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The Distribution of Wealth and Consumption Patterns in the UK

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Published: 11th Dec 2019

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The Distribution of Wealth and Consumption Patterns in the UK


Using Wealth and Assets Survey data, I calibrate an overlapping-generations life-cycle model to match the UK wealth distribution and provide structural estimates of UK consumption patterns over the life-cycle that are consistent with the empirical evidence. Furthermore, I estimate that the aggregate consumption response to fiscal stimulus targeted at households in the bottom wealth quintile is two to six times larger than to stimulus targeted at households in the bottom income quintile. A by-product of my analysis is a novel algorithm that almost halves the computation time of the calibration from 269 hours under standard methods to 137 hours.

Word Count: 7484


1 Introduction

Fiscal stimulus packages are one of the major policy tools used by governments to ameliorate weak or negative GDP growth. Governments also recognise that “the composition of fiscal stimulus is as critical as its size” (Group of Twenty, 2009). However, while fiscal stimulus programmes often target households according to their income, wealth is overlooked. For example, the $831bn American Recovery and Reinvestment Act of 2009, one of the largest peacetime fiscal stimulus programmes ever introduced (Congressional Budget Office, 2012), deliberately targeted stimulus at low-income households but had few provisions to target low-wealth households. This is surprising given both the level of wealth heterogeneity that we observe and the fact that wealth is more unequally distributed than income (Piketty, 2014). The lack of wealth-targeted fiscal stimulus in practice is matched in the literature; there is no research considering the relative effectiveness of wealth-targeted and income-targeted stimulus.1 This paper is a step towards filling this gap.

I calibrate an overlapping-generations life-cycle model with idiosyncratic income risk, hetero- geneous discount factors and liquidity constraints to match the UK distribution of wealth, using 2012-14 data from Wave 4 of the Wealth and Assets Survey (WAS4). The model is able to account for the full extent of UK wealth inequality and generates endogenous wealth heterogeneity through heterogeneous realisations of income shocks and precautionary and life-cycle savings motives that differ across households.

In this context, I make three contributions. First, I analyse how the aggregate consumption response to fiscal stimulus depends on the position of affected households in the wealth and income distributions. My results have two significant policy implications: (i) fiscal stimulus generates a larger consumption response when targeted at the unemployed or poor, whether by income or wealth and (ii) wealth-targeted fiscal stimulus is between two and six times more effective than income-targeted fiscal stimulus, depending on the specific policy. This is significant given that UK means-tested benefits make wealth-targeting possible and almost all fiscal stimulus programmes target income rather than wealth.

Second, I provide structural estimates of UK consumption patterns over the life-cycle and show that these are consistent with the empirical evidence on household consumption and saving. These estimates provide insight into the life-cycle dynamics of wealth accumulation through the precautionary and life-cycle savings motives: young households accumulate wealth in

1Japelli and Pistaferri (2014) consider income-targeting and cash-on-hand-targeting, but not wealth- targeting.


their 20s to self-insure idiosyncratic income risk, and households begin to accumulate wealth for retirement from their late 40s.

Finally, I introduce a new algorithm for solving single-constraint calibration problems, the Informed Calibration Algorithm (ICA). I prove that ICA is as robust as the primary method currently in use, Brent’s Method (Brent, 1973). I also show that ICA is simple to implement and approximately halves the calibration time of the model from 269 hours to 137 hours.2

1.1 Relation to the Existing Literature

Research linking consumption patterns and wealth accumulation has focused on the US (Gourinchas and Parker, 2002; Cagetti, 2003). However, given the large difference in capital- to-income ratios between the US and UK (Piketty and Zucman, 2014), it is possible that UK wealth accumulation dynamics differ from those in the US. Hence, the strength and timing of

the precautionary and life-cycle savings motives in the UK require further exploration.

However, UK data on wealth are “seriously incomplete” (Crossley and O’Dea, 2016) and hence econometric analysis is impossible. This is because cross-sectional data cannot be used to analyse life-cycle consumption behaviour (Banks et al., 1994) and WAS, the only UK panel dataset on wealth, contains insufficient data on household consumption (Alvaredo et al., 2016). Consequently, I adopt a structural modelling approach.

However, the majority of models that attempt to account for the distribution of wealth suffer from two key issues. First, they fail to generate a realistic distribution of wealth, substantially underestimating the level of wealth held by the top 5% (De Nardi, 2015). This is a problem with both the Bewley (1977) model and subsequent heterogeneous-agent incarnations such as the Krusell and Smith (1998) model. Second, they make implausible assumptions. For example, the highest productivity workers in Castaneda et al. (2003) have a risk that exceeds 20% of being 100 times less productive in the next period (De Nardi, 2015). Furthermore, Carroll et al.’s (2016) infinite-horizon specification relies on patient households being very close to the boundary at which they target infinite wealth-to-permanent-income ratios, which is not a sensible explanation of wealth accumulation.

To address these issues, I use a canonical life-cycle model with idiosyncratic income risk in the style of Gourinchas and Parker (2002), follow Cagetti (2003) in introducing liquidity constraints and model discount factor heterogeneity in the style of Carroll et al. (2016).

2Brent’s Method and ICA were run on a MacBook Air with a 1.4GHz Intel Core i5 processor and 4GB of 1600 MHz DDR3 RAM.


These features all serve to augment wealth heterogeneity, remain plausible and give rise to consumption patterns that match the empirical evidence (Fernandez-Villaverde and Krueger, 2007).

Furthermore, there is a large literature regarding the role of the income distribution (Parker et al., 2013; Broda and Parker, 2014), liquidity (Agarwal and Qian, 2014; Kreiner et al., 2016) and employment status (Japelli and Pistaferri, 2014) in determining the consumption response to fiscal stimulus. However, few authors consider the role the wealth distribution has to play in determining the efficacy of fiscal policy.

