The Multivariate Rational Addiction Model

The Multivariate Rational Addiction Model

Michael K. Wohlgenant

Department of Agricultural & Resource Economics, North Carolina State University, Raleigh, USA Email: [email protected]

Received July 26, 2012; revised August 27, 2012; accepted September 29, 2012

ABSTRACT

This paper generalizes the model of Becker, Grossman, and Murphy (1994) to the multivariate case. The multivariate model generates Frisch demand functions where current consumption is related to prices of all goods, and lagged and future consumption of all goods. The theoretical restrictions are that current price effects (holding lagged and future consumption constant) are negative definite, and lagged and future consumption are proportional to one another, the proportionality factor being the consumer’s discount rate. The conditions for dynamic stability are derived, and the so- lution to the matrix difference equation is derived. General formulas for multivariate Frisch price elasticities with re- spect to different lengths of time are also derived. Finally, alternative econometric specifications are derived, showing how theoretical restrictions can be imposed to test the theory and to reduce the number of estimable parameters. It is also shown how the model can be modified to account for different discount rates by commodity when estimating the model using aggregate data.

Keywords: Rational Addiction Model; Dynamic Frisch Demand Functions; Dynamic Consumer Demand; Habit

Formation

1. Introduction

The workhorse of empirical analysis of dynamic demand is the rational addiction model of Becker, Grossman, and Murphy [1]. This model has proved useful in estimating short-run and long-run demand elasticities, but it only allows for one commodity and one composite good. To the author’s knowledge, no one has rigorously formu- lated and analyzed the multivariate counterpart to the single-equation model. Bask and Melkersson [2] and Pierani and Tiezzi [3] extend the rational addiction model to two goods (and the composite good), but do not analyze the restrictions imposed by theory or derive the dynamic properties of the model. The purpose of this paper is to present the multivariate addiction model and analyze the restrictions imposed by theory as well as the dynamic properties of the solution to the matrix differ- ence equation.

2. The General Rational Addiction Model

The simple rational addiction model is extended by specifying that the consumer’s utility function for period t is given by the strictly concave, twice-differentiable function



C Ct, t 1, t

U _ Y



sume me period t, is an n-vector of quantities

the

(1)

where is an n-vector of quantities of goods con-

of goods consumed in previous time period t 1

t

C

d in ti Ct1

 , and Y_t is the quantity of a composite good at ti t rep- senting consumption of all other goods1_{. We shall as-}

sume that the individual consumer maximizes the utility of life-time consumption with utility discounted at rate

me re

. With P_t the vector of prices associated with C_t, th consum r’s problem is to maximize



1 _{C C}

t _{U Y}



 

e

(2)

(3) e

ect t

e 0



subj the intertemporal budget constr

wher

1

1 t, t , t

t 



o aint





1

t

 

0

1 P Ct t t

t   Y A



A is initial wealth and with initial conditions

0

0 

The fit rst-order conditions (F.O.C.) for utility za

Ct C 2.

maximi- tion are



t, t 1, t



₀ t

U C C Y

Y 





 

 (4a)



, 1,





1, ,



t t

C C C C C C P

 U _{t t} Y_t  U _t _t Y_t1  _t 0 (4b)

1_{Becker, Grossman, and Murphy [1] also include unobserved lifecycle}

variables in the utility function. These variables could easily be ac-commodated in the multivariate version but are not included to sim-plify the model. For econometric implementations, the main implica-tion is that we would need to assume both lagged and future quantities are endogenous variables because inclusion of lifecycle variables would imply the error term would have a moving average structure.

