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A Dynamic Analysis of Land PricesAuthor(s): Jean-Paul Chavas and Alban ThomasSource: American Journal of Agricultural Economics, Vol. 81, No. 4 (Nov., 1999), pp. 772-784Published by: Oxford University Press on behalf of the Agricultural & Applied Economics AssociationStable URL: http://www.jstor.org/stable/1244323 .

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A DYNAMIC ANALYSIS OF LAND PRICES

JEAN-PAUL CHAVAS AND ALBAN THOMAS

A dynamic model of land prices is developed. It derives arbitrage asset prices under nonadditive

dynamic preferences, risk aversion, and transaction costs. The model nests as special cases risk

neutrality, time-additive preferences, the static capital asset pricing model (CAPM), as well as the dynamic consumption-based CAPM. The model is applied to the analysis of U.S. land prices for the period 1950-96. The econometric results provide evidence showing that U.S. land price patterns are inconsistent with risk neutrality or with the static CAPM model. No strong evidence was found against time-additive preferences. The econometric findings indicate that both risk aversion and transaction costs have significant effects on land prices.

Key words: asset pricing, land, risk, transaction cost.

Agricultural land prices in the United States have fluctuated widely over the last decades. They have been subject to a sharp apprecia- tion in the 1970s, followed by a significant depreciation in the early 1980s. This has stim- ulated much research on the factors influencing land prices. A common approach has been to rely on capitalization formulas, where land prices are equal to the discounted expected value of the stream of future farm income or rent (e.g., Burt, Alston, Melichar). While it is clear that farm income can help explain a significant portion of land price movements, such a simple approach has left many issues un- settled. For example, Featherstone and Baker; Clark, Fulton, and Scott; and Falk have presented evidence that expected farm income (or rent) does not explain some farmland price variations. This indicates that simple capitalization formulas fail to provide an accurate representation of land prices. Similarly, Just and Miranowski found that real land values and recent changes in returns to land had op- posite trends. This suggests a need to examine other factors (besides expected farm income) that affect land pricing.

There is empirical evidence that land prices are also affected by risk and risk aversion (e.g., Just and Miranowski, Chavas and Jo- nes). The capital asset pricing model (CAPM)

allows for the effects of risk and risk aversion on asset prices (e.g., Barry, Robison, and Neartea). Yet, Bjornson and Innes have found that standard asset pricing models applied to land yield some anomalies. For example, they show that mean returns to farm assets are lower than those on comparable risky nonagricultural assets. Also, using a time-varying risk premium, Hanson and Myers found that CAPM models fail to explain adequately land price variations. This suggests that there are factors not captured by a CAPM representation of asset prices that play significant roles in the determination of land prices. A possible explanation for failures of arbitrage-based models to adequately predict land prices is that the capital market is segmented, hence leading to a costly arbitrage process between farm and nonfarm equities. In particular, transaction costs may act as barriers to nonfarm capital flow in agriculture.

Using a CAPM model, Shiha and Chavas uncovered statistical evidence that transaction costs have significant effects on land prices. Just and Miranowki argued that transaction costs are important and incorporated them in their land price model under risk aversion. However, they just assumed that transaction costs influenced land prices without testing for their statistical significance. This suggests a need for further research evaluating such effects.

There have also been concerns raised about the nature of intertemporal preferences and their effects on asset valuation (e.g., Koop- mans, Epstein, and Zin 1991; Barry, Robison,

The authors are, respectively, professor of agricultural and applied economics, University of Wisconsin, Madison, and assistant professor of economics, INRA, University of Toulouse, France.

This research was supported in part by a Hatch grant from the College of Agricultural and Life Sciences, University of Wisconsin, Madison. The authors would like to thank anonymous reviewers for helpful comments on an earlier draft of the article.

Amer. J. Agr. Econ. 81 (November 1999): 772-784

Copyright 1999 American Agricultural Economics Association



Chavas and Thomas A Dynamic Analysis of Land Prices 773

and Neartea; Knapp and Olson). For example, this has been illustrated in part by the so- called equity premium puzzle: Why is the his- torical return on stocks so much higher than the return on Treasury bills? Unless one is willing to assume very high levels of risk aversion, risk aversion alone does not seem able to explain this difference (see Kocher- lakota for an overview of this literature). It has become known that this puzzle can pos- sibly be resolved if one is willing to relax some key assumptions found in standard expected utility models that are time additive. In particular, Epstein and Zin (1989, 1991) have developed an asset pricing model that relies on nonadditive nonexpected utility models. This generalized specification is very flexible in the sense that it allows for a sep- arate characterization of intertemporal substitution and risk aversion (a feature not shared by the time-additive expected utility model). Epstein and Zin have investigated the implications of their model for asset pricing and presented empirical evidence showing that their dynamic representation of preferences can help solve the equity premium puzzle (Ep- stein and Zin 1991, Epstein and Melino, Ko- cherlakota). This raises the question, Could a proper understanding of land prices require dynamic models that are more general and less restrictive than the traditional time-additive expected utility models?

The objective of this article is to develop and apply a model of asset pricing that in- corporates risk aversion, transaction costs, and dynamic preferences that are not restrict- ed to the traditional time-additive expected utility models. The model generates arbitrage- pricing rules that are then used in an analysis of U.S. land values for the period 1950-96. This involves an econometric estimation of the derived asset pricing rules, generating useful information about the factors influencing land values in the United States.