Some of the existing literature has yielded counterintuitive results. In particular, Kaplan and Violante (2014) fail to find a strong relationship between wealth and the marginal propensity to consume out of transitory shocks to income (MPC), contrary to empirical evidence from Japelli and Pistaferri (2014) that the MPC is decreasing in levels of cash-on- hand. Indeed, recent work by Carroll et al. (2016) estimates that the MPC is declining in wealth percentile. Consequently, there is no consensus in the literature regarding the effectiveness of wealth-targeted fiscal policy. Moreover, no studies have been conducted that examine the relative effectiveness of wealth-targeting and income-targeting. This paper is a step towards addressing these two questions.

Finally, it is common for economists to use existing root-finding algorithms such as Brent’s Method (Brent, 1973) or Newton-Raphson in solving calibration problems (Judd, 1998). For problems where the zeros of a function need to be found once or the zeros of many unrelated functions must be found repeatedly, these techniques are entirely appropriate (Nocedal and Wright, 1999; Wilkins and Gu, 2013). However, for repeated root-finding of similar functions, discarding all information from previous solutions is likely to be inefficient. I develop ICA to use information from previous solutions of similar problems adaptively, formalising ‘hot start’ root-finding methods. To the best of my knowledge, this is a novel contribution.

The paper proceeds as follows: Section 2 presents the model; Section 3 describes how I solve the model and presents ICA; Section 4 discusses my estimates of life-cycle consumption patterns; Section 5 examines the implications of my results for fiscal policy; Section 6 concludes.


2 The Model

The model follows an overlapping-generations life-cycle specification similar to that of Gour- inchas and Parker (2002), Cagetti (2003) and Carroll et al. (2016). Agents differ ex-ante due to levels of education, permanent incomes, age, probabilities of death and discount factors. Furthermore, agents differ ex-post due to particular realisations of idiosyncratic permanent and transitory shocks to income, for which there exist no insurance markets. Wealth heterogeneity therefore emerges because of different realisations of income shocks and heterogeneous life-cycle and precautionary savings motives.

Households. There is a continuum of households of measure one, who maximise expected lifetime utility without bequests. Their instantaneous utility is isoelastic:

where ρ ą 0.

& x1 ́ρ ́1, ρ‰1,

uipxq “

1 ́ρ
lnpxq, ρ“1.


Household i enters the economy aged 24 with discount factor βi
∆ determines the magnitude of discount factor heterogeneity. i is endowed with a level of education j P t1, 2, 3u, an initial level of permanent income Pi0 and a quantity of capital Ki0, which is the only asset in the economy. Draws of i’s level of education and βi are independent.

Households retire at age 65 after sR periods. Each period before retirement, if i is employed its income is taxed at rate τt and it receives after-tax income Yit. If i is unemployed, it receives income-tax-funded welfare benefits and pays no tax. Once retired, i receives a pay-as-you-go state pension.

When household i with level of education j transitions into period t after having lived for s periods, they face probability of death φjs. Households die with certainty by the age of 120, period s ̄. Each year consists of four periods and the assets of deceased households are taxed fully and spent outside of the model.

Before making its consumption decision, i receives a transitory geometric shock to income ζit: $

& θit , with probability 1 ́ u, ζit “

% μ , with probability u.



„ U rβ ́ ∆, β ` ∆s, where

i.i.d. 1 2
where lnθit „ Np1 ́u,σθsq and u is the aggregate unemployment rate. Furthermore, i

i.i.d. 2 receives two permanent geometric shocks to income: a white-noise shock ln ψit „ N p1, σψsq;

and a deterministic shock ψ ̄js. The income process for i can therefore be specified by: Yit “ p1 ́ τtEitqζitPit

Pit “ ψitψ ̄jsPit ́1
where Yit is income, Pit is permanent income and Eit is equal to unity if i is employed at

time t and zero otherwise.3
Finally, household i is subject to dynamic budget and liquidity constraints:

rate of capital and r is the time-invariant real interest rate. A household’s value function Vis is therefore given by:4

Cit `Kit`1 “p1`r ́δqKit `Yit Ait ě λPit

(1) where λ P R, Cit is consumption, Kit is capital, Ait represents total assets, δ is the depreciation

VispMit, Pitq “ max Et tCis`nus ̄ ́s ĎRs ̄ ́s`1

”s ̄ ́s ́n ̄ı ÿź

βin p1 ́ φjs`kq uipCis`nq s.t. p1q k“0

where Mit represents household i’s market resources at time t.

n“0 `

I follow Carroll’s (2011) method of eliminating permanent income as a state variable, reducing the solution time of the model. I define xitPit ” Xit, where Xit is generic. Furthermore, I define the normalised consumption and value functions as:

cispmitqPit ” CispMit, Pitq v pm qP1 ́ρ ”V pM ,P q

is it it is it it

The household’s constraints can be rewritten in normalised form as: mit`1 “ 1 ` r ́ δ pmit ́ citq ` p1 ́ τt`1Eit`1qζit`1

ψt`1 ψ ̄j s`1 (2) ait ě λ

3Retirement is captured by both the deterministic shock ψ ̄js, which assumes values below one after the retirement age, and setting the variances of permanent and transitory shocks to income to zero.

4Derivations of all results are available on request.