2_{Because the price of the composite good is 1, each price of the}

where _Ct

 



on (4a) is is the marginal utility of wealth and

the gra ient vector with respect to d t. Equati , as in [1], is the condition that the marginal utility of the composite good equals the marginal utility of wealth. Equation (4b) generalize the univariate case to the mul- tivariate case where the marginal utility of current con- sumption of each good plus the discounted value of next period’s marginal utility of consumption equals the mar- ginal utility of wealth times the price of the good. As in [1], the model allows for both harmful addiction

C

1

0

U C

  



_ 

it

 

and beneficial addiction

1

0

it U

C 

  _ 

_ 

 

If the consumer takes the marginal utility of wealth constant in formulating decisions for the first-period of his planning horizon, then we can derive marginal utility of wealth constant (Frisch) demand functions showing how current period consumption responds to past, pre- sent, and future (expected) prices3_{. In keeping with a}

common assumption made when modeling intertermpo- ral demand behavior [4], I assume that the marginal util- ity of consumption of good i is independent of the quan- tity of consumption of good j (j ≠ i)of goods consumed in the previous time period

2 . ., U

i e C C

 _ 

1 0 it jt

j i

    

 

I also assume that the marginal utility o , while de- pendent on is independent o 5_.

ion at the F.O.C. ca

Pt

   4

f Yt

,

Ct t1

Assume that the current period utility function can be approximated by a quadratic funct so th

fC

n be expressed as6

0 0 0 0

0 0

a

a C C

t

Y Y 1 Y 1

1 C 1

A B B

t t

t t t

           

     

  

      _

 

 

     

  

          

2 2

2

2 2 2

1 ,

t

t t t t t

C A

C C C

a a

C C

t t

t t

t t

t

U U

Y Y

U U U

Y



 

   

 _{   } 

 _{  }



 _{    }



 _{   }

 

     

 

 



(5)

Where

2 0 0

1

0 0

0 0

t t

B

C C

U





 



 _{  } 

 _{ } 

 _{     }

 

. and

The vector is an 0 n-vector of zeros and 0 is its

transpose. The matrix pre-multiplying

t

Y

     Ct

is negative definite. Therefore, its inverse exists and has the following partitioned form:

1 _{1 1 1 1}

 _       

1

1  1

  

1 1

       

  

  _

a A  D a D  (6)

a a D a a D



_ 1





 _{ }  _

D 1 A aa 1



_ 1



where is a negative definite matrix, because _A _aa 1

is the second principal trix of

subma-

1

a

a A



       

which is negative definite. Given the partitioned inverse (6), the solution to is

1

t

C

1

1 1 1

Ct D BCt 1 D BCt 1 D a  D Pt

    

      

In contrast to the univariate rational addiction model, consumption of good in t

(7)

i he current period is related to lagged consumption of good i, as well as lagged of all ot

values her consumption goods. Moreover, current con- sumption of good i is related to consumption of all con- sumption goods in period t1. Because _D1

is nega- tive definite, current period price effects (holding future consumption constant) are negative definite. When B is diagonal

3_{The assumption that λ is constant over the planning horizon is}

pre-cisely what the model implies. With perfect certainty, the consumer would expect to choose life-time consumption allocations holding λ constant.

4_{In this specification, the effect of a change in lagged consumption of}

the jth variable on current consumption of the ith good (holding future

consumption constant) is 2 ₂

1 1

ij it

jt jt

C _U U

C C

 _{ } 

  , where is the i,jth

element of the inverse of the matrix

ij

U

2

it jt U C C

  

_{ }

 

 

2

1 1

. ., 0 ,sgn it sgn ij ,

j

1

and sgn sgn

j

ij it

jj jt

C

D b C  

   

3. Stability and General Solution of Matrix

Difference Equation

Proposition

dif expressed as

it jt jt

C U

i e j i D b

C C _ C _

 _  _

    

 

 _{ }  _

 

. Thus, current and

lagged consumption arestill interdependent when expressed in this form.

5_{This specification generalizes [2] and [3], who also make the marginal}

utility of t independent of Ct.

6_{The quadratic function is not necessary but is introduced in order to}

evaluate the stability of the system in the neighborhood of the station-ar

Y

y equilibrium. See [1] for a similar approach in the univariate case.

