Neutral Property Taxation Under Uncertainty - Springer Link

J Real Estate Finan Econ (2008) 37:211–231 DOI 10.1007/s11146-008-9132-4

Neutral Property Taxation Under Uncertainty Jyh-Bang Jou & Tan Lee

Published online: 7 July 2008 # Springer Science + Business Media, LLC 2008

Abstract In a framework where no uncertainty arises, Arnott (J Publ Econ Theor 7:27–50, 2005) investigates a neutral property taxation policy that will not affect a landowner’s choices of capital intensity and timing of development. We investigate the same issue, but allow rents on structures to be stochastic over time. We assume that a regulator implements taxation on capital, vacant land, and post-development property so as to expropriate a certain ratio of pre-tax site value as well as to achieve neutrality. We find that the optimal taxation policy is to tax capital and subsidize properties before and after development. We also investigate how this optimal policy changes in response to changes in several exogenous forces related to demand and supply conditions of the real estate market. Keywords Neutral property taxation . Real options . Tax revenues . Uncertainty JEL Classification G13 . H21 . R11

J.-B. Jou Department of Economics and Finance, Massey University Albany Campus, New Zealand J.-B. Jou Graduate Institute of National Development, National Taiwan University, No. 1 Roosevelt Rd. Sec. 4, Taipei 106 Taiwan, Republic of China e-mail: [email protected] T. Lee Department of Finance, Auckland University of Technology, Auckland, New Zealand T. Lee (*) Department of International Business, Yuan Ze University, 135 Yuan-Tung Rd., Chung-Li, Taoyuan 320 Taiwan, Republic of China e-mail: [email protected]

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Introduction Arnott (2005) investigates a neutral property taxation policy in a framework where no uncertainty arises.1 We investigate the same issue, but allow structures’ rents to be stochastic over time. We consider a landowner who chooses capital intensity and the timing of development to maximize the value of vacant land. The regulator taxes the landowner by implementing three policy instruments: taxation on capital, predevelopment land value, and post-development property value. Through the use of these instruments, the regulator not only expropriates a certain ratio of pre-tax site value, but also achieves neutrality, i.e., the landowner will develop properties at the same capital intensity and development timing both with and without these instruments. We assume that a landowner employs land and capital to provide a housing service that exhibits homogenous Cobb–Douglas production technology. The landowner irreversibly decides an optimal timing and intensity of development. Prior to development, the landowner receives no rents. After land is developed, however, the landowner, who is assumed to be risk neutral and faces a constant discount rate, receives stochastic rents that evolve according to a geometric Brownian motion. Prior to development, the decision to postpone development of vacant land confers upon the landowner a valuable option. The option’s value is expected to grow at a rate equal to the sum of the discount rate and the tax rate on the value of the vacant land. For the tax base to be finite, the tax rate on vacant land value must be negative, i.e., the regulator must subsidize vacant land. However, once the regulator subsidizes vacant land, the landowner will delay development and increase capital intensity. To exactly offset these effects, the regulator needs to use the following two policy instruments. First, the regulator must tax capital, which also delays development but reduces capital intensity, thus offsetting the stimulating effects on capital intensity induced by a subsidy on vacant land. Second, the regulator should subsidize post-development property, which accelerates development but does not affect the choice of capital intensity. As a result, neutrality with respect to both capital intensity and the timing of development are preserved. We employ plausible parameter values to quantify the magnitude of the neutral taxation policy. Our benchmark case indicates that prior to development, the regulator must subsidize vacant land value at a rate equal to 0.59%. At the instant land is developed, the regulator must tax development costs at a rate equal to 27.6%. After land is developed, the regulator must subsidize property value at a rate equal to 1.73%. We also investigate how the neutral property taxation policy changes in response to changes in various exogenous forces such as current demand conditions, the expropriation rate, the purchase cost of capital, the efficiency of capital, the expect growth rate of rents on structures, and the volatility of that growth rate. Our neutral tax policy implications differ from those of Arnott’s and those of advocates of the two-rate tax system. Unlike our results, Arnott shows that optimal 1 See also Arnott and Petrova (2006), which extends Arnott’s article to investigate the issue of deadweight loss.