The article is organized as follows. A model of asset pricing is first developed. It extends the Epstein-Zin approach by incorporating transaction costs into their dynamic preference specification. It shows how risk aversion, non-time-additive preferences, and positive transaction costs influence asset pricing. The model nests as (restrictive) special cases the following situations: risk neutrality, time-additive expected utility models, and zero transaction costs. This provides a basis for an investigation of the relevance and effects of such factors on land prices. Our analysis then

presents alternative asset pricing formulations that are particularly convenient for empirical analysis. The formulation also shows that our model can generate as special cases some pricing rules that are similar to those obtained from the static CAPM model as well as from the intertemporal consumption-based CAPM. This can provide useful insights on the empirical relevance of these two models. We then discuss the data and the econometric approach of our analysis applied to U.S. land values for the period 1950-96. The model is estimated in its structural form, using the generalized method of moments (GMM) method. The econometric results are presented next. First, we evaluate the validity of our econometric representation of land prices, looking at good- ness-of-fit measures and the orthogonality properties of the instruments used in the GMM method. Second, the econometric results provide statistical evidence about the effects of risk aversion, nonadditive preferences, and transaction costs on U.S. land prices. Implications of our findings for land pricing rules and the functioning of land markets are then discussed. Finally, concluding remarks are presented.

The Model

Consider an agent making consumption and investment decisions over time. Let y, > 0 be consumption at time t, and at =

(aot, alt, a2t,

. aKt) > 0 be a (K + 1) vector of assets, where akt is the kth asset held by the agent at time t, k = 0, 1, ..., K. Let ao, be a riskless asset, while (al,, a2t,, .... aKt) are risky assets whose future returns are not fully predictable. Denote by m, = (mo,, mt,, m2t, ..., mKt) a (K + 1) vector of investments, where mkt is the investment (or disinvestment if negative) in the kth asset made by the agent at time t. The state equation representing asset dynamics at time t is

(1) akt = ak,t-1 + mkt,

k = 0, 1, 2, ..., K.

Denote by q, the market price of the consumption good y,, and by (Pi,, p2t, ... PKt) the vector of market prices for the risky assets at time t. Let vkt be the transaction cost of buying or selling one unit of the kth risky asset akt at time t, k = 1, 2, ..., K. The riskless asset ao, is assumed to have price unity, and



774 November 1999 Amer. J. Agr. Econ.

to yield a next-period return of r,,, ao,, where rt+, is the risk-free rate of return. The risky assets (alt, a2t -...* aKt) generate an uncertain next-period return denoted by the differentiable function rr,+l(al,, a2t,, . , aK). The agent's budget constraint at time t is

(2) rao,~1 + Trt,(a1,1, ..... alt-, ) K

Sqy, + mor + [Pk, + Vkt(mk,)]mk,. k=l

Equation (2) states that, at time t, the return from riskless investment (rtao,t-_) as well as risky investment (Tr,) must equal consumption expenditures (q,y,), plus investment cost in the riskless asset (mo,) as well as risky assets

(kK=-1

[Pkt + vkt(mkt)]mkt). The function vkt(mkt) in equation (2) rep-

resents the unit transaction cost of buying or selling the kth risky asset mkt. The absence of transaction costs would correspond to a situation where vkt = 0, that is, where market transactions are done costlessly. Alternative- ly, we can expect vkt # 0 if market transactions in mkt are costly due to information cost, bro- kerage fees, time cost, etc. In this case, we will consider the following specification for vkt(mkt):

(3) vkt(mkt) = vk > 0 if mkt,> 0,

= 0 if mk = 0,

= vk < 0 if mkt < 0

where vk+ and (-vk-) are, respectively, unit transaction costs for investing (mkt > 0) and disinvesting (mkt < 0), k = 1, 2, ... , K. Such a specification has the following intuitive in- terpretation. When mkt > 0, transaction costs mean that the unit net price received by the agent for buying mkt is higher since (Pkt + Vk ) > Pkt. Alternatively, when mkt < 0, transaction costs mean that the unit net price paid by the agent for selling mkt is lower since (Pkt + Vk) < Pkt. This reflects a situation where transaction costs tend to reduce the income of market participants under any circumstance. This suggests that transaction costs provide a disincentive for agents to participate in markets.

Note that the budget constraint (2) is continuous everywhere. In the absence of transaction costs (vkt = 0), it is also differentiable everywhere. However, in the presence of transaction costs (vk, $ 0) given in equation

(3), it is differentiable everywhere except at the point mkt = 0. Indeed, at mkt = 0, equation (3) implies a different slope of the budget constraint with respect to mk, comparing an increase versus a decrease in mk, from zero. The implications of this "kinked" budget constraint under transaction costs will be explored below.

Assume that the agent receives utility from consuming y,, yt1, Yt+2, .... In period t, current consumption y, is deterministic, but future consumption y,?, Y,+2,

...., is uncertain. We

assume that the lifetime utility function representing the agent's preferences at time t has the following recursive form (Koopmans):

(4) U, = W(y,, yt+, yt,+29 ...) =

U(y,, M(U,?