When Pt is bounded, the general solution to the matrix



_{1 1}



1 _{1 1 1 1}

1 1 Q I 2 2Q B P

k

 L     

  

1  1

1 1 1 1

1 1 2

0

Q Q C Q  i Q B P

t t i

i          1

C_{t }Q Q C  _{t k}

t   where    







1



1 1 1 1 1

2 2   

     

 

k Q I   Q B a ,

matrix of positive, real eigenvalues which lie he unit circle, and 2 is the diagonal matrix

positive, real eigenvalues which lie outside the



1

 is the diagonal

within t of

unit cir-

e la cle.

Proof. Rewrite the system of Equation (7) in differ-

ence equation form using th g operator to obtain7_:



1 2 1_{B D}1 _I



1_C

1 1 1 1 1

B  Pt B a     Zt

   (8)

1 

t

L L L

 _  _ 

When B is a diagonal matrix with positive ele- ments, _B0.5_DB0.5

is symme there exists an orth

is positive definite an because tric negative definite

ogonal matrix such that

d symmetric

8_{. Therefore,}

D 0.5  B itive e  P P 0.5   _{ }

PB D P M, a diagonal matrix with all dis- tinct, pos lements. Define the transformation

0.5 _.





Q B Then

1 1 0.5 1 0.5

    _

Q B DQ PB B DB P M. Therefore,

1 

B D and _B0.5_DB0.5 _{are similar matrices [5].} Multiply both sides of (8) by _Q1 _{to obtain}

1_{C Q Z}1

t t

L  

or



_1_Q1 2_{L }1_{Q B D}1 1 _LQ1





1_Q 1 2 1_MQ 1 _Q1



1_{C Q} 1

t t

L L L

  _  _   _ _Z

ause . Define

eq written as (9) atrix pol as



L (10) and side of Equation (10) implies that bec

and

1 1 1

  _{ } 

Q B D MQ

1

t



Q Z . Then the above

1

ˆ

t t





C Q C

uation can be ˆ

t Z



1 2 1_{M Zˆ}

t t

L L

 _ _I



_L1_Cˆ _

The m ynomial on the left-hand side of (9) can be written



_1 2_{L }1_MLI



_



_{I L}



_I

1 2

The right-h



I1L



I2L



 I



12



L 1

2 ₍₁₁₎

2L

w atrix

polyno set -o

ference equations of the form

here 1

1 2 



  M

  and 1

1 2 



 I

  . The m mial in (11) is a of single, second rder dif-

1 2

 1 _{1 0}

i mii i

     

This matrix polynomial consists of individual charac- teristic equations of the form





2

1 1 1 ₂

1

4 ₄

2

ii ii _{ii ii}

i

m m _{m m}

   2          _{ }

Because the diagonal elements of are real, we

know that 0.5

  . (12)

M

2 _{4 0 2}

ii ii

m    m  

real and distinct. For stability, 1

and b

i mi

oth roots are 1 i 1

     and 1

  9_{. By the relationship among roots} 1

1 2  ,

  I   this means , 1 1

2i 1i 0

 _   _{ when}

1i 0

  . The matrix



I 2L



can be expressed as







1



2L 2L L

 

  I 1

2

   . Noting also that



I



1L



2L

 

I2L



1



Equation (9) can

L the matrix

 

I  I  I  ,

be written as



1







2 I 21 I 1 Cˆ ˆ

  L  L _{t Z}

t

    (13)

Multiplying both sides by the inverse of



1



2 L

 

 I 1

2

  yields

t Z

Substituting back in terms of an , and multi- plying both sides by we obtain

 d result.