Neutral Property Taxation Under Uncertainty

213

neutral property taxation is a mix of taxing post-development property, subsidizing capital, and neither taxing nor subsidizing vacant land.2 In contrast, advocates of the two-rate tax system ask for a switch from one-rate tax by increasing taxation on vacant land and decreasing taxation on improvements (i.e., decreasing taxation on either capital or post-development property value in our framework). However, our results suggest such a policy would lead to an increase in capital intensity instead of neutrality. The remainder of this article is organized as follows. We first present the basic assumptions in the following section. We then solve choices regarding capital intensity and development timing, and investigate how policy instruments affect these two choices. Next, we derive the optimal taxation policy and investigate how various exogenous forces affect it. We then demonstrate the theoretic implications of the previous section by employing numerical analysis. The last section concludes and offers extensions for future research.

Basic Assumptions Suppose that at the current date t a landowner has undeveloped land that is normalized at one unit. At any future time s, i.e., s≥t, the landowner is able to develop his unit of land. Denote the rent on one unit of housing as x(s), which follows a geometric Brownian motion as given by dxðsÞ ¼ ads þ sdΩðsÞ; s > 0; xð s Þ

ð1Þ

where the growth rate of x(s) has a constant expected growth rate α and a constant variance σ2, and dΩ(s) is an increment to a standard Wiener process. We assume that housing, Q, is produced with capital, K, and land, L. The production function is of a Cobb–Douglas type given by QðK; LÞ ¼ K g L1g, where 01 and β2t @M =@k @S=@x. We therefore need to impose the following constraint: k * > ½f g=ð1 g Þ1=ð1g Þ .

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Table 1 The impact of tk, tb and ta on x* and k* Exogenous variables

x* k*

tk

tb

ta

>0 1Þ is defined in Eq. 43 in Appendix 2. The first term on the right-hand side of Eq. 33 is the revenue collected from taxation on pre-development land value. This term is negative, suggesting that the regulator must subsidize vacant land. This is because the value of vacant land, Z(x (s)), follows the motion of xðsÞb1t . By Ito’s lemma, Z(x(s)) follows a geometric Brownian motion with a drift rate equal to ab1t þ 12 s 2 b1t ðb1t 1Þ, which is equal to ρ+ C b, by Eq. 15. To ensure that the tax base on pre-development land value is finite, this drift rate must be smaller than the discount rate ρ, thus leading to a negative value of C b. According to Proposition 1, a change in the tax rate on post-development property value does not exhibit any effect on the capital intensity. Given that subsidizing vacant land will increase capital intensity, the regulator thus must also tax capital to offset this stimulating effect. However, subsidizing vacant land and taxing capital will both lead to excessively late development. To correct this, the regulator must subsidize post-development property value. The following proposition thus follows. Proposition 2 The optimal taxation policy is a mix of taxing capital and subsidizing properties both before and after development. Proof From Eqs. 28 and 29, we find that if C k is positive, then both C a and C b must be negative. QED. Unlike our results stated in Proposition 2, Arnott (2005) shows that the neutral property taxation policy is characterized by taxing the net value of post-development property, subsidizing capital, and neither taxing nor subsidizing pre-development land. This divergence arises from the different underlying assumptions. We find that the regulator must subsidize vacant land. This is because we not only assume that structures’ rents are uncertain and follow a geometric Brownian, but also assume that the housing production technology is of a homogenous Cobb–Douglas type, which implies that the elasticity of substitution between land and capital is equal to