III))

where M(U,,+II,)

is an aggregator function representing the certainty equivalent of future consumption given the information I, available to the agent in the planning period. Fol- lowing Epstein and Zin (1989, 1991), we will assume the following specification for equation (4):

(5a) U, = [(1 - P)y," + 3MtP]"P,

for 0 # p < 1,

= (1 - P)log(y,) + p log(M,),

for p = 0

where

(5b) M, = M(U,,,+ I) = (EUt+1,)l()

if 0 xot < 1, = exp[E,log(U,.,)] if xo = 0

yt > 0, 3 = 1/(1 + 8), and E, is the expectation operator based on the information available at time t. When future consumption is deterministic, this specification is a utility function exhibiting a constant intertemporal elasticity of substitution u = 1/(1 - p) and a rate of time preference 8. Also, the parameter o can be interpreted as a relative risk aversion coefficient with the degree of risk aversion increasing as o falls (Epstein and Zin 1989, 1991). The specification (5) is flexible since it permits the separation of intertemporal sub- stitutability issues from risk aversion (Epstein and Zin 1989, 1991). Also, when o = p # 0, equation (5) reduces to U, = [(1 - P3)E, •j>o




dollars

marginal value of investment (MVI) MV2

Pkt + Vk zone of "no investment"

S+ Vk

0 mk mkt(3) mt(1) mk(2)

Figure 1. Investment under transaction costs

3jyo,+J]"1, which corresponds to the familiar expected time-additive utility specification.

We assume that the function U(y,, M(Ut+,,,)) is differentiable and bounded for all feasible (Yt, at, mt). The individual's decisions can then be represented by the following recursive op- timization problem:

(6) V,(at-1) =

max{U(yt, M,(Ut+l)):

equations (1) and (2)},

= max U

rtao,t_

+ 7r,

- (ao,

- ao,t1)

K

- (Pkt + Vk,)(ak, t- akt-1) k=l

- qt, Mt(Vt+,(at))

where V,(at,_) is the indirect objective function (or value function) at time t, and ar, =

Wr,(al,,_ , ..., a,,_1). Under appropriate differentiability assumptions, the first-order necessary conditions for a, in equation (6) are

(7a) a0,: (JU/lay)/q, = (aU/8M,)(aM,/tao,)

(7b) ak: (U/ayt)(pkt + vkt)Iq, =

(aU/Mt)(aM,/lakt), if mkt 5 0

(7c) (aU/ay,)(pkt vk )/qt

- (aU/aMt)(aMt/akt)

- (OU/ay,)(Pkt + v)/qt, if mki = 0

for k = 1, 2, ... , K. Equations (7b) and (7c) reflect the kinked budget constraint at mkt = 0 under transaction costs. As illustrated in figure 1, the presence of transaction costs (vk, t 0) in equation (3) generates three possible scenarios. Scenario 1 is obtained under equation (7c), when the marginal return to investment is moderate and no investment takes place (mkt = 0). Scenarios 2 and 3 correspond to equation (7b). Scenario 2 is obtained when the marginal return to investment is high, stim- ulating investment (mk, > 0). And scenario 3 is obtained when the marginal return to investment is low, inducing disinvestment (mk, < 0). Figure 1 illustrates that the zone of "no investment" tends to increase with higher transaction costs.

The value function V,(a, 1) in equation (6) is continuous everywhere. In the absence of transaction costs (vk, = 0), it is also differentiable everywhere under appropriate regu- larity conditions. And in the presence of transaction costs given in equation (3), it is differentiable everywhere except at the boundary points where mink, switches away from zero. Indeed, at the boundaries between market par- ticipation and nonparticipation (i.e., at the switching points between

mkt, > 0 and

mkt, =

0, and between mk, < 0 and mk, = 0), the




budget constraint is kinked and so is the value function. However, in the language of measure theory, these points of nondifferentiabil- ity of

V,(a,_l) are "few" and occur "with

measure zero." Thus, the value function V,(a,_-) is differentiable "almost everywhere," that is, everywhere except at the few points where a kink exists.

At points of differentiability (i.e., almost everywhere), the envelope theorem applied to equation (6) yields

(8a) aV,/aoa,_1 = (U/lay,)(1 + r,)/q,

and

(8b) V,laaka,_ = (aU/ay,)[(aOr,/aak,_-l) + (Pk, +

Vk,)]/q,, if mkI # 0

(8c) = (aU/&y,)[(Tr,/&ak,_t-) + pktlq,,

if mt, = 0.

Equations (8) link the marginal utility of the assets a,_, to the marginal utility of consumption y, at time t. Also, we need to evaluate the derivatives aM,l/aa, in equation (7). From equation (5b), the derivative of M, with respect to some argument y of

U,+, is M,/lay

= M-"Et[U;-,'(aU,t+,ay)]. Assuming that mk,,

0 0, use this expression after substituting equations (8) into equations (7) to obtain

(9a) (U/lay,)/q, = (aU/aM,)

x (M -oE,[U+-,'(aU/ay,+,)(1 + r,+,)/q,+,])

(9b) (U/ly,)(pk, + vkt,)q, = (aU/aM,)

x {M,-oE,[U,O+,'1(aU/ay,+,)

x [(aI,+,l/aak,) + (Pk.t+ + Vk,t+)]/qt+ll]},

if mkt,

0

(9c) (3U/3y,)(pk, + v )/q,

- (aU/aM,)

x {M: -,E,[U?+.,'(aU/ay,,

)

X [(aTI,+ ,l/ak,)

+ (Pk.t+ + Vkt+l)]/qt+I]}

- (aU/Iy,)(Pkt + vj)/q,, if mk, = 0

k = 1, ..., K. Equations (9) are Euler equations characterizing the optimal price dynamics corresponding to optimal investment under rational economic behavior. Equation (9a) is the arbitrage equation related to investments in the riskless asset and equations (9b)-(9c) are the price arbitrage equations for the kth risky asset. Note that the effects of monetary values in equations (9) are in real terms: they are all deflated by the consumer price q, or

q,+,, thus expressing all monetary effects in

terms of their purchasing power. Equations (9) provide the basic specification for the economic analysis presented below.