4. Elasticities

ived from eter estimates. From the above Pro-

that the price derivatives of the dy-



I1L



Cˆt  



I2L



2

1

1 1_ˆ

1 

  

t

C dZt

Q









1 1

1 2

I Q Q_  L C_{t }Q I_ L Q Z _t,

which immediately leads to the desire

1

1 1 1

2 

  

Short-run and long-run elasticities can be der the structural param

position we see

namic demand functions are

  1 1 1 2

1

i

C

Q Q B

P  t t i           

 (14a)

L is defined as follows:

1 1 1 Z t t t nt nt Z Z L L Z Z            _{  } _        

  holding all past prices constant for a temporary price _{change. For an expected permanent future}

7_{The lag operator}

1

and for any arbitrary vector t.

1 1 1 1 1 Z t t t nt nt L L            _{  }        0.5  price change  





1 1 1

2 0

1 _{1 1 1}

1

Q Q B

P

Q I Q B

  t i t 1 2 i

C _{ }

             



(14b)        1    Z ‘s ii Z Z Z Z 

8_{The matrix is the diagonal matrix whose elements are the}

recip-rocals of the square roots of the b_ii . This means then that the ma-trix _B0. _{the diagonal matrix whose elements are the square roots}

of the bii ‘s

B

s .

5 _i

9_{To see this, note that} 1 2 1 _{1 0} 1 _{. I}

i mii i i i m

  _ _{   }  _{ f} 1

ii

m   then i1 using (12). So because

1

i i

Finally, the long-run price effects are



₁

 

1



1 _{1 1 1}

Assuming a discount rate of  0.95 1

 , the matrices of

eigenvalues associated with B D are

1

1 2

C

I Q Q Q I Q B

P    

 _(14c)

 

   



   



5.

As an example to illustrate the method of calcula

solution to the matrix difference equation and formulas der the two good case which might ddictive goods such as alcohol and

1 2

0.1495 0 7.0412 0

, .

0 0.3558 0 2.9588

 _ _  _ _

   

An Example

The matrix consisting of the two eigenvectors associ- ated with the set of eigenvalue is as follows

ting the

for elasticities consi

correspond to two a 0.7746 0.7746

0.6325 0.6325

Q_{ }  _

 

tobacco. Specify the matrices B and D as follows

0.4 0 2 1

, .

0 0.6 1 3

B_ _ D_ _



   

Given these numbers, the solution to the matrix dif-ference equation shown in Proposition is

 1

1 1 1 1

0

2 2 1 2

0.25 0.13 0.7746 0.7746 7.0412 0 1.6987 1.3869

0.08 0.25 0.6325 0.6325 0 2.9588 1.6987 0.2526 k

i

t t t i

i

t t t i

C C

C C

 

  

 



 

 

 _  _      

  _       _  

       

   



 

P P  The matrices of price elasticities shown in Equations (14a)-(14c) are as follows:

 

1 1

1

1 2

2t 2t 0.6325 0.6325 0 2.9588 1

C C      

   _{    }

 

1 2

0.7746 0.7746 7.0412 0 1.6987 1.3869 .6987 0.2526

t t

i t i t i

t i t i

C C

P P

P P

 

 

 

 

 

_{ }  _{      }

  _



 

 

(14a’)

1 1

1 2

2 2

1 2

3.5211 0.3706 0.3706 2.3474

t t

t t

t t

t t

C C

P P

C C

P P

 

 

_{ }  _{ }

 _{   }

   _  _

_{ } 

 

(14b’)

1 1

1 2

2 2

1 2

4.8882 1.0466 1.0466 3.2588

C C

P P

C C

P P

 

 

_{ }  _{ }

 _{  }

   _ 

_{ } 

 



 (14c’)

This example shows that the solution to the matrix difference equation is stable because the matrix 1con-

sists of positive real roots all within the un

the matrix consists of positive real roots al side th

an

ct

rational addiction behavior. The simp

be to start with the F.O.C. from Equation (5), after eliminating from the set of equations related to to obtain

t

lest approach would

t

Y Ct,

it circle, and l out

2



e unit circle. The solution also shows that both goods are interrelated in consumption through lagged qu tities and current and future prices. Note also that the matrices of price effe s as shown in (14a’)-(14c’) indicate that all own-price effects are negative, and all cross-price effects are symmetric and positive. This numerical illustration indicates that we should expect changes in current and future price effects to exhibit complementary effects when both goods exhibit habit formation. In addition, all long-run price effects (in absolute value) should be larger than short-run price effects.