221

one.10 By contrast, Arnott assumes that structures’ rents grow at a positive constant rate and that the housing production technology is of a CES production type with the elasticity of substitution between land and capital being smaller than one. As a result, to ensure the tax base to be positive, Arnott does not need to impose that tb, the tax rate on pre-development land value, be negative. In fact, he finds tb to be zero when adding the two neutrality conditions. Our neutral taxation policy also differs from that of advocates of the two-rate tax system. In the United States, while most communities have uniform tax rates on land value and land improvements, several communities have adopted a two-rate tax system. Most notably is Pittsburgh, which applies a tax rate of land that is five times the rate applied to improvements (Anderson 1999). Within our framework, land improvement taxes may be represented by either taxation on post-development property value (ta) or taxation on capital (tk). We then find that a switch to a two-rate tax system will lead to the following result: (1) the development process will speed up, but the capital intensity will be ambiguously affected as a result of a decrease in the tax rate on vacant land, tb, and an increase in tk; (2) the development process will speed up, while the capital intensity will decrease as a result of a decrease in tb and an increase in ta. In sum, the switch will speed up the development process, and hence will not be able to achieve neutrality.11 We now discuss how the optimal neutral taxation policy changes in response to changes in several exogenous forces. To address this issue, we need to compare the development timing and capital intensity choices in the presence of taxation with their counterparts in the absence of taxation. The results are stated in Proposition 3. Proposition 3 For comparison, we consider the case in which no neutral taxation policy is imposed. (1) With this policy an increase in either the volatility of structures’ rents (σ) or capital efficiency (γ) will delay development and increase capital intensity. (2) With this policy an increase in either the expected growth rate of structures’ rents (α) will increase capital intensity and ambiguously affect development timing. (3) With this policy a change in the purchase cost of capital (c) or the fixed construction costs (f) does not exhibit any impact on development timing or capital intensity. Proof See Appendix 3. We use Fig. 1 to explain the reason for Proposition 3(1), which states that there is an increase in the volatility of rents on structures. Suppose that the initial equilibrium with and without neutral property taxation coincides at point A, the intersection of

10

Note that we also assume that a landowner incurs fixed construction costs, while Arnott abstracts from these costs. 11 In a non-stochastic framework, Anderson (1999) also reaches a similar conclusion. He shows that a shift to a two-rate tax system (a decrease in τb and an increase in τk) will speed the development process and increase the capital intensity if capital and time are substitutes in the land-profit function (which applies to a declining urban area). The results become ambiguous if capital and time are complements (which apply to a growing urban area).

222

J.-B. Jou, T. Lee x*

S C

xt* x0*

M

M0

B

t

Mt

M0

M A

xe* M

S

ke*

k0*

kt*

k*

Fig. 1 The impact on development timing, x*, and capital intensity, k*, when the volatility of rents on structures, σ, increases. Note that line SS graphs the condition for the choice of capital intensity regardless of the existence of neutral property taxation. Line SS will not shift when σ increases. Line MM graphs the condition for the choice of development timing regardless of the existence of neutral property taxation. When σ increases, line MM will shift to line MtMt if neutral property taxation is implemented. Otherwise it will shift to line M0M0

MM and SS, where the former applies to Conditions 21 and 21′ and the latter applies to Conditions 22 and 22′. An increase in the volatility of rents on structures increases the value of vacant land, thus increasing M(x*,k*) in Eq. 21 (where the neutral taxation policy is imposed) and M 0 ðx*; k*Þ in Eq. 21’ (where the neutral policy is not imposed). However, the former effect is more significant than the latter because vacant land is subsidized when the neutral taxation policy is imposed. This is shown by an outward shift from MM to MtMt and M0M0, respectively. On the other hand, an increase in the volatility of rents on structures will not affect the condition for the choice of capital intensity regardless of the existence of neutral taxation policy. Consequently, the new equilibrium with and without this policy will be at points C and B, respectively, or the intersections of MtMt and SS and of M0M0 and SS, respectively. As a result, an increase in the volatility of rents on structures will delay development and increase capital intensity with this policy relative to without it. Similar reasoning can be used to explain the results for the changes in the other variables stated in Proposition 3. However, all exogenous forces except the purchasing costs of capital and the fixed construction costs will ambiguously affect the optimal neutral taxation policy (see Appendix 4 for an example of an increase in the volatility of rents on structures). An increase in these two costs will not affect the neutral taxation policy because it will affect neither the two neutral conditions given by Eqs. 28 and 29, nor the revenue constraint given by Eq. 33. In the next section, we employ simulation analysis to investigate how the neutral taxation policy changes in response to changes in all the other exogenous forces.


