_{1}

The implications of the intertemporal state adjustment model (ISAM) are evaluated. The ISAM accounts for the effect of current purchases on future utility through a state variable that can either reflect habit formation or inventory holding. The model is shown to be forward looking with purchases depending upon beginning-of-the period state variable as well as the present discounted value of future user costs of the state variable. In this way, the model accounts for the speculative motive for inventory holding. The myopic state adjustment model, which depends on the beginning-of-the period state variable and current price, is a special case of ISAM when the discount rate is zero. Other special cases of the ISAM are identified and alternative representations of it for empirical analysis are presented.

There is a large literature on inventories and consumer behavior, but virtually all studies assume utility depends only on consumption and not beginning-of the- period inventories^{1}. Making the current period utility function dependent on inventories is in the spirit of the classic approach of Houthakker and Taylor [

The main problem with the myopic state adjustment model is that it does not account for the speculative motive for inventory holding. In particular, no provision is made for the influence of anticipated capital gains (losses) from holding the stock. Moreover, if the consumer is forward looking, he will take into account the effect of purchases on future marginal utility. Also with inventory behavior, consumer purchases can differ from consumption. Thus, an intertemporal version of the state adjustment model is necessary to accommodate these features and to extend the myopic state adjustment model to a model amenable to empirical specification and analysis. The purpose of this note is to extend the model to its intertemporal version and show how it accounts for both demand with habit formation and stock holding.

The consumer is assumed to maximize the present discounted value of future utility subject to an intertemporal budget constraint and the equation of motion defining how inventories change over time. The utility function for period t is

v ( q t , s t ) + φ ( y t )

where q t is purchases of the good in question, s t is the level of the state variable at the beginning of time t, and y t is the quantity of the composite good of all other goods purchased. We assume that the utility function is strongly separable with respect to choice of the good and the composite good. The state variable is assumed to change over time according to the relationship:

s t + 1 − s t = q t − δ s t

where the parameter δ is the (assumed) constant rate of depreciation of the state variable. If the state variable on net reflects habit formation, the parameter shows how the stock of habits depreciates with respect to time; if the state variable reflects durability of the product, then the parameter shows the rate of depreciation of the asset over time. For a commodity that is storable, the parameter can be interpreted as the rate of consumption per time period. Finally, the budget constraint is

w t = ∑ j = t T β j − t ( p j q j + y j )

where w t is the present discounted of wealth at time t, β is the one period discount rate, and p j is price of the good, with the price of the composite good taken to be unity^{2}.

The Lagrangian function for maximizing utility subject to the budget constraint is

L = ∑ j = t T β j − t [ v ( q j , s j ) + φ ( y j ) − λ ( p j q j + y j ) − μ j ( s j + 1 − ( 1 − δ ) s j − q j ) ] + λ w t

The first-order conditions are:

∂ L ∂ q t = ∂ v ( q t , s t ) ∂ q t − λ p t + μ t = 0 (1)

∂ L ∂ y t = d φ ( y t ) d y t − λ = 0 (2)

∂ L ∂ s t + 1 = β ∂ v ( c t + 1 , s t + 1 ) ∂ s t + 1 − μ t + β ( 1 − δ ) μ t + 1 = 0 (3)

∂ L ∂ λ = − ∑ j = t T β j − t ( p j q j + y j ) + w t = 0 (4)

∂ L ∂ μ t = − s t + 1 + ( 1 − δ ) s t + q t = 0 (5)

Following Becker et al [^{3}. With λ taken as constant, we can substitute (1) into (3) and use (5) to obtain:

β v s ( s t + 2 − ( 1 − δ ) s t + 1 , s t + 1 ) − β ( 1 − δ ) v q ( − ( 1 − δ ) s t + 1 , s t + 1 ) + v q ( s t + 1 − ( 1 − δ ) s t , s t ) = λ p t − β ( 1 − δ ) λ p t + 1 (6)

where subscripts denote first-order partial derivatives.