Thus, the value function can be redefined in terms of normalised variables for s ă s ̄:

! ” ı)

vpmq“max upcq`βp1 ́φqψ ̄1 ́ρEψ1 ́ρv pm q s.t.p2q

cit PR`

c ́ρ “max βp1 ́φ qψ ̄ ́ρ p1`r ́δqE ψ ́ρ c ́ρ ,pm ́λq ́ρ it i js js`1 t it`1it`1 it

and for s “ s ̄:

cit “ mit ́ λ

Intuitively, the LHS is the marginal utility of normalised consumption in period t, and, when the liquidity constraint does not bind, the RHS is the marginal utility of saving in period t. Furthermore, the liquidity constraint binds if the marginal utility of normalised consumption in the current period is greater than the marginal utility of saving (Deaton, 1991).

Macroeconomic Dynamics. There is population growth such that each cohort’s mass is a factor of 1 ` N larger than the previous cohort’s and technology grows at a constant rate Γ. Households are weighted such that there is, at all times, a unit mass of households.

Owing to the difficulty of accounting for aggregate state variables in the presence of overlapping generations, I follow the existing literature and ignore the presence of shocks in the aggregate (Gourinchas and Parker, 2002; Cagetti, 2003; Carroll et al., 2014). This is a limitation of the

model as I cannot investigate household behaviour over the business cycle.

Government. The government’s role is purely redistributive; the tax rate in the model is time-invariant and given by (Carroll et al., 2016):5

is it

i it i js js`1 t it`1 is`1 it`1

and for s “ s ̄:
Hence, household i’s Euler equation can be expressed in normalised variables for s ă s ̄:

vis ̄pmitq “ uipmit ́ λq !”ı)

”ı ř ̄řs ̄ “ ́j ́j␣śj ` ̄ ̆(‰

iPt1,2,3u θiPi0 j“sR p1 ` Nq p1 ` Γq k“0 ψikp1 ́ φikq τ“uμ`” ̄s ́1 j ̄ ı

ř řR“ ́j ́j␣ś` ̆(‰ iPt1,2,3u θiPi0 j“0 p1 ` Nq p1 ` Γq k“0 ψikp1 ́ φikq

where the θi are the share of the population of each level of education and P ̄i0 are the average

5A household with education level j that has lived for s periods is weighted by: wjs ” θjp1 ` Nq ́sp1 ` Γq ́s śsk“0p1 ́ φjkq. This weighting captures the proportion of each educa- tion level in the population, the fact that older cohorts are smaller, the proportion of each cohort that has not died and factors out the component of ψ ̄js that stems from technological growth.


initial levels of permanent income for each level of education. The intuition behind this formula is that the first term captures the tax rate needed to finance unemployment insurance payments and the second term captures the tax rate needed to finance pension payments.

3 Solving the Model

Using WAS4 data, I simulate the model and calibrate it by using ICA (see Section 3.2 and the Technical Appendix) and matching the model-generated and UK wealth distributions. In so doing, I estimate β and ∆ and find plausible values for both parameters. I perform sensitivity analysis and show that my estimates are robust to alternative parameterisations of the model.

3.1 MethodologySimulation. As there is no closed-form solution to the model, it must be simulated. This

consists of simulation of individual households and aggregation of those households.6

Households’ idiosyncratic shocks are drawn from discrete approximations to their log-normal distributions. I use the method of endogenous gridpoints to simulate households’ normalised consumption functions. This speeds simulation by computing only those expectations that are used in approximating normalised consumption functions and avoids the pitfalls of log-linearised solution methods (Carroll, 2006).

In aggregating simulated households, I follow Carroll et al. (2016). Individual households

represent population masses and are broken down into 30 types: for each of the three levels

of education, discount factors are drawn uniformly from the set tβ ̆ 2k`1 ∆u4 . For each 11 k“0

type, I simulate 2000 households for 384 periods. Hence, the simulated aggregate capital stock K is:

60000” s ̄ ́ n
ÿ ÿ ́n ́nź ̄


where jpiq is the education level of household i. Furthermore, aggregate output Y is approxi- 6Python code is available on request.

K “ θjpiqPi0 i“1

kinp1 ` Γq p1 ` Nq t ψisψjpiqsp1 ́ φjpiqsqu s“0



mated by:

”sR ́1 j ı ÿ ̄ÿ“ ́j ́j␣ź` ̄ ̆(‰

p1`Nq p1`Γq ψikp1 ́φikq k“0

Parameter Values. As few life-cycle models have been explicitly calibrated to match the UK economy, I introduce a set of parameter values that are consistent with the empirical evidence (see Table 1).

I construct the φjs by using the UK National Life Tables for 2013-15 (Office for National Statistics (ONS), 2016b) and adjusting by socioeconomic status using the National Statistics Socio-economic Classification and the most recent ONS data on mortality by socioeconomic status (ONS, 2017b). As is standard in the life-cycle literature, I assume that all households have the mortality characteristics of women (Carroll et al., 2016).

Average initial incomes and the ψ ̄js determining average earnings trajectories by education are taken from the 2013 Labour Force Survey (ONS, 2013) for 24-60 year-olds, where I take the values of ψ ̄js for 60 year-olds as the values for 61-65 year-olds. I take post-retirement earnings trajectories from the Department for Work and Pensions (2016) and extrapolate up to the age of 120. As the information is not available, post-retirement the ψ ̄js are not differentiated by level of education.

For the UK’s quarterly capital-output ratio, I divide the 2013 ONS estimate of the UK capital stock of £3.9 trillion (ONS, 2014) by one-quarter of 2013 real GDP (ONS, 2017c). This generates a capital-to-income ratio consistent with recent evidence on the UK (Piketty and Zucman, 2014).