Note that substituting (9a) into (9b) yields (Pkt +

Vkt).E,[Ul(aU/ay,+l)(1 + r, l)/q, l],

(10a) = E, { Ut'(aU/lay,+,)

x [(a7rt,?/ak,) + pk,,t + vk,,t+]lq,+1

k = 1, 2, ..., K. An interesting special case of equation (10a) is worth noting. Assuming that a = p = 1, we obtain the standard time- additive model under risk neutrality. Indeed, in this case, Ut+0-1 (aU/ay,j+) = 1 - 13, and equation (10a) becomes

(10b) (pkt + Vk,)

= E,{[(a, t+/aakt) + Pk,t+ + vk,t+ ]q,?1} + E,[(1 + rt,1)/q,1+,]

k = 1, ..., K. This is a standard arbitrage pricing formula. For example, in the case where the rate of return

r,+, and prices q,+,

are known at time t, equation (10b) states that current asset prices must equal the discounted value of next-period return (awr,+/laak,) plus next-period asset prices (Pk,t+I + vk,t+1), the discount factor being 1/(1 + r,+ ). This is the intuitive result that, at the optimum, marginal discounted return plus marginal capital ap- preciation equals zero. However, it is worth stressing that equation (10b) holds only under restrictive conditions. Indeed, comparing equations (10a) and (10b), it is clear that assuming time-additive model under risk neutrality does affect arbitrage pricing.

In the more general case allowing for risk aversion and nonadditivity over time, equation (10a) provides the appropriate arbitrage pricing formulation. It implicitly involves the covariance between the utility

U+4,' and the

marginal utility of income at time (t + 1). While this covariance is zero under risk neutrality (or

= 1), it becomes nonzero under risk




aversion (a < 1). Then, risk aversion can provide incentive for consumption and/or income "smoothing." Also, implicitly through the term U, 1, equation (10a) allows the parameter p (reflecting the elasticity of intertemporal substitution) to affect asset price dynamics. The economic assessment of these effects is discussed next.

Specification

Equations (9) provide a basis for the analysis of dynamic arbitrage pricing under general conditions. However, they are not in a form that can be used directly. The main issue is the empirical tractability of the term U t+I(aU/ay,t+) in equations (9). In this sec- tion, we derive relationships that express this term in a form that is econometrically tractable.

From equation (5a), note that aU/ay,?, =

UI+((1 - P3)yr 1. It follows that U;+-1(aU/ay,+1)

= U,-f(l - P)y,11.

Note that the term U,•-p = 1 whenever ao = p, which corresponds to

the expected utility specification. Thus, in this (restrictive) special case, there is no need to evaluate U,+, in the econometric specification of equations (9a) or (9b). However, any departure from expected utility (i.e., . # p) re- quires an evaluation of U,+ . One way to do this is to solve numerically the dynamic programming problem (6) for the value function

V,+,(a,). However, this can be complex. An

alternative is to try to obtain analytical results on the value function without obtaining an explicit solution to equation (6). This second alternative is explored here.

Denote the agent's wealth at time t by A,, where

K

(1la) A, = ao, + 1 (Pk, + kt) ak,. k=l

Define the gross return at time t as G, = (1 + r,)a0,t-l + EI (Pk, + Vk,)ak,t + -t(al,,t-,

.. aK,,). Then, the gross rate of return on wealth at time t is R, = G,/A,_I. It follows that the budget constraint (2) can be alternatively written as

(lib) A, = R,A,_I - q,y,.

Note that equation (1 la) involves only variables at time t, implying that all the dynamics are captured by equation (1 ilb). In the language of dynamic programming, treating

(1 lb) as the state equation, this means that the variables (y,, ao,

..... aK,) can be treated

as "control variables," while A, is the "state variable." This implies that the dynamic programming problem (6) can be alternatively expressed as

V,(A,_i) = max U

R,A,_tI - ao,

At,at

K - (Pk, + vkt)akt

k=l

+ q,,

M,[V,t+(A,)]},

where the value function is now written

V,(A,_,). Using the envelope theorem at points of differentiability (i.e., almost everywhere), we obtain aV,/aA,_, = (aU/ay,)(R,/q,). Using equation (5a), it follows that

(12) aV,I/IaA,

= (aUt+llayt+,)(R,+l/qt+) = Utl+f(1 - 3)yt,+P1(Rt+,/qt+,).

We now make the following assumption.

ASSUMPTION Al. The return function 7r,(al,, a2,t ..* aKt) is linear homogenous in (al,, a2,,

?. . ,

akt).

Note that the utility function specification U, in equation (5) is linear homogeneous in the consumption stream. Under assumption (Al), the budget constraint in equation (2) is also homogenous in (y,, a0,, al,,

a2,,...., aK,). Giv-

en the homogeneity of equations (Ila) and (1 b), this implies that the value function V,(A,_1) is linear homogeneous in A,_,, that is, that V,(A,_-) is proportional to

A,_l.' Under

assumption (Al), this means that the value function can be expressed as V,1, =

y,,lA,. Combining this result with equation (12) yields

V,+, =

U+p(1l -

p)yp(R,+l/q,+l)At,,or, for p # 0,

(13) U,+,

= [(1 - f3)y,y+1A,R,,+/q,+ll/P.

In the case where p # 0, it follows that

' To see that, let V(A) = maxb (b, A), where f(b, A) is linear homogenous in (b, A) and satisfies Xf(b, A) = f(Xb, XA), for all X > 0 and all feasible (b, A). It follows that V(XA) = {maxb, f(b, XA)} = {X maxbf(b, A)} = XV(A), thus proving that V(A) is linear homogenous in A.