223

Numerical Analysis We begin numerical analysis by establishing a set of benchmark parameter values as follows. A landowner incurs a fixed cost of $1 (f=1) and a proportional cost of $1 (c=1) to install capital in per unit of land. The rent per unit of developed property, x, is expected to increase at 2% per year, α=2%, the volatility of this growth rate, σ, is equal to 12% per year, and the risk-free rate, ρ, is equal to 10% per year. Additionally, the output elasticity with respect to capital, γ, is equal to 0.6, and the expropriation ratio, ε, is equal to 0.25.12 Finally, the current x is set to 0.97x*. On average, it takes 2.38 years for the rent to reach x* from its current level.13 Given these benchmark parameter values, Table 2 reports the results for the optimal taxation rates on capital, t k* , pre-development land value, t *b , and post-development property value, t a* . They are respectively equal to 27.6%, −0.59%, and −1.73%. These results conform to the results of Proposition 2. Table 2 also reports the effects of the ratio of the initial rent over the rent that triggers development (m), the expropriation rate (ε), the proportional cost (c), the output elasticity with respect to capital (γ), the expected growth rate of the rent of developed property (α), and the volatility of this growth rate (σ) on the optimal taxation rates, t *k , t *b and t *a . We explain each effect in turn. We see that a decrease in ε and an increase in either m or γ exhibit the same qualitative impacts on t k* , t *b , and t *a . Specifically, the regulator needs to decrease the subsidy rates on properties both before and after development (t *b and t *a increase), and the tax rate on capital (t *k decreases). As indicated by Proposition 2, such policy changes can achieve neutrality, and thus explain the results for a decrease in ε and an increase in m because both ε and m are unrelated to the neutrality conditions shown by Eqs. 22 and 23. The same policy changes can also apply to an increase in γ even though the regulator is required to accelerate development and reduce capital intensity for an increase in γ, as suggested by Proposition 3(1). This is because as suggested in Proposition 1, the regulator can decrease the subsidy rate on pre-development land value to meet this requirement. Such policy changes can also satisfy the revenue constraint because two of them reduce expenditures on subsidies (t *b and t *a increase), while the remaining one reduces revenues collected from taxation (t *k decreases). We see that an increase in either σ or α exhibits the same qualitative impacts on t k* , t *b , and t a* . Specifically, the regulator needs to increase the subsidy rate on predevelopment land value (t *b decreases), decrease the subsidy rate on postdevelopment property value (t a* increases), and increase the tax rate on capital (t *k increases). Recall from Propositions 3(1) and 3(2) that the regulator needs to accelerate development and reduce capital intensity for an increase in σ and needs to reduce capital intensity for an increase in α, in order to preserve neutrality. Such policy changes can work because as suggested in Proposition 1, through decreasing 12

The effective rate of property tax for the residential property in the USA ranges from 0.4% to 2.9% (Bird and Slack 2004). We divide this tax rate by the discount rate (ρ=10%), and thus derive an expropriation rate of between 4% and 29%. Our benchmark parameter value for the expropriation rate, 25%, is within this range. 13 From Cox and Miller (1970), the expected hitting time for the stochastic rent from to x* is given by x ln x ln x ln 0:97x ¼ 2:38 years: EðT t Þ ¼ lnas 2 =2 ¼ 0:02ð0:12Þ2 =2

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Table 2 Optimal taxation rates on capital, pre-development land value, and post-development property value Tax rates