The variances of transitory and permanent income shocks are taken from Carroll et al.’s (2016) extension to the estimates from Sabelhaus and Song (2010). These parameter values pertain to the US, rather than the UK. However, there are no UK estimates available of sufficient granularity to calibrate the model. This is not a significant issue as the majority of Sabelhaus and Song’s (2010) estimates fall within the confidence bands provided by Blundell

et al. (2013) for the UK from the late 1970s to the late 1990s.
For comparability to the existing life-cycle consumption literature, I set λ equal to zero and

in so doing suppose that households can never borrow, but can instantaneously liquidate

7I simulate 23 million population masses and so the approximation is almost exact. 9

Y « θiPi0 iPt1,2,3u

where the approximation is valid if the realised draws of ψis and ζis average to approximately

unity (Carroll et al., 2016).7



Table 1: Parameter Values in the Model Value

Gandelman and Hernandez-Murillo (2014) Holston et al. (2016)
Den Haan et al. (2010)
ONS (2016a)
Goodridge et al. (2014)
ONS (2011)
ONS (2011)
ONS (2011)
ONS (2013)
ONS (2013)
ONS (2013)
Den Haan et al. (2010)
ONS (2017a)
Author generated
Author generated
Cagetti (2003)
Author generated
Carroll et al. (2016)
Carroll et al. (2016)
Author generated


Coefficient of relative risk aversion
Real interest rate
Depreciation rate of capital
Population growth rate
Technological growth rate
Proportional of education level 1 households Proportional of education level 2 households Proportional of education level 3 households Average initial income, level 1
Average initial income, level 2
Average initial income, level 3
Unemployment insurance payment Unemployment rate
Labour income tax rate
Capital-to-income ratio
Liquidity constraint parameter
Growth factor of income by education and age Variance of transitory shocks by age
Variance of permanent shocks by age Probability of death by education and age

1 0.0037 0.025 0.00175 0.0030 0.23 0.50 0.27 3990 4430 5290 0.15 0.054 0.0937 9.75
On request9 On request On request On request








P ̄10

P ̄20

P ̄30





λ ψ ̄ j s

σ2 θs

σ2 ψs


8 All parameters are defined on a quarterly basis.

9 As the φjs, ψ ̄js, σ2 and σ2 comprise 3880 separate parameter values, they are available on request. θs ψs

all of their assets (Cagetti, 2003). Initial levels of capital are allocated such that household wealth-to-permanent-income ratios are drawn uniformly from the set t0.17, 0.50, 0.83u (Carroll et al., 2016).

Data. I use data on the net household wealth of the 20,200 households surveyed in the 2012-14 wave 4 of the longitudinal UK Wealth and Assets Survey (WAS4). This is because WAS4 provides the most recent and comprehensive data regarding the entire distribution of household wealth in the UK (Crawford et al., 2016). Using WAS4, I construct a Lorenz curve for net household wealth and use this to calibrate the model. To correct for different household sizes, I divide total household wealth by the number of adults in each household. Furthermore, to make the sample representative, each observation’s weight in constructing the curve is given by the relevant sampling weight in the WAS4 data.

However, there are two substantive issues with the WAS4 data. First, the survey’s definition of total household wealth includes net property wealth, net financial wealth, physical wealth and private pension wealth but includes neither business assets nor rights to state pensions (ONS, 2015). This is highly likely to lead to underestimation of wealth held by households in the upper ranges of wealth, who are more likely to own large amounts of business assets

(Alvaredo et al., 2016).

Second, non-response is likely to be higher among high wealth groups (ONS, 2009). WAS4 oversampled wealthier households using income tax records and FTSE350 dividend yields such that households likely to be in the 90th percentile of shareholding value were 2.5 to 3.0 times more likely to be sampled (ONS, 2012). However, this is unlikely to eliminate bias and Vermeulen (2014) estimates that WAS underestimates the share of the wealthiest 1% by between 1 and 5 percentage points.

As a result, WAS4 cannot “provide a fully satisfactory representation of the upper tail of the UK wealth distribution” (Alvaredo et al., 2016). To address this issue, my calibration does not target the wealth share of the wealthiest 1%. Hence, I avoid using the synthetic WAS data constructed by either the Credit Suisse Research Institute (2014) or Vermeulen (2014), which suffer from issues with the underlying data and imposing untestable distributional assumptions on wealth.


Estimation. I estimate β and ∆ by choosing them to minimise the sum of squared distances between the model-generated Lorenz curve and the Lorenz curve that I generate from the WAS4 data, subject to the constraint that the model economy’s capital-to-income ratio matches that of the UK:

ÿ ́K ̄K

tβ ̊, ∆ ̊u “ arg min pωipβ, ∆q ́ wiq2 s.t. β,∆ iPI



where I “ t20, 40, 60, 80, 90, 95u is the set of percentiles of the Lorenz curve that I seek to

match, ωipβ,∆q is the cumulative wealth share of the ith percentile in the model, wi is the

cumulative wealth share of the ith percentile in the WAS4 data, and `K ̆ is the model’s YM

quarterly capital-to-income ratio.

This approach augments Castaneda et al. (2003), who match a large set of wealth percentiles, with Carroll et al.’s (2016) constraint on the capital-to-income ratio which prevents an unreasonable level of wealth holding in the economy. It also avoids matching purely mean wealth (Gourinchas and Parker, 2002) and median wealth (Cagetti, 2003), approaches which suffer from their propensity to generate unrealistic wealth distributions.

3.2 The Informed Calibration Algorithm

I calibrate the model using ICA, a new algorithm I have developed for single-constraint constrained optimisation problems. ICA is as robust as the method currently in use, Brent’s Method (Brent, 1973), and performs 48.2% fewer iterations in calibrating the model, almost halving the calibration time from 269 hours to 137 hours. Technical details are available in Appendices A and B.