(14) Upll (a Ulayt l)

= U,"+P(1 - P)y+I1

= [(1 - P)yP,-IARt+l Iq,+

I](O-P)/

x (1 - 13)ytpI.

Using equations (5a), (13), and (14), equations (9) can be alternatively written as (see the derivation in the appendix)

(15a) 1 = E,(({qtq/qt+I)Y(yt+ I/yt)Y(P 1)

X (Rt+1) -Y'(1 + rt+,)}

(15b) pkt + vkt = E,{(p3q,/q,+I)-(y, l/y,)(P-'I)

X (R, l)-

X (arr,+ laakt + Pk,t+ 1

+ Vk,+1)},

if mkt, 0,

pkt + V k E,{(pq,/q,+,)-(yt+ ly,)Y-1)

X (R,+,))-I

X (awt+,/aakt + pk,t+ +

vk,t+1)

(15c) Pk + Vk, ifmkt = 0

k = 1, ..., K, where y = a/p. Again, equations (15b) and (15c) are illustrated in figure 1, where the marginal value of the kth asset is MV =

E,{(Pq,/q,+,)/(y,+m/y,)Y(P-)(R,m)Y-I (ar~+,,laak, + Pk,t+l + vkt+1)}. As discussed above, this identifies three possible regimes: regime 1 corresponding to a moderate MV where no investment takes place (mk = 0); regime 2 corresponding to a high MV and positive investment (mk > 0); and regime 3 corresponding to a low MV and disinvestment (mk < 0).

Expressions (15) give general asset pricing relationships where the term

[(pq,/q,+,)Y(y,,1/ y,)Y(P•-)(R,+,)-)Y] is used to discount future payoffs in the determination of current asset prices. This includes as special cases some well-known formulas for asset pricing. For example, having y = 0 generates the static capital asset pricing model (CAPM) where the discount factor becomes

1/R,+1, which de-

pends only on the gross return on wealth and not on the growth rate of consumption (y,+,/ y,). In this case, the riskiness of an asset is measured in equations (15) by the covariance

of its return with the overall return (see Jen- sen). Alternatively, having y = 1 gives the intertemporal consumption-based capital asset pricing model (CCAPM) where the discount factor is [(pq,/q,, )(y,,I/y,)'P•-], which depends only on the growth rate of consumption (Y,+i/y,) (and not on the return R,,+). In this case, the riskiness of an asset is measured in equations (15) by the covariance of its return with the growth rate of consumption (see Merton, Breeden). This is a situation where risk management under risk aversion clearly involves "consumption smoothing" activities.

In the general case where y # 0 and y # 1, equations (15) give asset pricing formulas that generalize the CAPM model and the CCAPM model. Indeed, in equations (15), the factor

[(pq,/q,,m )-(y,, /y,)-Y'P)(R,,,)V-']

used to discount future payoffs in the determination of current asset prices depends both on overall return R,,, and on consumption growth (y,,I / y,). In this case, the riskiness of an asset is measured by the covariance of its return with both the overall return (as measured by R7; ') and consumption growth (as measured by

(Yt+l/Yt)yp-"1)). Finally, note that equations (15) are con-

sistent with the results obtained by Epstein and Zin [e.g., equation (16) in Epstein and Zin 1991, p. 268)]. Our article extends their analysis in two main directions. First, while Epstein and Zin assume implicitly that q, =

1 for all t, we incorporate explicitly the role of inflation in the determination of asset pricing. In equations (15), this is measured by the term (q,/q,+,)- which reflects the relative change in consumer prices. Second, we intro- duce transaction costs in Epstein and Zin's analysis and derive their implications for dynamic asset pricing. This is reflected by the terms vkt, and vk,t+I in equations (15). It gives a basis for investigating empirically the role of transaction costs in the determination of asset prices.

Equations (15) provide a parametric model of dynamic asset pricing that can be estimated econometrically. The empirical investigation of this model in the context of land prices is presented next.

Data and Estimation

The above model was developed at the microeconomic level. Unfortunately, time-series data on investment, consumption, and asset




prices are quite rare at the micro level. The best time-series data available for analyzing the determinants of land prices are at the aggregate level (e.g., USDA). Such data con- straints are the main reason why we focus our empirical analysis here at the aggregate level. This raises the issue, Are the microeconomic implications derived above also valid at the aggregate level?

First, consider the situation in the absence of transaction costs (vk, = 0). As one moves from a micro to an aggregate analysis, we expect to find significant differences in wealth across agents. However, if we adopt the assumption that agents are similar otherwise, then homothetic preferences given in equation (5a) imply that aggregate behavior is consistent with the behavior of a representative agent (e.g., Epstein and Zin 1989, p. 956). In this case, all decisions are proportional to wealth and the pricing rule in equations (15) depends only on relative quantities or wealth. This means that equations (9) or (15) would apply as well at the aggregate level, as if they were associated with a representative agent. Thus, in the absence of transaction costs (where Vk, = 0), equations (15) can be expected to provide a useful basis for aggregate analysis.

