth of the bubble.In our benchmark calibration, the asset trading e¤ect represents a welfare cost that is equivalent to 1.6% of consumption.The asset trading e¤ect has no …rst-order consequences for welfare, since every gain from selling the overpriced asset is o¤set by a loss (in another household) from buying the overpriced asset.However, second order e¤ects don't cancel.Concavity in the utility function causes the gains in marginal utility to be more than o¤set by the losses in marginal utility.Finally, the asset trading e¤ect vanishes as heterogeneity is reduced -i.e. as V ar k i W i ; the cross-sectional variance of asset purchases, goes to zero.
Second, we identify a boom/bust e¤ect.The welfare cost arising from this channel -expressed as a fraction of permanent consumption -is given by The new variables include: p, the fraction of households that hold zero claims to the aggregate capital stock; K, the aggregate stock of capital; K i , the net claim to the capital stock of household i; and W , the aggregate net worth of the agents in the economy (including human capital).The boom/bust e¤ect is increasing in risk aversion, in the square of the magnitude of the asset bubble, in the concentration of holding of the capital stock, in the square of the normed size of the capital stock, in the cross-sectional variance of normed capital holding, and in the growth of the bubble.In our benchmark calibration, boom/bust e¤ects account for welfare costs equal to 1.3% of permanent consumption.
The boom/bust e¤ect also has no …rst-order consequences, since every …rst-order gain from overconsumming (relative to the true annuity value of wealth) during the bubble is o¤set by a …rst-order loss from underconsuming (relative to the pre-bubble annuity value of wealth) after the bubble bursts.However, second order e¤ects again survive due to concavity.Finally, the boom/bust e¤ect drops by an order of magnitude as heterogeneity is reduced -i.e. as p goes to zero and as V ar ln K i W i K i > 0 goes to zero.At this homogeneous limit, the boom bust e¤ect is 0.5% of consumption in our benchmark calibration.Hence, heterogeneity is also a key contributor to this e¤ect.
Third, we identify a covariance e¤ect, which arises because of interactions between the previous two e¤ects.
The covariance e¤ect is increasing in risk aversion, in the square of the magnitude of the asset bubble, in the cross-sectional covariance of normed capital holding and normed capital trading, and in the growth of the bubble.In our calibration the covariance e¤ect represents a cost that is equivalent to -0.6% of permanent consumption.In other words, this third e¤ect turns out to partially o¤set the other two.Mean reversion in household portfolio allocations induce a negative covariance between K i W i (household i's original holdings of the capital stock) and k i W i (household i's net purchases of the capital stock).This negative covariance reduces the welfare costs.Intuitively, households that tend to allocate the greatest portfolio share to domestic capital, K i ; when the bubble begins, and hence are likely to raise their consumption the most during the bubble, are also the households that probabilistically sell the most capital during the bubble period, thereby partially cushioning the fall in consumption when the bubble bursts.The covariance e¤ect vanishes in the homogeneous case.
Fourth, we identify an aggregation e¤ect, which arises since the social welfare cost is not a linear weighted average of the individual welfare costs.Speci…cally, because the utility function is concave, welfare costs aggregate in a way that overweights the households with the largest welfare losses.Specifically, the social welfare cost is equal to the average value of the householdlevel welfare costs (the sum of the three terms above) plus an adjustment term for aggregation: where i is the welfare cost of household i: Since E i is close to zero, this aggregation e¤ect is essentially =2 times the variance of the individual households'welfare costs.In our calibration this aggregation e¤ect is equivalent to 1.4% of permanent consumption.This aggregation e¤ect also vanishes in the homogeneous case.
The paper proceeds as follows.In Section 2 we review the related literature.In Section 3 we describe our model.In Section 4, we work out the welfare costs of an asset bubble, using both exact methods and a secondorder Taylor expansion.In Section 5, we calibrate the model, including a discussion of micro-level data from the Heath and Retirement Study and the Survey of Consumer Finances.In the empirical analysis, we provide a new distributional result: the ratio of equity+housing wealth to total net worth is well-approximated in the cross-section by a log-normal distribution plus mass at zero.Section 6 presents our welfare cost results -both exact results and approximations.Section 7 discusses potential extensions of our model and section 8 concludes.