Consider the following general calibration problem:
tx ̊u “ argminfpx;yq s.t. gpx;yq “ 0


for n ě 2, where x “ px1,x2,…,xnq P Rn is a vector of parameters to be calibrated, y “ py1,y2,…,ykq P Rk is a vector of fixed parameters, f : Rn`k ÞÑ R is an objective function, g : Rn`k ÞÑ Rl is an l ˆ 1 vector of constraint functions and x ̊ P Rn is a vector of calibrated parameters.

Consider the class of single-constraint constrained optimisation problems (l “ 1).10 A 10For l ě 2, ICA can be generalised. However, owing to profound issues with multivariate root finding

(Judd, 1998), a number of other numerical constrained optimisation techniques are preferred.


current solution method observes that, given a value of the first n ́ 1 variables, xn must be chosen to satisfy the constraint. Hence, authors use a numerical minimisation algorithm to guess a vector px1, x12, …, x1n ́1q and use Brent’s Method to find x:n P I “ ra, bs to satisfy gpx1, …, x1n ́1, x:n; yq « 0. After evaluating fpx1, …, x1n ́1, x:n; yq, the process continues until a minimum is found to a given tolerance.

ICA changes this by saving all past values of the guess vectors px1, x12, …, x1n ́1q and corre- sponding roots x:n. Then, ICA locates the x:n that corresponds to the closest previous guess to the current guess in terms of Euclidean distance. Finally, ICA uses x:n to update the interval I “ ra, bs to either I1 “ ra, x:ns or I2 “ rx:n, bs, depending on whether the root lies in I1 or I2, and then follows Brent’s Method on the updated interval.

The intuition behind ICA is that discarding information from all previous solutions is likely to be inefficient when finding roots of several similar functions. Therefore, ICA selects the most similar problem that has already been solved and uses this to update the interval used by the root-finding algorithm. This is likely to be more efficient than Brent’s Method as it searches for a root on not only a smaller interval, but also one where an endpoint may be very close to a solution, giving rise to superlinear convergence much sooner than under Brent’s Method.

However, it is Brent’s Method’s robustness that has made it the world’s most used root-finding algorithm (Wilkins and Gu, 2013). Hence, Theorem 1:

Theorem 1. ICA can solve any calibration problem that Brent’s Method can solve. However, there exist calibration problems that ICA can solve and Brent’s Method cannot.

Proof. See Appendix B.
Furthermore, ICA is easy to implement; it requires only 14 lines of code in addition to Brent’s

Method in my Python 2.7 implementation.

The relative speed of the two algorithms cannot be established analytically. This is because ICA and Brent’s Method may take different paths to the root. To demonstrate ICA’s efficiency gain, I compare the solution times of calibrating the model using both ICA and Brent’s Method. For consistency, I use the same calibration and priors. As measuring the number of iterations controls for environmental factors which affect computation time, it is a more reliable measure of efficiency gain than clock time. In this respect, ICA outperforms Brent’s Method by 48.2%, almost halving the calibration time from 269 hours to 137 hours.


Figure 1: Model Fit and the Significance of Discount Factor Heterogeneity

(a) Model with ∆ “ ∆ ̊ (b) Model with ∆ “ 0 3.3 Results

I estimate tβ ̊, ∆ ̊u “ t0.988, 1.21 ˆ 10 ́3u. Therefore, households in my calibrated model have discount factors drawn uniformly from the interval r0.987, 0.989s. These values imply a moderate degree of discount factor heterogeneity and are of magnitudes consistent with the empirical evidence on time preference (Frederick et al., 2002).

As I target 6 points of the wealth distribution and have only one free parameter, a heuristic test for the model is provided by comparing the model-implied and WAS4 wealth distributions (see Figure 1a). Up to around the 60th percentile, they match almost exactly. However, above the 60th percentile, the model generates more wealth inequality than we observe in the data. This notwithstanding, the WAS4 data are likely to systemically underestimate the amount of wealth holding in the upper percentiles of the wealth distribution, limiting the

extent to which this compromises the predictions generated by the model.

Furthermore, while the difference in discount factors across individuals may seem small at the quarterly frequency, it has a substantial impact on behavioural differences across patient and impatient households. Figure 1b shows the deleterious impact on model fit of forcing ∆ “ 0 and calibrating β. This is because discount factor heterogeneity leads to heterogeneity of the precautionary and life-cycle savings motives, which augments wealth inequality.

Sensitivity Analysis. I examine whether my estimated parameters are robust to alternative values of the parameters about which I am most uncertain. As a result, I investigate how tβ ̊, ∆ ̊u varies as I change: (i) the coefficient of relative risk aversion ρ, (ii) the unemployment insurance parameter μ, (iii) the growth rate of technology Γ and (iv) the interest rate r (See Figure 2).


Figure 2: Sensitivity of tβ ̊,∆ ̊u to Parameter Variation

g=(1+Γ)4 −1 R=(1+r)4 −1 1.00 1.00 1.00 1.00

0.99 0.99 0.99 0.99 0.98 0.98 0.98 0.98 0.97 0.97 0.97 0.97 0.96 0.96 0.96 0.96



0.010 0.010 0.010 0.010

0.008 0.008 0.008 0.008 0.006 0.006 0.006 0.006 0.004 0.004 0.004 0.004 0.002 0.002 0.002 0.002

0.5 2.3 4.0 0.0 0.1 0.2 0.3 0.00 0.01 0.02 0.03 0.00 0.01 0.02 0.03



0.000 0.000
0.5 2.3 4.0 0.0 0.1 0.2 0.3 0.00 0.01 0.02 0.03 0.00 0.01 0.02 0.03

0.000 0.000

g and R are the annual technological growth rate and interest rate, respectively. The dot indicates my baseline calibration of each of the parameters.

tβ ̊, ∆ ̊u is insensitive to μ, Γ and r over the ranges I examine. However, ρ has a large impact on tβ ̊,∆ ̊u. This is to be expected as the degree of risk aversion is a key determinant of the strength of the precautionary and retirement savings motives. Despite this, over the interval r1, 3s, in which ρ is conventionally estimated to lie (Gandelman and Hernandez-Murillo, 2014), tβ ̊,∆ ̊u appears robust to changes in ρ. Overall, my estimates of β and ∆ are robust to parameterisation.