Variation

t k t b t a

Variation in m 0.95 27.9% −0.597% −1.75%

0.96 27.8% −0.592% −1.74%

0.97 27.6% −0.590% −1.73%

0.98 27.3% −0.587% −1.72%

0.99 27.2% −0.582% −1.71%

t k t b t a

Variation in ε 0.200 24.8% −0.546% −1.59%

0.225 26.3% −0.568% −1.67%

0.250 27.6% −0.590% −1.73%

0.275 28.7% −0.606% −1.79%

0.300 29.7% −0.623% −1.83%

t k t b t a

Variation in γ 0.58 30.6% −0.834% −1.88%

0.59 28.9% −0.709% −1.80%

0.60 27.6% −0.590% −1.73%

0.61 26.3% −0.478% −1.66%

0.62 25.2% −0.373% −1.61%

t k t b t a

Variation in α 0.015 20.9% −0.818% −1.47%

0.0175 23.8% −0.709% −1.59%

0.02 27.6% −0.590% −1.73%

0.0225 32.1% −0.459% −1.88%

0.025 37.6% −0.312% −2.05%

t k t b t a

Variation in σ 0.1 23.5% −0.770% −1.52%

0.11 25.5% −0.686% −1.62%

0.12 27.6% −0.590% −1.73%

0.13 29.8% −0.478% −1.84%

0.14 32.0% −0.354% −1.94%

The benchmark parameter values are as follows: m=0.97, ε=0.25, γ=0.6, α=0.02, σ=0.12, c=1, f=1, ρ= 0.10, t k ¼ 27:6%, t b ¼ 0:590%, t a ¼ 1:73%. The terms m, ε, γ, α, σ, c, f, and ρ denote the initial rent over the rent that triggers development, the expropriation rate, the elasticity of output with respect to capital, the expected growth rate of rents on structures, the volatility of the growth rate of rents on structures, the proportional cost on capital, the fixed cost, and the riskless rate, respectively. The terms t k , t b , and t a stand for the optimal levels of the tax rates on capital, pre-development land value, and postdevelopment property value, respectively

the subsidy rate on pre-development land value, the regulator can accelerate development and reduce capital intensity. Such policy changes can also satisfy the revenue constraint because one of them reduces expenditures on subsidies (t *a increases), another one increases expenditures on subsidies (t *b decreases), and the third one increases revenues collected from taxation (t *k increases).

Conclusion We investigate the issue regarding neutral property taxation policy in a framework where demand for developed properties is stochastic over time. We consider a


225

landowner who irreversibly chooses capital intensity and the timing of development. The regulator employs three kinds of property taxation, namely, taxation on capital, predevelopment land value, and post-development property value, so as to expropriate a certain ratio of pre-tax site value as well as to achieve neutrality. We investigate how the regulator changes these policy instruments in response to changes in exogenous forces such as current demand conditions, the expropriation rate, the elasticity of output with respect to capital, the expected growth rate of demand for developed properties, and the volatility of that growth rate. When examining the neutral property taxation policy, Arnott (2005) abstracts from four important considerations. Specifically, they are (1) land use externalities; (2) zoning and other land use controls; (3) redevelopment; and (4) uncertainty. Given that our article incorporates the last consideration, future research may incorporate the other three considerations. Furthermore, we consider one type of uncertainty: rents on structures follow a geometric Brownian motion. We might alternatively assume that rents on structures follow an arithmetic Brownian motion, given that these rents are distributed more similar to a normal than a log-normal distribution, as suggested by Capozza and Li (1994). Finally, we abstract from uncertainty in the tax policy, which could be incorporated by following Hassett and Metcalf (1999). Acknowledgements We would like to thank the guest editors (Richard Buttimer and Kanak Patel), one anonymous reviewer, Richard Arnott, Edward Coulson, Steven Ott, Dean A. Paxson, Brenda A. Priebe, and participants at the 2007 Cambridge-UNCC Symposium, the 2007 Joint AsRES-AREUEA International Conference, and the National Taiwan University for their helpful comments on earlier versions of this manuscript. Jyh-Bang Jou acknowledges financial support from the Social Policy Research Center, College of Social Sciences, National Taiwan University.

Appendix 1 Differentiating k* and x* in Eqs. 23 and 24, respectively, with respect to τk yields dk * k* < 0; ¼ dt k ðc þ t k Þ

ð34Þ

dx* gx* > 0: ¼ dt k ðc þ t k Þ

ð35Þ

Differentiating k* and x* with respect to tb yields dk * k * @b1t

¼ < 0; 2 @t b 1 dt b 1 b1t g b1t

ð36Þ

dx ð1 g Þx @b1t

¼ < 0: dt b 1 b11t g b21t @t b

ð37Þ

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J.-B. Jou, T. Lee

Differentiating k* and x* with respect to ta yields dk ¼ 0; dt a

ð38Þ

dk x ¼ > 0: dt a t a

ð39Þ

QED.