Literature Review
Real Business Cycle models imply that policy-makers should not adopt countercyclical policies, since ‡uctuations are optimal responses to changing fundamentals (e.g., Kydland and Prescott 1982;Prescott, 1986).2However, some economic models do imply that policy-makers should adopt counter-cyclical policies.Lucas (1987) calculates an upper bound for the bene…ts of such counter-cyclical policies.Lucas considers a representative agent with constant relative risk aversion.Lucas models consumption as a log linear trend with noise: Here A is a scaling parameter, g is the long-run growth rate, " is an iid Gaussian random variable with standard deviation .This implies that the welfare cost of ‡uctuations is approximately (1=2) 2 ; where is the coe¢ cient of relative risk aversion.Lucas uses US data from the period after World War II to calibrate the model and to estimate how much the representative agent would be willing to give up as fraction of consumption in order to set = 0. Lucas shows that, for reasonable values of the coe¢ cient of risk aversion, the welfare cost of economic ‡uctuations is very low, in fact not more than 0.1% of consumption. 3he most frequent critique of this result assumes heterogeneous agents and incomplete markets.Such assumptions can raise the welfare costs relative to Lucas'estimates, but the results are mixed and the average welfare cost is still generally estimated to be no more than 1% (e.g., Atkeson and Phelan 1994, Imrohoroglu 1989, Krusell et al., 2009, Krusell and Smith, 1999, Krebs, 2007, Mukoyama and Sahin, 2006and Storesletten et al., 2001).

Model
We analyze a continuous-time, small open economy that faces …xed world factor prices (cf.Laibson and Mollerstrom, 2009).Heterogeneous households, with an index i that is temporarily suppressed, maximize an exponentially weighted integral of utility ‡ows: Here is the exponential discount rate, K t is domestic capital, Y L t is …xed labor income, C t is consumption, r is the real interest rate, D t is net foreign debt (so D t is net foreign assets), and B t is net domestic debt.Across households, B t adds up to zero.We assume that r = , which is a standard steady state restriction.Finally, we assume that households have constant relative risk aversion, .
We assume that an asset bubble begins an instant after date t = 0 and ends at date N .In other words, the bubble exists when t 2 (0; N ]: The pre-bubble state of the economy will be the benchmark to which we keep referring.Hence, we adopt special notation for all variables at date zero.Speci…cally, whenever we drop the time subscript, we are implicitly referring to the date 0 (pre-bubble) value of the variable.For example, we let C = C 0 : We assume that the asset bubble immediately raises the notional value of …xed capital by an increment K: We conceptualize the bubble as the discounted value of productivity gains that are anticipated to occur N periods away. 9Agents expect a unit of domestic capital to yield returns of r from date 0 to date N; and returns of thereafter.So a physical unit of domestic capital has price (1 + ) when the bubble begins an instant after date t = 0: The marginal propensity to consume out of wealth is r; so that the K increase in notional wealth leads consumption to rise by r K: Since r = ; pre-bubble consumption is equal to the annuity value of wealth: Bubble consumption is the annuity value of bubble-inclusive wealth: Recall that assets appreciate at rate r; so throughout the bubble period households hold wealth with notional value K + K D B: Capital gains and dividends are exactly o¤set by consumption.
9 These productivity gains need to be anticipated to occur in the future to enable the bubble to persist in the meantime.The bubble bursts on the date that the anticipated productivity gains fail to be realized.In this way our model is similar to that of Beaudry and Portier (2007) where news about the future changes expectations in a way that impacts consumption, investment and growth.Even more relevant is Christiano et al (2007), which investigages what happens when such expectations turn out to be wrong.
We also allow our agents to trade assets.This will only matter during the bubble period: agents who buy domestic capital will be harmed (since the asset is over-valued) and agents that sell domestic capital will bene…t.
To track this trading, we introduce a new variable k; which represents the net change in the physical units of bubble asset accumulated by an agent.Negative values of k represent a net reduction in the physical units of the bubble asset.If we norm the pre-bubble real price of the bubble asset to 1, then an agent who pays 1 + dollars to buy domestic capital during the bubble period will take possession of 1 extra physical unit of K: For this illustrative example, k = 1: Since domestic capital is only held by domestic agents, it follows that the average value of k; across all households, is 0.
Agents who buy domestic capital (during the bubble period) are left with a rude shock when the bubble bursts.They experience an additional capital loss exp(rN ) k at the time the bubble bursts: This is the additional capital loss that arises from acquiring k physical units of domestic capital.Without loss of generality we assume that domestic agents …nance purchases of incremental domestic capital with domestic borrowing.Likewise, a domestic agent who sells an incremental unit of domestic capital uses the proceeds of this sale to make domestic loans.For example, an agent who buys k units of physical capital at date 2 [0; N ) borrows B B = [1 + exp(r ) ] k on the domestic market to do so.We assume that the agent uses her income from k to …nance this new domestic debt.However, this income is not su¢ cient to …nance all of the new domestic debt, since the period of higher productivity has not yet arrived.We assume that the residual domestic debt is rolled over, so the domestic debt is equal to During the bubble period, households have a gap between their desired level of consumption C 0 = Y L + r(K D B) + r K and the level of physical income from domestic assets Y L + r(K D B): The gap, r K; is borrowed as a ‡ow from abroad. 10The resulting change to the trade de…cit (which is the same as the initial change to the current account de…cit) is r K: since the k 0 s average out to zero.
The trade de…cit continues at this level throughout the bubble period.By contrast, the current account de…cit grows, since the foreign debt is growing: households must also pay interest on the accumulating shortfalls.Integrating these ‡ows yields the net accumulation of foreign debt during a sub-period [0; ] of the bubble period (where N ): So the (change in the) current account de…cit from date 0; an instant before the bubble starts, to date < N; is During the bubble, assets at the household level can be decomposed into domestic assets valued at and debt to domestic agents valued at When the bubble bursts (at date N ), the household is left with net assets: In other words, consumption falls by We can decompose this into three e¤ects: Accumulation e¤ect : The direct bubble e¤ect is the reversal in the initial consumption boom.The trading e¤ect is the additional reduction in wealth associated with loss (or gain) on assets that have been acquired during the bubble period.The accumulation e¤ect is the consequence of growing the bubble at rate r over an interval of length N: As the duration of the bubble goes to zero, the accumulation e¤ect ceases to matter: lim N !0 exp(rN ) = 1:

Distributional assumptions
We now study inter-household di¤erences.Consequently, we will start using household subscripts.We assume that the initial distribution of capital follows a two-part distribution.A mass p of consumers have K i = 0: The remaining mass 1 p has a log normal distribution of K i levels (see the calibration section for empirical evidence that validates this assumption).Speci…cally, ) with probablity 1 p where " i is normally distributioned with mean zero and variance 2 " .Moreover, " i is independent of C i and iid across households.This implies that, We will exploit this relationship when we calibrate the economy in section 5.In that calibration, we also truncate the right-hand-tail of the log-normal density to prevent extreme welfare costs for households with large positions in the bubble asset.Finally, we need to characterize the distribution of trades, k: To do this, we …rst characterize the Markov process that relates K i to K 0 i : Suppose that at each iteration of this Markov process, a fraction of the population 1 stays at their old level of domestic capital, and the remainder adjusts their domestic capital.For simplicity, we assume that the adjusting households are randomly dropped into the same ergodic distribution.Speci…cally, We use this ergodic assumption to (numerically) back out the distribution of the transaction variable 4 Welfare calculations We …rst characterize the welfare of a single agent in this economy.We then show how to aggregate these agents into a social welfare cost.
An individual agent has consumption of during the bubble and after the bubble.So utility is given by We can compare this to the counterfactual of a bubble-free economy.Without the bubble, lifetime utility would have been

Exact welfare calculations
We …rst provide an exact calculation of the welfare costs engendered by the bubble.As noted before, we are only measuring the welfare costs associated with ‡uctuations (and ignoring any welfare costs associated with the loss of resources or output).For any individual agent, we can solve for the factor i that equates the welfare they received as a result of the bubble epsiode and the welfare they would have received had they instead simply scaled their pre-bubble consumption by i : The implicit equation for i is given below.
Rearranging this expression, yields a closed-form expression for i : " We can also relate the individual i coe¢ cients to an aggregate that has the property that if every agent's consumption pre-bubble were uniformly scaled down by ; then the equilibrium path welfare (with the bubble) would equal the counterfactual welfare resulting from scaled consumption (without the bubble).More formally, is given by the following equation: Constant relative risk aversion implies that When all of the households are ex-ante identical, so that C i = C j for all i; j pairs, this reduces to