∆∗ β∗

Figure 3: Life-Cycle Consumption Patterns










20 25 30 35 40 45 50 55 60 65 Age

Average Education Level 1 Education Level 2 Education Level 3

Education Level 1 corresponds to those with only GCSEs or lower levels of qualification, Level 2 to those with A-Levels and apprenticeships and Level 3 to university graduates.

4 Consumption Patterns and Wealth Accumulation

I estimate UK life-cycle consumption patterns over households’ working lives that are consistent with the empirical evidence. My estimates provide insights into the dynamics of wealth accumulation. In particular, I find that the precautionary motive is strong and active up to the age of 30 and that the life-cycle savings motive leads to increased levels of savings from the late 40s until retirement.

I simulate normalised consumption patterns by performing 1200 simulations of 10,000 house- holds using the calibrated model. I convert these into approximate consumption patterns by appealing to the weak law of large numbers: the average permanent income across all consumers of a certain education level is given by the implied education-specific deterministic earnings trajectory. Given that I simulate 36 million population masses, this approximation is almost exact. I provide estimates up to the age of 65 as the model omits medical expenses, bequest motives and housing factors that are highly important in analysing the consumption of retired households (Banks et al., 2016).

Average household consumption follows a hump-shaped pattern (see Figure 3): increasing from £16,100 at the age of 24 to a peak of £25,700 at the age of 39, remaining roughly constant up to the age of 50, and declining from £25,000 at the age of 50 to £16,200 at the age of 65. This hump-shaped pattern obtains for all education groups and is consistent with the evidence on household consumption patterns from the US (e.g. Carroll and Summers, 1991; Gourinchas and Parker, 2002). Furthermore, high-education households have steeper consumption profiles than low-education households and the average peak-trough consumption


Annual Consumption (Thousand Pounds)

Figure 4: Precautionary and Life-Cycle Saving Behaviour

30 28 26 24 22 20 18 16

20 25 30 35 40 45 50 55 60 65 Age

Average Consumption Average Income

20 15 10

5 0 5

20 25 30 35 40 45 50 55 60 65 Age

ratio is approximately 1.6. These facts are fully consistent with Fernandez-Villaverde and Krueger’s (2007) findings regarding US consumption patterns.

These estimates provide insight into saving behaviour and wealth accumulation. Figure 4 shows that, on average, households save a significant portion of their income in their 20s, late 40s, 50s and 60s, while households gradually dissave in their 30s and early 40s. Moreover, the magnitudes of household saving implied by these estimates are consistent with the UK data on household saving, which show that UK households have an average saving ratio of approximately 6% (ONS, 2016c).

This behaviour can be understood through a combination of the precautionary and life-cycle savings motives. Through their 20s, households wish to self-insure against idiosyncratic income risk. Once they are optimally self-insured, they gradually dissave their buffer-stock as they age and their exposure to income risk declines. This is in line with Cagetti’s (2003) finding in the US that the majority of wealth held by the middle-aged can be attributed to the precautionary savings motive. Finally, from their late 40s, households save to smooth their consumption through retirement. My result differs from Gourinchas and Parker’s (2002) US result that households begin to accumulate retirement assets in their early 40s. This is likely due to my omission of a bequest motive and different parameterisations of the models.

The interpretation that saving behaviour in the model stems from precautionary and life- cycle savings motives can be evaluated by considering how savings differ across patient and impatient households. The precautionary savings theory predicts that households who are more patient will engage in greater levels of precautionary saving at young ages as they place more value on future consumption smoothing. In addition, as more patient households self-insure to a greater extent, they then run down their buffer-stock of savings at a faster rate as they age. Furthermore, the life-cycle savings theory predicts that patient households


Annual Consumption and Income (Thousand Pounds)

Average Annual Saving (% of Annual Income)

Figure 5: Discount Factor Heterogeneity and Saving Behaviour

20 15 10

5 0 5

20 25 30 35 40 45 50 55 60 65 Age

Most Patient Least Patient

4 3 2 1 0 1 2 3

20 25 30 35 40 45 50 55 60 65 Age

will save a greater amount to smooth consumption across their working and retired lives.

These predictions are borne out by my results. Figure 5 compares the life-cycle saving behaviour of households with discount factors of 0.987 and 0.989, the least and most patient types of household in my calibrated model. Relative to impatient households, patient households have greater rates of saving through their 20s, on average a negative and lower rate of saving from the early 30s to late 40s and a much greater rate of saving from their late 40s until the period immediately preceding retirement.

To summarise, these results are interesting for two reasons. First, they show that the overlapping-generations life-cycle model with idiosyncratic risk, liquidity constraints and discount factor heterogeneity is able to match the key empirical facts of life-cycle consumption and saving.

Second, they provide a tractable and plausible explanation of wealth accumulation through the precautionary and retirement savings motives. Hence, wealth inequality results from realisations of idiosyncratic income shocks and heterogeneity of the precautionary and life- cycle savings motives across households. In the model, this owes to heterogeneous expected growth rates of income, mortality rates, realisations of income shocks and discount factors.