The comparative statics of this relationship can be ascertained through differentiating the first-order conditions (6). For simplicity, assume the utility function is quadratic:

u t = 0.5 v q q q t 2 + 0.5 v s s s t 2 + v q s q t s t + 0.5 y t 2 (7)

where for convenience the linear terms (which produce constant derivatives) have been omitted. Taking derivatives of (7), substituting into (6), and combining like terms in s t results in the expression:

β ( v q s − ( 1 − δ ) v q q ) s t + 2 + [ β δ v s s + β ( ( 1 − δ ) 2 + 1 ) v q q − 2 β ( 1 − δ ) v q s ] s t + 1 + ( v q s − ( 1 − δ ) v q q ) s t = λ p t − β ( 1 − δ ) λ p t + 1 (8)

The solution in terms of s t + 1 is:

s t + 1 = θ s t + β θ s t + 2 + θ 1 [ λ p t − β ( 1 − δ ) λ p t + 1 ] (9)

where θ 1 = [ β δ v s s + β ( ( 1 − δ ) 2 + 1 ) v q q − 2 β ( 1 − δ ) v q s ] − 1 and

θ = θ 1 [ ( 1 − δ ) v q q − v q s d ] . The parameter θ 1 is expected to be negative due to the fact that the utility function is strictly concave. The parameter θ will be positive if v q s > 0 , which is what we would expect if habit formation dominates. On the other hand, if durable good behavior dominates, v q s < 0 , and θ < 0 if | v q s | > | ( 1 − δ ) v q q | , which is more likely the larger the depreciation rate δ .

Upon substituting Equation (9) into Equation (5) we obtain the demand equation for the good in question:

q t = − [ ( 1 − δ ) − θ ] s t + β θ s t + 2 + θ 1 [ λ p t − β ( 1 − δ ) λ p t + 1 ] (10)

Demand for the good depends upon the level of state variable at the beginning of the time period, the level of state variable at the end of period t + 1 ( s t + 2 ) , and the user-cost of the state variable, λ p t − β ( 1 − δ ) λ p t + 1 ^{4}.

Proposition 1.

The solution to the second-order state adjustment Equation (9) is stable and has the form (provided λ p t − β ( 1 − δ ) λ p t + 1 does not grow at a rate larger than | λ 2 − 1 | ):

s t + 1 = ( 1 − λ 1 ) s t + θ 1 β θ λ 2 ∑ j = 0 ∞ λ 2 − j ( λ p t + j − β ( 1 − δ ) λ p t + 1 + j ) (11)

where for θ > 0 , 0 < λ 1 < 1 , and λ 2 > 1 ; for θ < 0 , − 1 < λ 1 < 0 , and λ 2 < − 1.

Proof:

Rewrite (9) as follows:

( 1 − β − 1 θ − 1 L + β − 1 L 2 ) L − 1 s t + 1 = − β − 1 θ − 1 θ 1 z t (12)

where L is the lag operator and z t = λ p t − β ( 1 − δ ) λ p t + 1 . Obtain the factorization,

( 1 − λ 1 L ) ( 1 − λ 2 L ) = ( 1 − β − 1 θ − 1 L + β − 1 L 2 ) (13)

implying that λ 1 + λ 2 = β − 1 θ − 1 and λ 1 λ 2 = β − 1 . The characteristic equation is

λ 2 − β − 1 θ − 1 λ + β − 1 = 0

The solution to the characteristic equation is

λ 1 , λ 2 = β − 1 θ − 1 ± ( β − 2 θ − 2 − 4 β − 1 ) 0.5 2

The roots of this equation are real [

− ( 1 − λ 2 − 1 L − 1 ) ( 1 − λ 1 L ) λ 2 s t + 1 = − β − 1 θ − 1 θ 1 z t

Multiplying through both sides by − λ 2 − 1 and ( 1 − λ 2 − 1 L − 1 ) − 1 , and substituting for z t yields the desired result.

Proposition 2.

The intertemporal state adjustment model is:

q t = − [ ( 1 − δ ) − θ ( 1 − ( 1 − λ 1 ) ) β ] s t + + θ 1 λ 2 ∑ j = 0 ∞ λ 2 − j ( λ p t + j − β ( 1 − δ ) λ p t + 1 + j ) (14)

Proof:

Substitute Equation (11) into Equation (5) and combine like terms to obtain result.

Proposition 3.