:
To gain intuition for this magnitude, it is helpful to rewrite it as (3) It follows that is the certainty equivalent 'consumption'of an 'agent'with constant relative risk aversion and stochastic 'consumption' i :

Taylor Approximation of Welfare Cost
We now use a second-order Taylor expansion to calculate the welfare loss from the asset bubble.We want to …nd such that We are interested in solving for which is the permanent percent reduction in pre-bubble consumption that produces a reduction in welfare equivalent to the experience of the bubble.Hence, We expand the argument of the utility function around C = Y L + r(K D): Since this expansion is algebraicly intensive and conceptually routine, we provide the details in the appendix.For an individual household the second-order expansion yields, We can also integrate across households to produce an average value for i : This derivation exploits the fact that the average value of k i is zero.The average welfare cost, This can be broken down into three terms, using the fact that C i = rW i : First, there is a consumption boom/bust e¤ect: Second, there is a covariance e¤ect: Third, there is an asset trading e¤ect: Finally, we need to aggregate these e¤ects.As already noted, aggregation is not linear.Concavity implies that the aggregate is the certainty equivalent of the individual i (cf equation 3).Using a second-order Taylor expansion, In other words, the aggregate social welfare cost is the sum of equations 4, 5, and 6, plus an aggregation e¤ect, 2 Since E i is close to zero, this aggregation e¤ect is essentially =2 times the variance of the individual households'welfare costs.
We calibrate and compare our measures of welfare loss in the next section of the paper.

Calibration of the model
For tractability, we study the steady state, in which r = = 0:05.Our baseline value for the coe¢ cient of relative risk aversion is = 3.We also consider CRRA values from 1 to 5.
We need to calibrate the (plausible) magnitude of the U.S. asset bubbles.Figure 1 plots the U.S. ratio of household wealth minus government debt4 all divided by GDP. 11This series ‡uctuated historically in a range roughly between 2.5 and 3 units of GDP.Starting in the late-1990's, however, the series broke from this historical range and rose sharply.At its peak in the 4th quarter of 2006, the series reached a value of 4.1 units of GDP.By the 4th quarter of 2008, the series had fallen back to its historical range.These comparisons imply an estimated peak bubble value of about 1.1 units of GDP.However, the ratio of household wealth to GDP misses part of the value of the bubble, since it nets out the value of debt accumulated to …nance consumption during the bubble years.Household debt increased from 0.3 units of GDP from the late 1990's to 2006:4.This analysis implies that had U.S. households not consumed some of their bubble wealth, the ratio of economy-wide net worth would have risen 1.65 units of GDP.Since, For this calculation we take g = 0:02; r = 0:05; Here g is the real growth rate in GDP.This calculation implies that the total value of the bubble is 0.33 of the value of all domestic capital (including land).To err on the side of conservativism, we set = 0:3 in our benchmark calibration.
We calibrate the distribution of K i from the Health and Retirement Study (HRS) and the Survey of Consumer Finances (SCF).Though the HRS has the limitation that it only covers respondents who are middle-aged or older, the HRS has an o¤setting advantage.Respondents are surveyed longitudinally every two years, whereas the SCF is purely cross-sectional. 5 First, we use the HRS to construct a household-level estimate of K i W i ; where W i is total household resources, including an estimate of the value of human capital. 6In our model it should be the case that This procedure is explained in our second appendix.Figure 2 plots the distribution of K i W i ; for seven di¤erent waves of HRS surveys (starting in 1992/93 and ending in 2006). 7The distributions all have two distinct components.First, there is substantial mass at zero (about 15% of the HRS households have no K assets).Second, a log-normal distribution …ts the data that is greater than zero.To con…rm this second parametric property, we analyze the households with K > 0, and plot the natural log of their K i W i ratios.Figure 3 plots these distributions for the same seven waves of HRS surveys.Figure 3 also superimposes a Gaussian density to con…rm our parametric assumptions.In our actual simulations (for exact calculations of welfare losses), we truncate the log normal distribution so that K i C i 30: Furthermore, we distinguish between the mass-at-zero and greater-than-zero sub-distributions by partitioning the data at the 18th (for the HRS waves) and 30th (for the SCF waves) percentiles.This is done to rule out extreme welfare losses (for households with extreme values of K): Our natural log plots from the HRS have associated standard deviations 5 The HRS asset data is of higher quality than the asset data in the Panel Survey on Income Dynamics. 6Both the HRS and the SCF bias down this ratio by collapsing private businesses into a net equity summary statistic.For example, a family business with $1 million of assets and $800,000 in debt would be recorded in these data sets as a private equity position of $200,000.This understates the household's leverage. 7The data in Figures 2 and 4 are displayed between ratio values of 0 and 1.5.In Figures 3 and 5 we display all data points with log ratio values within 2.5 of the sub-distribution mean.(f) Mean = -1.2709,SD = 0.5617 that range from 0.72 to 0.82 (depending on the wave of the HRS).In Figures 3 and 4 we plot analogous …gures generated by the SCF.The SCF plots in Figure 4 have standard deviations that range from 0.56 to 0.72.We therefore adopt a benchmark standard deviation, " ; of 0.70 for our calculations.
The U.S. Census Bureau reports that the household homeownership rate ranged from a low of 65.9% (1998q1) to a high of 69.2% (2004q2) during the bubble period.We assume that p = 0:30: i.e. 30% of households own neither a home nor equity.
Finally, we turn to asset ownership dynamics.We assume that 1 = 0:5 of the households did not change their physical claims K i during the duration of the bubble period.The remaining households (mass ) trade their claims during the N -year bubble period, replacing their initial ratio K i =W i with a new iid draw from the ergodic distribution of K i =W i : This assumption produces a simulated correlation between K i =W i and K 0 i =W i ; of 0.50 (a numerical coincidence).Note that K 0 i =W i is unit claims to domestic capital of household i at the end of the N -year bubble divided by initial net worth of household i.The actual empirical correlation (using the HRS) is much lower: 0.26.If we raised to match this empirical correlation, our imputed welfare costs would be even higher (since more trading increases the magnitude of the asset trading e¤ect).However, we believe that the low empirical correlation is partly due to measurement error in the HRS.For this reason we would be biasing our imputed welfare costs up if we picked a value that was high enough to match the empirical correlation of 0.26.