In the presence of transaction costs (vk, t 0), aggregation becomes a more complex issue. The problem arises if we aggregate across both buyers and sellers that face different objective economic conditions. As reflected in equation (3) under transaction costs, buyers pay a higher net purchase price while sellers receive a lower net selling price. Because they face different net prices, buyers and sellers may have different decision rules, suggesting that aggregate behavior would de- pend on each group's decision rules. Also, the implications of these decision rules for aggregate asset pricing can be complex. Such complexities create significant challenges for empirical modeling. Here, we propose to handle such complexities as follows. As argued above, in the context of a representative agent, we expect equations (15) to provide a good basis for an aggregate analysis of asset pricing in the absence of transaction costs. We also expect equations (15) to give some basis for investigating the role of transaction costs at the aggregate. As long as decision rules are not too different among market participants, one can expect the parameters x, I3, p, and y in equations (15) to represent some "average behavior" of participants in

asset markets. However, one expects the im- pact of transaction costs on aggregate behavior to differ from equations (15). The reason is that, given equation (3), equation (15b) differs between buyers (where vk, = vk > 0 when mk, > 0) and sellers (where vk, = v~ < 0 when mk, < 0). Since both buyers and sellers are present in the market, the aggregate effects of transaction costs on asset pricing should be some (weighted) average of their micro effects across market participants. For example, one might expect transaction costs to affect asset prices in a way similar to v +

in equations (15) in a "buyers' market" (when buyers' behavior dominates), but similar to vk in a "sellers' market" (when sellers' behavior dominates). We will rely on such arguments in the specification, estimation, and testing of transaction cost effects in the aggregate econometric analysis presented below.

Our analysis focuses on the aggregate analysis of annual land prices in the U.S. between 1950 and 1996. Data on land values (p,, measured in 1,000 dollars per acre), aggregate land quantity (Q,, measured in million acres), gross rate of return on farm equity (R,), and net farm income per acre (rr/ a, measured in 1,000 dollars per acre) were obtained from USDA publications. Under assumption (Al), constant return to scale means that we can measure the marginal return arlaa by the average return rr/a. Also, the price of consumption goods (q,) is measured by the Consumer Price Index (CPI) reported by the Bureau of Labor Statistics. The risk-free interest rate r, is measured by the interest rate on U.S. Treasury bills. Fi- nally, we calculated consumption y, by in- terpreting (q,y,) as disposable personal income of the farm population (measured in billions of dollars), as reported by the USDA.2 These data, summarized in table 1, provide a basis for estimating equations (15).

We estimate model parameters using the system of equations (15a)-(15b) [rather than equation (15b) alone], where equation (15b) represents arbitrage pricing for land. Such an approach allows us to obtain more efficient parameter estimates by exploiting all the model information. This implies that parameter restrictions across equations are imposed and

2 Disposable personal income is calculated from net farm income, plus nonfarm income, minus taxes.




Table 1. Descriptive Statistics

Standard Variable Mean Deviation Minimum Maximum

q, (= 1 in 1982-84) 0.66 0.42 0.26 1.52 y, (billion dollars) 60.43 8.92 40.19 93.89 Q, (million acres) 1,085.11 79.02 974.00 1,206.00 p, (1,000 $/acre) 0.37 0.26 0.07 0.76 a, (billion dollars) 416.09 263.13 115.70 816.40 R, 1.04 0.07 0.86 1.24 7r,/a, (1,000 $/acre) 0.023 0.014 0.009 0.055 r, 0.054 0.028 0.009 0.141

Note: Number of observations is 45.

are easily tested for by investigating possible model misspecification.

The Generalized Method of Moments (GMM) is the most popular estimation procedure when dealing with Euler equations. GMM estimates are fairly easy to obtain and are consistent provided an adequate set of instrument variables is used. It is known (e.g., Epstein and Zin 1991, Hansen and Singleton) that the choice of instruments is an important issue with GMM, as parameter estimates can be sensitive to competing sets of instruments. This suggests a need to assess the relevance of instruments used in GMM estimation.

Based on the orthogonality condition E u(O) 'W = 0, the GMM estimation method involves minimizing the distance:

(16) u(O)'W(W'QW)- W'u(0)

where 0 is the (p x 1) vector of parameters, W is a (2T x k)-instrument matrix, 11 is the (2T x 2T) variance-covariance matrix associated with the above orthogonality condition, and u(0) is the (2T x 1) vector of residuals defined as

(17) u(O) = (u,,(0), u,12(), . . . U,(),

u21(0), u22(0), .. 2T. )) where3

ut, = 1/q, - (Pq,1/qI)-,(y,,Ily,)t -

X (R,+,)Y'-I(1 + r,+,)/q,

u2t = P, + vt (tqtq,,+I) (yt+, /y,)vct-1

X (R,+ )Y-

X (a7r,+t/ aa+ +

p,+ +

v,+ ).

Several remarks are in order. First, the general form of the GMM estimator defined above allows for different instruments in each equation. We choose to use the same set of instruments for equations (15a) and (15b); hence, the instrument matrix W is constructed by du- plicating the original (T x k) set of instruments. Thus, there are a total of (2 x k) identification restrictions. In general, one wants to choose instruments among variables that are known to both the econometrician and the economic agents at the time of their decisions. Such instruments would satisfy the orthogonality condition Eu(O)'W = 0 and reflect the nature of price and economic expectations used in decision making. The relevance of the instruments and the associated orthogonality conditions can be assessed by the Hansen test, evaluating the validity of the overidentification restrictions. Second, consistent and robust parameter estimates obtain using a two-stage procedure. In the first stage, the variance-covariance matrix 11 is consistently estimated by a robust procedure allowing for heteroskedasticity and serial correlation of residuals (Newey and West). The matrix fi is then replaced by its estimate in the GMM criterion and final parameters are estimated.