6. Econometric Implications

There is more than one approach to take for quantifying

1

t t  t1 It t

DC BC EC c P U (15) where c is a vector of constants, It

hus

is income, and

am

negat definite. T , the symmetry restriction

t

U is a vector of disturbance terms . The advantage of this specification is that it simplifies imposing and testing for the theoretical restrictions. The testable restrictions are that the matrix D, which represents intra-period substitution ong the individual consumption goods, is symmetric and

10

ive

 

D D could be imposed linearl s,

y and tested. Because D _{is a matrix of constant} one could also impose negative definiteness on the contemporary sub-stitution matrix using one of the several methods avail-able in the literature (e.g., [7]). The other restriction that one may wish to impose is EB. With diagonal ma-trices, this means the restriction is eii bii for each

equation.

From an econometric point of view, it is straight for- ward to estimate the model using generalized method of

10_{We follow [1] in approximating the product of the marginal utility of}

moments finding instruments such that the orthogo- nality condition



by



t t

E U Z 0 holds, where Zt is a

vector of instrumental variables. In this case, as in [1], we could use current, lagged, and futures prices as in- struments, in addition to inco t is not reasonable to assume that consumers know future prices with high probability

me. If i

then we could use current and enough lagged pr

one good ([2], [ o ices sufficient enough to identify the parameters of the F.O.C.

The few attempts to extend the rational addiction mo- del to more than 3]) specify the m del as follows

1 1 1 1 1 1

1 1

ˆ ˆ ˆ ˆ ˆ ˆ

t

D DC _tD BC _t D EC _t D c I D P _t D U _t (16) where _Dˆ1 _{is the diagonal matrix whose diagonal ele-} ments are 1

ii

d so that the ith equation can be written as

Cit 



j iπijCjtiiCit1iiCit1i tI ii itP it

(17) In light of (16), the symmetry constraint, which is now nonlinear, would be imposed as follows:

1 1

πij πji

ii jj

 

The param r ii

 .

ete  could then be estimated separately and an estimated value of  obtained by dividing the estimate of ii by ii.

h t

The othe o est ation is what [2] an call the reduced-form approach and is indicated by Equa- r approac im d [3] tion (7), generalized below as follows

1 1 t

CtFCt GCt gI HPtt (18)

where

f all goods in the current period simultaneously. It is notable tha

pted to utilize these restrictions fro

Equation (16) or Equation (18) in their em- pirical work.

We typically do not have the luxury to ork with panel data at the individual household level. Therefore, it is

onsumption of good i, when the discount rate is allowed to be different for each consumer, can be

w

1 1

1 1 1

,

nd , a

, ,

F D G D B

g D c H D  D U

B 

 

  

   

   _t  _t

This set of equations like (16) has nonlinear restrict- tions. Equation (18), however, is consistent with the view that the consumer chooses quantities o

t neither [2] m theory or [3] attem

implied by

w

clear that the estimates of the discount factor may dif- fer from one commodity to another. This is particularly true with aggregate data as in [3], where it is shown that the discount rates for alcohol and tobacco are quite dif- ferent. To see why estimates based on aggregate data could produce divergent estimates by commodity, note that average c

ritten as follows (over-bars denote simple averages over the total population of consumers):

1 1

π

it j i ij jt ii it ii it i t ii it it

C 



 C  C   C  I  P 

(19)

where 1

1

k kit

k it

C C

 