Results
Table 1 reports our benchmark calibration values for the key parameters that we will vary.Table 1 also reports a low/high range for each variable.Table 2 reports three additional variables that we will hold …xed in all of our simulations.3 reports welfare costs using the benchmark values and varying each variable independently.Several properties stand out.First, our social welfare costs are typically one to two orders of magnitude larger than Lucas' welfare costs.However, this comparison is misleading, since our welfare costs are not discounted.Our calculations derive the welfare evaluation from the instant before the bubble begins.In the next section, we generate a timeless perspective with a recursive argument.
Second, our welfare costs are highly sensitive to the calibration values of the key parameters.The size of the bubble turns out to be particularly important.For example, a bubble equal to 40% of the value of the capital stock is cataclysmic.
Third, the Taylor approximation is usually signi…cantly larger than the exact calculation.So the Taylor expansion should only be used as a pedagogical tool and not as a close numerical approximation.
Fourth, the asset trading, boom/bust, and aggregation e¤ects are usually of similar magnitude.
Fifth, the covariance e¤ect is about half as large (and of opposite sign) as the other e¤ects.

The timeless perspective
To make our calculations directly comparable to those of Lucas (1987Lucas ( , 2003)), we now analyze a timeless perspective.Let V be the value function of an economy with a constant probability of entering a bubble event and a welfare loss of 1 fraction of permanent consumption in the event of a bubble.Then,  2.0% †For the benchmark case, the coefficient of relative risk aversion is γ=3, the bubble is ζ=0.3 proportion of the value stock of physical capital, the duration of the bubble period is N=10 years, the cross-sectional household-level standard deviation of the ratio of K/C (for K>0) is sigma=0.7, the fraction of households that trade in the N year bubble period is φ=0.5, the interest rate and discount rate are both equal to 0.05, the aggregate K/C ratio is 5, and the fraction of households with no claims to the capital stock is 0.3.
Let 1 represent the timeless social welfare cost of having a stationary likelihood of entering a bubble event.Then, If = 0:05; = 0:02; and = 0:97; then 1 = 0:987: Hence, the timeless welfare cost of bubbles is 1.1% of permanet consumption.Moreover, this calculation overlooks the fact that welfare costs are convex in the magnitude of the bubble.Even a tiny chance of a bubble that is signi…cantly larger than = 0:3 will have potentially large e¤ects on the timeless welfare cost.Recall that a bubble of magnitude = 0:4 implies a welfare cost of 33% of permanent consumption.If bubbles of this magnitude are added to the distribution of bubbles, and bubbles of this magnitude occur with a probability of only 1/1000, then the total timeless welfare cost doubles to 2.3% of permanent consumption.
8 Conclusion Lucas (1987Lucas ( , 2003) ) estimates the welfare costs of economic ‡uctuations and bounds them above at only 1/10 of 1% of permanent consumption.Motivated by his analysis, we estimate the welfare costs of an asset bubble.Our model identi…es two types of welfare costs: consumption ‡uctuations due to active asset trading and consumption ‡uctuations due to passive asset ownership.
To calibrate the model we use the Health and Retirement Survey and the Survey of Consumer Finances.With a coe¢ cient of relative risk averion of 3 (our benchmark assumption) the asset bubbles of the last decade have a welfare cost equal to 3% of permanent consumption.With calibrated values for key parameters, the welfare costs generally lie between 1% and 10% percent of permanent consumption.Even from the timeless perspective, the welfare costs in our model are greater than 1% of permanent consumption.