Appendix 2 Define g ðZ Þ ¼ Et into Eq. 14 yields

R T t

Z ðxðsÞÞerðstÞ ds. Substituting A1 in Eq. 19 and A2 in Eq. 20

Z ð xð s Þ Þ ¼

k *g x * ðr þ t a aÞb1t

xð s Þ x*

b1t

:

ð40Þ

Given that x(s) evolves according to the geometric Brownian motion given by Eq. 1, Z(s) evolves according to a geometric Brownian motion given by dZ ðsÞ s2 ¼ ab1t þ b1t ðb1t 1Þ dt þ sb1t dΩðsÞ: ð41Þ Z ðsÞ 2 Thus, g(Z) will satisfy the differential equation 1 2 2 00 s2 s b1t g ðZ Þ þ ab1t þ b1t ðb1t 1Þ Zg0 ðZ Þ rgðZ Þ þ Z ¼ 0: 2 2

ð42Þ

This has the general solution gðZ Þ ¼ B1 Z q1 þ B2 Z q2 þ

Z r ab1t

s2 2

b1t ðb1t 1Þ

;

ð43Þ

where θ1 and θ2 are, respectively, the larger and smaller roots of the quadratic equation 1 s2 s 2 b21t qðq 1Þ ab1t þ b1t ðb1t 1Þ q þ r ¼ 0: 2 2

ð44Þ

The boundary conditions are given by

g ð0Þ ¼ 0 and g Z x* ¼ 0:

ð45Þ

Therefore, B2 ¼ 0 and B1 ¼

1q1 Z x* r ab1t s2 b1t ðb1t 1Þ 2

:

ð46Þ


227

After detailed calculations, this yields

g ð Z ð xÞ Þ ¼

f

x b1t h x

1

x b1t ðq1 1Þ i x

:

t b ½ð1 g Þb1t 1

ð47Þ

Furthermore, Z 1 Et ½Wa ðxðsÞ; k Þ ðc þ t k Þk f erðstÞ ds T

¼ Et e

rðT t Þ

Z

1

T

xðsÞk g ð1 þ t k Þck f erðsT Þ ds; ðr þ t a a Þ

ð48Þ

where Z

1

T*

¼

*

r sT xðsÞk g ds ð1 þ t k Þck f e ðr þ t a aÞ

x * k *g ð1 þ t k Þck * f r r ðr þ t a a Þðr a Þ

¼

f 1 b11t g

1 g ra r

ð49Þ

r f and Et e r

T * t

¼

x x*

b10

:

Finally, the pre-tax site value at development, discounted back to the current time t is given by

V ¼ Et e

r T * t

"

x * k *g ck * þ f ðr aÞ

#

¼

x x*

b10

f : b10 ð1 g Þ 1

ð50Þ

QED.

Appendix 3 Differentiating k* and k0 with respect to σ yields @k * 2ðb1t 1Þk * ¼ > 0; @s s ½ð1 g Þb1t 1ð2b1t 1Þ

ð51Þ

@k0 2ðb10 1Þk0 ¼ > 0: @s s ½ð1 g Þb10 1ð2b10 1Þ

ð52Þ

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J.-B. Jou, T. Lee

Differentiating x* and x0 with respect to σ yields @x* 2ð1 g Þðb1t 1Þk * ¼ > 0; @s s ½ð1 g Þb1t 1ð2b1t 1Þ

ð53Þ

@x0 2ð1 g Þðb10 1Þk0 ¼ > 0: @s s ½ð1 g Þb10 1ð2b10 1Þ

ð54Þ

Differentiating k* and k0 with respect to α yields @k * 2k * ¼ > 0; @a s ½ð1 g Þb1t 1ð2b1t 1Þ