Before estimation, we need to specify a parametric structure for transaction costs in equation (15b). At the micro level, equation (3) implies that transaction costs v, are positive (negative) when assets are sold (bought). As argued above, one expects this pattern to hold on the average at the aggregate. As a result,

3 For simplicity, we suppress the subscript k on asset equity and price.




Table 2. GMM Estimation Results, 1950-96

Param- Standard eter Estimate Error

p 0.7561** 0.0352 y 1.0622** 0.0633 p 0.9779** 0.0053 cm, 0.0172* 0.0064

C, 0.1366 0.1086 Oa 0.8031** 0.0744 R2 equation (15a): 0.9980

equation (15b): 0.9303

Notes: * and ** denote a parameter significantly different from zero at the 5% and 1% levels, respectively. Instruments used: (1, p,_t, y,/y,_,, q,/ q,1_, Rt,).

in aggregate analysis, we hypothesize that transaction costs are proportional to the variation in the volume traded. This generates the following specification for transaction costs:

Model MI: v, = cAQt if AQ, > 0,

ScmMAQt if AQ, < 0,

where Q, is land quantity at time t, AQ, = Q, - Qt,_l, and c, and cm are parameters. Hence, marginal transaction cost is proportional to a positive change in land quantity, with parameter c, > 0, and is proportional to a negative change in land quantity, with parameter cm > 0. Asymmetry in the marginal transaction cost is allowed for when the parameters c, and cm are different. Al- ternative models obtain by appropriate restrictions on model M 1:

Model M2:

v, = cAQ, (symmetric transaction costs, proportional to changes in quantity);

Model M3:

v, = 0 (no transaction cost).

Table 3. Hypothesis Testing

Model M2 is obtained by imposing the restriction

cp = cm = c on model Ml. The hy-

pothesis of no transaction costs of model M3 obtains through the restriction c, = Cm = 0 imposed on M1.

Parameters to be estimated are therefore 0 = (., p, "y,

Cp, Cm) for the unconstrained model Ml. Note that we follow here the notation of Epstein and Zin (1991) for structural parameters, where the risk aversion coefficient ae is embedded in the parameter y = o/p.

Estimation Results

The model is estimated using the two-stage GMM procedure described above. Several sets of instruments were considered to obtain consistent parameter estimates. The selected set of instruments is (1, pt-1, y,/yt-1, qt/q,_1,

R,_I) for equations (15a) and (15b). It was

found that increasing the size of these instrument sets with any other combination of instruments improved efficiency of parameter estimates but often failed the Hansen overidentification-restrictions test. Hence, we exploit ten orthogonality restrictions for the model, leaving five overidentification restrictions. Model performance was also en- hanced when serial correlation of residuals was taken into account in the computation of the variance-covariance matrix ni. We set the number of lags to four in the Newey and West robust estimation procedure.

Parameter estimation results are in table 2, and tests of the various hypotheses are presented in table 3. Using the Hansen test of the overidentification restrictions, we do not find statistical evidence against the orthogonality conditions between the instruments and the error terms. This suggests that the model specification and the choice of instruments appear appropriate. By using a rather large number of overidentification restrictions, parameter efficiency is increased

Test Statistic p-Value

Overidentifying restrictions (Hansen test) X2(5) = 3.0769 0.6881 No transaction costs (c, = Cm = 0) X2(2) = 7.5591 0.0228 Symmetric transaction costs (c, = Cm) X2(1) = 1.2407 0.2653 Expected utility (y = 1) X2(1) = 0.9643 0.3261 p = 1 X2(1) = 47.7823 0.0000 P= 1 X2(1) = 17.0069 0.0000 Risk neutrality (a = 1) X2(1) = 7.0020 0.0081




while orthogonality restrictions are not rejected. Statistical fit is good for both equations, with an R2 equal to 0.9980 for equation (15a) and 0.9303 for equation (15b). Thus, our model has a high explanatory power for land prices.

The structural parameter a (= y-p) is estimated together with its asymptotic standard error, using consistent estimates for the parameters y and p. Risk neutrality is tested by considering the null hypothesis Ho:

a- = 1. A Wald test statistic rejects this hypothesis at the 5% significance level, hence pro- viding strong evidence that market participants are risk averse. Recall that risk aversion increases as parameter ax decreases away from 1. Estimates of xo and p are very close, indicating that the hypothesis of expected utility may not be rejected. This is confirmed by a Wald test statistic [distrib- uted as a X2(1)] which fails to reject the null hypothesis (Ho: y = 1) at the 5% significance level. Our results therefore indicate that there is no strong evidence of departure from the expected utility hypothesis. This con- trasts with Epstein and Zin (1991) who reported strong rejection of this hypothesis for most of their instrument sets. This difference may be due to the fact that Epstein and Zin used more disaggregate consumption commodities in their analysis. Finally, our estimate of the parameter p is close to 1. It is larger than the one estimated by Epstein and Zin. The intertemporal elasticity of substitution is accordingly higher: a = 1/(1 - p) = 4.10. Again, this difference may be related to the fact that it may be easier to uncover substitution possibilities while working with more aggregate commodities. The hypothesis that parameter p is equal to 1 is strongly rejected at the 5% level.

Our estimate of the parameter y also indicates that, while y is close to 1, it is sta- tistically different from zero at the 5% level. As discussed above, this suggests a strong rejection of the static CAPM model in favor of the dynamic consumption-based CAPM model. This suggests that the effects of risk and risk aversion can be more appropriately characterized through "consumption smoothing" rather than "income smoothing." Such a result can help explain previ- ous findings by Bjornson and Innes or Han- son and Myers concerning the inadequacy of the static CAPM model in explaining land price variations.