Our model predicts that asset bubbles should give rise to increased consumption inequality in their wake. 8This prediction needs to be tested.
Our model also has several policy implications.Our results imply that the welfare costs of bubbles are highly convex.This suggests that policy makers consider leaning against the wind only in the presence of large asset price movements that may be bubbles.A small policy intervention is likely to have small (second-order) welfare costs if no bubble is present, and enormous (…rst-order) welfare gains in the presence of a large bubble.Our results imply that it is not important to eliminate asset bubbles, just to reduce or constrain their size.

Appendix A
We now use a second-order Taylor expansion to calculate the welfare loss from the asset bubble.We want to …nd such that We expand the argument of the utility function around C: Hence, These expansions imply that We ignore terms in 2 : Adding this equation up across all agents, the …rst order terms vanish.The households that lose are exactly o¤set by the households that gain (in …rst order terms).(1 p)

#
The …rst term in brackets follows from Appendix B We estimate the ratio of ownership of bubbly assets to total net worth, Here, E represents equity and A = E + (A E) are total …nancial assets.L represents …nancial liabilities, including mortgages.Physical assets are denoted by H (housing) and Z (all other assets, e.g.vehicles).K h , human capital, is the net present value of future income (excluding capital income).
To calculate this ratio we use data from the Health and Retirement Survey (HRS) and the Survey of Consumer Finances (SCF).
Our unit of analysis for both the HRS and SCF is a single household, which means that ownership and income values are summed over the primary respondent and, where applicable, the respondent's spouse.9To calculate the above ratio, we use the following procedure.First, all variables except human capital are constructed from raw survey variables as detailed below.They are then converted into real terms using CPI data from the Department of Labor. 10ince the HRS-data are in panel format, we calculate human capital at time t as where R is the discount rate, which we calibrate as 1.05.Within the survey timespan, we set Y L s equal to its realized value when possible.When data are missing for year s, but are available for s + 1 and s 1 we impute If a respondent is alive in the last survey wave, we project Y L s by setting it to the average of Y values recorded over the course of the survey, and

Table 1 :
Critical Parameters

Table 3 :
Calibrated social welfare loss

Table 3 :
Calibrated social welfare loss

Table 1 :
Composition of ratio components in the HRS Variable(s) Survey Variable Survey Variable Description E = HwASTCK Net value of stocks and mutual funds + HwABSNS Net value of businesses + HwAIRA Net value of IRA and Keogh accounts (only 100 age percent are counted as E) H = HwAHOUS Value of the primary residence + HwAHOUB Value of the secondary residence (not reported in 1996) + HwARLES Net value of real estate, besides primary and secondary residence A E = HwACHCK Net value of checking, savings, and money market accounts + HwACD Value of CDs, government savings bonds, and treasury bills + HwABOND Net value of bonds or bond funds HwITOT Sum of all income in household HwICAP Business or farm income, self-employment earnings, business income, gross rent, dividend and interest income, trust funds or royalties, and other asset income

Table 2 :
Composition of ratio components in the SCF Variable(s) Survey Variable Survey Variable Description E = NMMF Total value of directly held pooled investment funds + STOCKS Total value of directly held stocks + RETQLIQ Value of IRAs, Keoghs, thrift-type accounts, and accounttype pensions (only 100 age percent counted as E)