Average Annual Saving (% of Annual Income)

Differenced Average Annual Saving Across Most and Least Patient Types (% of Annual Income)

5 Wealth, Consumption and Fiscal Stimulus

While fiscal stimulus targeted at households with low levels of income is common, wealth- targeted stimulus is not. This is likely because data on household wealth are poor in most advanced economies (Crossley and O’Dea, 2016). However, means-tested benefits make wealth-targeted stimulus possible in the UK.

This is pertinent for policy as there is prima facie reason to suppose that households with some of the highest MPCs will have little wealth. This is because liquidity-constrained households, impatient households and those with the experience of the most adverse income shocks are likely to be overrepresented at the bottom of the wealth distribution. This notwithstanding, low-wealth households will have some of the highest precautionary and life-cycle savings motives. Consequently, given the lack of agreement in the literature over the relationship between wealth and the MPC, further analysis is required. I contribute to the literature by estimating that the MPC is decreasing in wealth percentile.

Furthermore, there is a considerable gap in the literature regarding the relative effectiveness of wealth-targeted and income-targeted fiscal stimulus. I take the first steps towards filling this gap by estimating that, depending on the particular stimulus package, the aggregate annual MPC out of wealth-targeted fiscal stimulus is two to six times larger than out of equivalent income-targeted stimulus.

5.1 Fiscal Stimulus Experiments

I conduct a series of fiscal stimulus experiments in which I consider two types of policy:

1. Debt-financed transfers to various wealth quintiles and the unemployed

2. Balanced-budget redistributive transfers, wherein the top quintile of the wealth distri- bution is subject to lump-sum taxes which finance transfers to various wealth quintiles

To compare wealth-targeted and income-targeted fiscal stimulus, I also consider the same policies when targeted at various quintiles of the income distribution, in the style of Japelli and Pistaferri (2014).

These experiments are performed as follows: at date t1, the government makes an unexpected announcement that a certain fiscal stimulus package will be introduced immediately.11 For the debt-financed stimulus all debt is paid back at any date t such that t ě t1 ` s ̄ ` 1. This

11All transfers have the value of _1 per household. 19

Table 2: Effect of Wealth-Targeted Fiscal Stimulus

Policy Aggregate MPC
Transfer to All Wealth Quintiles Transfer to Bottom Wealth Quintile Transfer to Middle Wealth Quintile Transfer to Top Wealth Quintile Transfer to the Unemployed Transfer to the Employed 0.30 0.74 0.19 0.11 0.59 0.26
Transfer to All Wealth Quintiles
Transfer to Bottom Wealth Quintile Transfer to Bottom Two Wealth Quintiles Transfer to Bottom Three Wealth Quintiles Transfer to Bottom Four Wealth Quintiles 0.20 0.64 0.45 0.33 0.25

Policy Type Debt-financed

Relative Effectiveness 1
2.45 0.63 0.35 1.93 0.86

Balanced-budget 1 3.21

2.25 1.64 1.25

The aggregate MPC is the proportion of the transfer that is spent in the aggregate in the year following period t1. The relative effectiveness of all policies is the aggregate MPC out of each policy normalised by the aggregate MPC out of the uniform transfer.

simplifies the analysis by eliminating any Ricardian offsets that may otherwise arise (Carroll et al., 2016).

5.2 The Effectiveness of Wealth-Targeted Fiscal Stimulus

Targeting the bottom wealth quintile is more effective than both non-targeted and unemployment- targeted policy (see Table 2). I estimate that a uniform debt-financed £1 transfer to all consumers will lead to stimulus in the year following period t1 of £0.30 per consumer. Using this as a benchmark, the same transfer to the poorest 20% of consumers by wealth is 2.45 times more effective in stimulating aggregate consumption, larger than the stimulus effect of such a policy targeted at the unemployed. Furthermore, taxing the wealthiest quintile to finance a transfer of £1 to all consumers leads to a stimulus of £0.20 per consumer, making

it less effective in stimulating aggregate spending than the equivalent debt-financed policy. However, the balanced-budget transfer to the bottom wealth quintile is nevertheless 3.21 times more effective in stimulating spending than the uniform balanced-budget transfer.

The reason for the high effectiveness of policy targeted at the least wealthy quintile of consumers is the concentration of high MPC households in the bottom end of the simulated wealth distribution: of the third of the population with the highest MPCs, 49% are in the bottom wealth quintile. Low-wealth households are likely to have higher MPCs for a number of reasons in the model: on average, they have lower discount factors and lower growth rates


Table 3: Comparison of Income-Targeted and Wealth-Targeted Fiscal Stimulus

Wealth-Targeted Aggregate MPC Income-Targeted Aggregate MPC
0.30 0.74 0.30 0.36
0.20 0.64 0.05 0.11

Policy Type/Policy


Debt-financed/All Quintiles 1.00 Debt-financed/Bottom Quintile 2.06 Balanced-budget/All Quintiles 4.08 Balanced-budget/Bottom Quintile 5.82

The aggregate MPC is the proportion of the transfer that is spent in the aggregate in the year following period t1. Wealth/Income gives the aggregate MPC of the wealth-targeted policy relative to the income-targeted policy.

of income; and they are more likely to face a binding liquidity constraint and to have been subject to adverse permanent income shocks. As a result, despite having potentially large precautionary and life-cycle savings motives, low-wealth households have high MPCs.

There is empirical support for the finding that fiscal stimulus is more effective when targeted at low-wealth households when compared to non-targeted policy. In particular, Japelli and Pistaferri (2014) estimate that balanced-budget fiscal policy of the kind simulated here is significantly more effective when targeted at consumers with low levels of cash-on-hand.