The rate of time preference is computed at

8 = 1/P - 1 = 0.0226 for model Ml, about 2.2%. This seems reasonable, recalling that the effects of all monetary values in our model are in "real terms." Although parameter p is very close to 1, the hypothesis that this parameter is equal to 1 is strongly rejected, because it is estimated with sufficient pre- cision.

We now turn to the results concerning parameter estimates related to transaction cost. Using a Wald test, the absence of transaction costs (c, =

cm = 0 in model M3) is rejected

at the 5% level against the alternative that transaction costs are proportional to changes in quantity (model Ml). In particular, the parameter Cm (related to negative changes in land assets) is significantly different from zero at the 5% level. This provides statistical evidence that transaction costs have significant effects on land prices. To evaluate these effects quantitatively, consider the 1980s and the 1990s periods. During these periods, aggregate farm land decreased over time. Esti- mated at mean values for each period, our model shows that transaction costs were on average 12.98% of land value in the 1980s, and 7.54% in the 1990s.

Finally, the hypothesis of symmetry in marginal transaction cost is also tested (c, = c, in model M2). The Wald test statistic indicates that symmetry cannot be rejected at the 5% level. Thus, we do not find strong evidence that transaction costs differ between periods of land increase versus land decrease. Such a result may be due to the fact that the estimated value of cm in model Ml is rather small in magnitude, whereas parameter c, is not significantly different from 0 at the 5% level. This latter finding may be due to the fact that only a few sample observations were relevant for the estimation of

cp: increases in aggregate

land quantity took place only during the 1950s. Further empirical analysis may be needed to confirm whether our evidence of symmetry in transaction costs holds true in different situations and with different data.

Conclusions

In this article, we analyze land price variation using a general pricing model, including as special cases most popular specifications found in the literature. This approach is mo- tivated by the need to consider more flexible pricing models that can better explain land price movements, as capitalization-based or




CAPM models have often been rejected when applied to land prices. Regarding farmers' behavior, risk neutrality and time-additive preferences are easily tested. Transaction costs are explicitly introduced in the pricing formula and are specified as asymmetric functions of quantity changes.

Our empirical findings uncover statistical evidence that both risk aversion and transaction costs affect land prices. The fact that transaction costs are a significant component of land prices suggests the potential for capital market segmentation. Transaction costs would affect the functioning of the capital market in agriculture as well as the return to farm assets (e.g., Shiha and Chavas, Bjornson and Innes). Also, we find statistical evidence against the static CAPM model. This can help explain the inadequacies of the static CAPM model in explaining land price variations (e.g., Hanson and Myers). This stresses the need to capture adequately the effects of risk, risk aversion, and transaction costs on land prices in order to understand better the functioning of land markets.

Finally, we do not uncover strong evidence against either the time-additive preferences or the dynamic consumption-based CAPM. Con- trary to Epstein and Zin, we do not find the additive expected utility model overly restrictive. This may be due to the fact that our analysis relies on a consumer good that is more aggregate than that in Epstein and Zin (1991). Epstein and Zin worked with more disaggregated commodities and obtained a negative relationship between durability of goods and intertemporal substitution. Because we use aggregate data on consumption, possibilities of substitution may be more important, explaining why the intertemporal rate of substitution is higher in our case.

Our results supporting the dynamic CAPM model indicate that agricultural households manage risk through consumption smoothing activities. This stresses the importance of consumption data in analyzing the effects of risk on land values. While such information exists at the aggregate level, it is often not available at the micro level. Our findings stress the need for collecting better micro data in the farm sector. This would allow for more refined empirical analyses. For example, building on our conceptual model, it would be interesting to model transaction costs at the micro level by considering a regime-switching framework. At every time period, individual farmers would be in any of three situations: sell, buy,

or do nothing. The presence of transaction costs could then explain who participates in the land market, with implications for land transactions and variations in land prices. In- vestigating such issues appears to be a good topic for future research.

[Received July 1998; accepted January 1999.]

References

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Appendix

Derivation of Equations (15)

When p # 0, note that equation (5a) gives aU,/aM, = U-P3MP-', and aU,/ay, = Uh-P(1 - P)y-'l, implying that

(Al) (aU,IaM,)I(aU,Iay,)

= P3M-'I[(1 - P)y-'].

Also, equation (5a) gives MP = [Uf - (1 - p)yp]/ p. Using equation (13), this yields

(A2) M, = {[(1 - P)yp-'A,_,R,/q,

- (1 - 13)y,]/3}l'Ip.

Combining equations (Al) and (A2) with equation (14) gives

[( U/laM,)/( U/lay,)]M, -

Utl ( U/layt+, 1 )q,/q, +1

= {1/[(1 - P3)y,-']}P3M-'

x [(1 - P)yp-_lAR,, /qt+l]-" ')(1 - P)

using equations (Al) and (14)

= {1/[(1 - P)y-'] x {[(1 - 3)y•'A,_R,/q,

- (1 - f3)y]/Pf3}(P'-Y)'P

x [(1 - )yp-_'ARt l/ql]'" •-,'P

x (1 - P)y-' q,/q,+

,

using equation (A2)

= (q/q ,t+l)c"/"(ytlyt+

)"x/P)-x

x [(R,A,_, - q,y,)/(R,+lA,)1

"-(/''

= (pq, /q,+ l)c"P(ytl/y,+ )I-op) *(Rt+, )'('P-) ,

using equation (11b).

Substituting this expression into equation (9) gives equation (15).



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