 _





 _,

k

 is the discount factor for consumer k, Ckit1 is con-

sumption of consumer k for good i at time t+1,and Cit1

is aggregate consumption of good i. The significant fea- ture of Equation (19) is that the aggregate discount factor

 _{is a weighted average of discount fact}

n relativ t be th the sam

ncome on good

rent d ors, eac ghted by consumer k’s consumptio e to to

ption. Clearly these weights need no e consumers. For example, even with

ction, a consum with a higher i e a different mix of all consumpti s t

h tal

e util- could

han a wei

consum for all

con

same

ity fun er

consum

sumer with a lower income. With diffe iscount rates, the average discount rate  could be differen different goods. Therefore, for aggregate data, Equation (19) should be modified as follows:

t for

1 1

π

it ij jt ii it i ii it i t ii it it

j i

C C  C _   C _ I  P 







    

(20) where i is indexed by the particular good consumed.

This means different discount rates can be accommo- dated by the model, while preserving symmetry and negative definiteness in own quantity effects. Note that all the above results still hold when _1_{I is replaced}

with _β1_{, where β}1 _{is the diagonal matrix with di-}

agonal elements 1

i

 11_.

7. Concluding Remarks

pe

e d nite, holding lagged and future ent and past consumption the proportionality factor This pa r formulates and analyzes the multivariate ver- sion of the rational addiction model of Becker, Goldman, and Murphy [1]. The multivariate counterpart to the uni- variate model is that consumption of a specific good in the current period depends on prices of all goods, lagged consumption of all goods, and future consumption of all goods. The theoretical restrictions are that current price effects are negativ efi

consumption constant, and curr are proportional to one another,

being the consumer’s discount rate. These results indi- cate that the main restrictions of the univariate model are preserved in the multivariate model.

The conditions in which the model is shown to be dy-

11_{Of course, we assume that distribution of effects across individual}

consumers is relatively constant over time so that _i can be taken as constant. Otherwise, we would need to modify the model to make _i

elasticity form fo

Service, Raleigh, North Carolina, 27695.

NCES

Review, Vol. 84, No. 3, 1994, pp. 396-418.

[2] M. Bask and nally Addicted to Drinking and S ,Vol. 36, No. namically stable are derived. When the model is stable,

the solution will have exactly 2n realroots, n of the roots falling within the unit circle and n falling outside the unit circle. The smaller roots can be used to solve the problem backward in time, or to express the current-period solu- tion conditional on the levels of consumption of all goods in the previous period. The set of larger roots are used to express current consumption as a linear function of all

future prices. Short-run and Long-run ulas _{4, 2004, pp. 373-381.}

doi:10.1080/00036840410001674295 r the multivariate version are derived and are shown to

be generalizations of the univariate version.

Estimation can be undertaken on one of three different forms: 1) The first-order conditions directly, Equation (15); 2) the so-called structural form, Equation (16); or 3) the reduced form, Equation (18). Which of the above approaches to estimation is best can only be determined through further empirical work. Regardless of the ap- proach taken for estimation, the theoretical framework developed in this paper should prove useful to researchers modeling addictive goods that are interrelated in con-sumption.

8. Acknowledgements

Research supported in part by the North Carolina Agri-

cultural Research

REFERE

[1] G. Becker., M. Grossman and K. Murphy, “An Empirical Analysis of Cigarette Addiction,” American Economic

M. Melkersson, “Ratio moking,” Applied Economics

[3] P. Pierani and S. Tiezzi, “Addiction and Interaction

be-8-0220-3

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doi:10.1007/s00181-00

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North-Hol-Vol. 8, No. 1, 5383

[4] L. Phlips, “Applied Consumption Ana land Publishing Co., Amsterdam, 1983.

[5] R. Lucas, “Optimal Investment Policy and Flexible Ac-celerator,” International Economic Review,

1967, pp. 78-85. doi:10.2307/252

/1911156

[6] T. Sargent, “Macroeconomic Theory,” Academic Press, New York, 1979.