The myopic, static state adjustment model

q t = θ s t + θ 1 λ p t (15)

results as a special case of the intertemporal adjustment model, Equation (14), if the discount rate β = 0.

Proof:

Using Equation (5) in Equation (6) with the quadratic utility function (7) when β = 0 yields the desired result for θ = − v q s v q q .

Another special case of the intertemporal model can be isolated when the depreciation rate δ = 1.

Proposition 4.

The intertemporal state adjustment model with δ = 1 becomes

q t = β θ [ 1 − ( 1 − λ 1 ) ] q t − 1 + + θ 1 λ 2 ∑ j = 0 ∞ λ 2 − j λ p t + j (16)

where θ 1 = [ β v s s + β v q q ] − 1 and θ = − θ 1 v q s .

Proof:

Set δ = 1 in Equation (14) and use the fact that s t + 1 = q t .

In this special case, we have the model of Becker et al [

Proposition 5.

When the depreciation rate δ = 1 , Equation (9) becomes

q t = θ q t − 1 + β θ q t + 1 + θ 1 λ p t (17)

Proof:

Set δ = 1 in Equation (9) and we immediately obtain the result.

Therefore, we see directly how the model becomes identical to that of Becker et al [

The intertemporal state adjustment model, which is forward looking, produces a much different specification of demand than the static, myopic model. In particular, the intertemporal demand model, Equation (14), from Proposition 2 shows that demand depends on future expected prices, the user cost of capital, and beginning-of-the period state level. In contrast, the static model, Equation (15), only depends on current period price and the beginning-of-the period state level. Therefore, in contrast to the myopic model, the intertemporal model accounts for the speculative motive in stock holding.

Long-run effects of the intertemporal state adjustment model can be computed either using Equation (11) with Equation (14), or directly using (9) and (14). From Equation (9), the long-run price derivative of the state variable is

∂ s * ∂ p * = λ [ 1 − β ( 1 − δ ) ] θ 1 [ 1 − θ ( 1 + β ) ] (18)

where asterisks denote long-run steady-state levels. In contrast, the short-run effect, with steady-state price changes, is just the numerator of (18). This means, as is the case with myopic model, long-run effects exceed (are less than) short- run effects as habit formation (inventory) behavior dominates. That is, when θ > 0 , long-run price effects are larger in absolute value than short-run price effects; when θ < 0 , long-run price effects are smaller in absolute value than short- run price effects.

An alternative representation of the intertemporal state adjustment model that may also be more useful for econometric analysis can be obtained by multiplying both sides of Equation (9) by [ 1 − ( 1 − δ ) L ] and substituting for q t from Equation (5) to obtain:

q t = θ q t − 1 + β θ q t + 1 + θ 1 [ λ p t − β ( 1 − δ ) λ p t + 1 ] − ( 1 − δ ) θ 1 [ λ p t − 1 − β ( 1 − δ ) λ p t ] (19)

Long-run price derivatives can be computed from Equation (19) directly

∂ q * ∂ p * = λ [ 1 − ( 1 − δ ) ] [ 1 − β ( 1 − δ ) ] θ 1 [ 1 − θ ( 1 + β ) ] (20)

As in the case of the state variable, Equation (16), short-run effects will be larger (smaller) as the parameter θ is larger (smaller) than zero.

A full discussion of econometric strategies for estimating the intertemporal state adjustment model (19) is beyond the scope of this paper. However, it should be clear that the approach of Becker et al [

In contrast to the myopic state adjustment model, the intertemporal state adjustment model produces a forward-looking demand specification. Demand for the good in question can be characterized by Equation (10), Equation (14), or Equation (19). As shown in Equation (10), the static state adjustment model should be extended to include end-of-the period state variable and the price variable should be the user-cost variable, which includes the impact of both current and future (expected) price on demand. Equation (14) is the solution to this equation, expressing demand as a function of future expected user-cost variables on demand, conditioned on the beginning-of-the period state variable. Equation (19) shows that the demand equation of Becker et al. [

Research supported in part by the North Carolina Agricultural Research Service, Raleigh, North Carolina 27695.

Wohlgenant, M.K. (2017) The Intertemporal State Adjustment Model. Theoretical Economics Letters, 7, 582-588. https://doi.org/10.4236/tel.2017.73043