My estimate of an annual MPC of 0.30 is also consistent with the empirical evidence on the magnitude of the MPC. Empirical studies on the UK MPC are extremely limited; the only recent work comes from Bunn et al. (2015) who use NMG Consulting data to calculate that the UK average MPC is 0.48 for borrowers and 0.09 for savers. However, these estimates suffer from the fact that they use survey data where households respond to hypothetical transitory income shocks. This notwithstanding, Bunn et al.’s (2015) results are consistent with those from the US, where the aggregate MPC is typically estimated to lie between 0.2 and 0.6 (Carroll et al., 2016). Consequently, my estimates are consistent with the empirical evidence on the magnitude of the MPC in both the UK and the US. This is particularly valuable given that most representative agent macroeconomic models generate an aggregate MPC ranging from 0.02 to 0.04 (Carroll et al., 2016).

Furthermore, I consider the impact of introducing fiscal stimulus that is targeted at income instead of wealth to compare the efficacy of wealth-targeting and income-targeting (see Table 3). The debt-financed and balanced-budget transfers to the bottom wealth quintile are 2.06 and 5.82 times more effective, respectively, than the equivalent income-targeted policies. Consequently, my results indicate that wealth-targeting is more effective than income-targeting.


The model generates these results because wealth holding is more strongly related to a household’s liquidity and degree of patience than their level of income, while both wealth and income are highly related to realisations of idiosyncratic income shocks. As a result, the relative impatience and low liquidity of low-wealth households, compared to low-income households, is sufficient to outweigh relatively higher precautionary and life-cycle savings motives. This is demonstrated by the fact that 49% of the households with the top third of MPCs are in the bottom wealth quintile, while only 27% are in the bottom income quintile. Once again, this finding is supported empirically by the results of Japelli and Pistaferri (2014), who show that the response to fiscal stimulus is greater when targeted at households with low levels of cash-on-hand than households with low levels of income.

5.3 Discussion and Policy Implications

These results have two significant implications for the design of fiscal stimulus policy. First, targeting unemployed, low-wealth or low-income households is likely to increase the stimulus effects of fiscal policy greatly. This gives an efficiency rationale to the normal equity-based arguments for targeting poor households. Second, the stimulus induced by wealth-targeted policy is significantly larger than that by equivalent income-targeted policy. This implies that stimulus should target wealth rather than income.

Furthermore, wealth-targeted fiscal stimulus is possible, despite the fact that UK data on wealth are very poor (Crossley and O’Dea, 2016). Because UK Housing Benefit and Council Tax Reduction are both means-tested, requiring that total wealth12 is less than £16,000,13 a simple version of the policies analysed here could be implemented by targeting stimulus at the recipients of these benefits. In addition, using WAS data, it is possible to forecast wealth based on factors observable to the government. However, such analysis is outside the scope of this paper.

Moreover, while my finding of a negative relationship between wealth and the MPC corre- sponds to that of Japelli and Pistaferri (2014) and Carroll et al. (2016), it contrasts with Kaplan and Violante (2014). One reason for this is that the model of Kaplan and Violante (2014) does not include transitory idiosyncratic shocks, which are likely to significantly affect the MPC (Deaton, 1992; Carroll et al., 2016). Another reason is that Kaplan and Violante (2014) model both a liquid and an illiquid asset, generating 7%-26% of consumers that are

12The means-testing for these benefits does not include some sources of wealth. In particular, the means- testing ignores the value of property in which recipients live.

13The exact level is at the discretion of local councils for Council Tax Reduction.


liquidity-constrained and wealthy: the wealthy hand-to-mouth (Kaplan and Violante, 2014). These consumers have high MPCs, damping the correlation between wealth and the MPC. This notwithstanding, one-third of the consumers in my estimates are hand-to-mouth, which is approximately consistent with the estimate that 40% of UK consumers are hand-to-mouth (Kaplan et al., 2014). Hence, given the difference between my results and those of Kaplan and Violante (2014), more research into hand-to-mouth consumers and their impact on the MPC is required.

A major caveat with these estimates is that I do not consider general equilibrium effects on asset prices, production or labour supply and assume away Ricardian effects. As with the estimates of Japelli and Pistaferri (2014), my estimates are therefore best seen as an upper bound of the true impact of fiscal policy.

6 Conclusion

I have demonstrated that the UK’s distribution of wealth can be generated using a suitably calibrated overlapping-generations life-cycle model with liquidity constraints, idiosyncratic income risk and a small degree of discount factor heterogeneity. Furthermore, my estimates of UK consumption patterns are consistent with the empirical evidence and allow us to understand the dynamics of wealth accumulation through the precautionary and life-cycle savings motives.

My results have two implications for the design of fiscal stimulus. First, fiscal stimulus is more effective when targeted at the unemployed or poor, whether by income or wealth. Second, wealth-targeting is significantly more effective than income-targeting. This finding is highly significant given that UK means-tested benefits make wealth-targeting possible and almost all fiscal stimulus programmes target income rather than wealth.

In the future, research into the relationship between wealth and consumption should aim to integrate heterogeneous-agent life-cycle models into larger general equilibrium models, as Krusell and Smith (1998) do with infinite-horizon models. This would enable analysis of fiscal policy over the business cycle and the role that general equilibrium effects and macroeconomic shocks have on consumption. Furthermore, it would be interesting to extend the model to examine wealthy hand-to-mouth consumers, as in Kaplan and Violante (2014), consumption patterns of retired households, as in Banks et al. (2016), and test whether my results are robust to making labour supply endogenous, as in Low and Pistaferri (2015).

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