ð55Þ

@k0 2k0 ¼ > 0: @s s ½ð1 g Þb10 1ð2b10 1Þ

ð56Þ

Differentiating x* and x0 with respect to α yields @x* x* 2ð1 g Þx* > ¼ 0; þ @a ðr þ t a aÞ s ½ð1 g Þb1t 1ð2b1t 1Þ
0: ¼ þ @a ðr aÞ s ½ð1 g Þb10 1ð2b10 1Þ
0; @g g 1 b11t g

ð59Þ

@k0 k0 k0 ¼ þ > 0: @g g 1 b110 g

ð60Þ

Differentiating x* and x0 with respect to γ yields 2 3 @x* 4 ðc þ t k Þ ðg 1Þ 1 f fg 5 *>

ln 1 ¼ ln x g þ þ 0; @g ðc þ t k Þ ðc þ t k Þ f b1t < 1 1 g b1t

ð61Þ 2 3 @x0 4 c ð g 1Þ 1 f f g5 >

ln 1 þ x0 0: ¼ g þ ln f b10 c c < @g 1 1 g b10

ð62Þ


229

Differentiating k* and k0 with respect to c yields @k * k * ¼ < 0; @c c

ð63Þ

@k0 k0 ¼ < 0: @c c

ð64Þ

Differentiating x* and x0 with respect to c yields @x* gx* ¼ > 0; @c c

ð65Þ

@x0 gx0 ¼ > 0: @c c

ð66Þ

Differentiating k* and k0 with respect to f yields @k * k * ¼ > 0; @f f

ð67Þ

@k0 k0 ¼ > 0: @f f

ð68Þ

Differentiating x* and x0 with respect to f yields @x* ð1 g Þx* ¼ > 0; @f f

ð69Þ

@x0 ð1 g Þx0 ¼ > 0: @f f

ð70Þ

Given that k*=k0, x*=x0, and β10 > β1t, it then follows that @k * =@s > @k0 =@s, * @x =@s > @x0 =@s, @k * =@a > @k0 =@a, @k * =@g > @k0 =@g, @x* =@g > @x0 =@g, @k * =@c ¼ @k =@c, @x* =@c ¼ @x =@c, @k * =@f ¼ @k =@f , and @x* =@f ¼ @x =@f . 0

0

0

0

QED.

Appendix 4 Suppose that we explicitly write x* and k* as a function of ta, tb, tk, and σ, that is, as x*(ta, tb, tk, σ) and k*(ta, tb, tk, σ), respectively, and x0 and k0 as a function of σ, that is, as x0(σ) and k0(σ), respectively. Suppose that at s ¼ s, k * ðt a ; t b ; t k ; s Þ ¼ k0 ðs Þ and x* ðt a ; t b ; t k ; s Þ ¼ x0 ðs Þ. Also suppose that s is increased to s 0 . Then Proposition 3(1) indicates that k * ðt a ; t b ; t k ; s 0 Þ ¼ k0 ðs 0 Þ and x* ðt a ; t b; t k ; s 0 Þ ¼ x0 ðs 0 Þ. Our problem then reduces to finding a combination of t 0a ; t 0b ; t 0k such that

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J.-B. Jou, T. Lee

k * t 0a ; t 0b ; t 0k ; s 0 ¼ k0 ðs 0 Þ and x* t 0a ; t 0b ; t 0k ; s 0 ¼ x0 ðs 0 Þ. However, we are unable to ascertain the relationship between ta and t 0a , tb and t 0b , and tk and t 0k . We can only rule out several possibilities. For example, we can rule out the combination such that t 0a > t a , t 0k > t k , and t b < t 0b because x* t 0a ; t 0b ; t 0k ; s 0 > x ðt a ; t b ; t k ; s 0 Þ > x0 ðs 0 Þ. We can also rule out the combination that t 0b < t b , t 0k > t k and either t 0a > t a or such * * 0 0 0 0 0 t a < t a because k t a ; t b ; t k ; s > k ðt a ; t b ; t k ; s 0 Þ > k0 ðs 0 Þ. QED.

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