Lumpy Heterogeneity in Groundwater Service Values and Time Preferences

Kent F. Kovacs; Heather Snell; Brandon R. McFadden; Rodolfo M. Nayga

doi:10.3368/le.101.1.040524-0033

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Figure 1
Discount Factors for Best-Fitting Estimates of Class Membership Weighted Exponential and Hyperbolic Discounting Behaviors
Note: The exponential, HM, Harvey discounting forms have two classes, and the QH discounting form has three classes.

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Table 1

Criteria for Selecting the Optimal Number of Classes by Discounting Model

Discounting Model	No. of Classes	Log-Likelihood	AIC	BIC	Parameter
Exponential	1	–4,185.30	8,388.61	8,445.00	9
	2	–3,571.86	7,195.73	7,358.61	18
	3	–3,557.42	7,200.84	7,470.23	27
	4	–3,482.19	7,084.39	7,460.29	36
	5	–3,654.94	7,463.88	7,946.28	45
Hyperbolic (HM)	1	–4,199.30	8,416.61	8,472.99	9
	2	–3,579.80	7,211.60	7,374.48	18
	3	–3,939.42	7,964.85	8,234.24	27
	4	–4,011.48	8,142.96	8,518.86	36
	5	–4,022.11	8,198.23	8,680.63	45
Hyperbolic (Harvey)	1	–4,197.39	8,412.78	8,469.17	9
	2	–3,623.60	7,299.21	7,462.09	18
	3	–4,115.93	8,317.87	8,587.26	27
	4	–4,038.28	8,196.57	8,572.46	36
	5	–3,639.52	7,433.04	7,915.44	45
Quasi-hyperbolic	1	–4,185.31	8,390.62	8,453.27	10
	2	–3,571.87	7,199.73	7,375.15	20
	3	–3,489.97	7,071.94	7,360.12	30
	4	–3,479.66	7,087.33	7,488.28	40
	5	–3,762.20	7,688.41	8,202.13	50

Note: N = 777 in total (3 alternatives per question) × (5 questions per person) = 11,655 observations. Bootstrap standard errors using 250 bootstrap samples for the exponential form: l (log-likelihoods are 10.31 for one class and 9.22 for two classes); Akaike information criteria (AIC) are 21.35 for one class and 19.45 for two classes; Bayesian information criteria (BIC) are 20.87 for one class and 19.70 for two classes.

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Table 2

Time Parameter Results by Discounting Model for the Optimal Number of Latent Classes

	Exponential	Hyperbolic (HM)	Hyperbolic (Harvey)	Quasi-hyperbolic (QH)
No. of classes	2	2	2	3
Log-likelihood	–3,571.86	–3,579.80	–3,623.60	–3,489.97
AIC	7,195.73	7,211.60	7,299.21	7071.94
BIC	7,358.61	7,374.48	7,462.09	7360.12
Time parameter
r	0.430*** (0.095)			0.159 (0.158)
	0.188*** (0.044)			0.656*** (0.095)
				0.186*** (0.044)
w		1.4585*** (0.338)
w		0.5364*** (0.174)
u			2.5978*** (0.2952)
u			0.0989 (0.0725)
β				0.97 (0.961)
				0.96*** (0.194)
				0.98*** (0.168)

Note: Standard errors are in parentheses. Bootstrap standard errors of the BIC using 250 bootstrap samples are 19.70 for the exponential form, 22.45 for the HM form, and 19.85 for the QH form. AIC = Akaike information criterion; BIC = Bayesian information criterion.
*, **, *** Significance at the 10%, 5%, and 1% levels, respectively.

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Table 3

Results for Exponential Discounting Latent Class Model in Willingness-to-Pay Space

Variable/Class	A (Reference)	B
Variable/Class	Policy Motivated	Business as Usual
Utility function
ASC1 (C/T)	8.490*** (2.2717)	–6.811*** (1.8702)
ASC2 (SWI)	10.461*** (2.4728)	–6.423*** (1.8339)
Buffer	9.926*** (3.8444)	0.622 (0.9661)
Quality	13.103*** (7.6828)	9.094** (4.1318)
Jobs	8.770*** (5.0589)	2.441 (1.6997)
Infrastructure	1.735 (5.8754)	6.561*** (3.4596)
Wildlife	4.769 (5.9324)	1.959 (2.8899)
Scale (λ)	0.358*** (0.044)	0.774*** (0.18)
r	0.430*** (0.0952)	0.188*** (0.044)
Class membership function
Constant	—	–0.525 (0.518)
INC	—	–0.004 (0.029)
EDU	—	0.158*** (0.059)
AQU	—	–0.104 (0.121)
PRESS	—	–0.591** (0.263)
EMP	—	–0.281 (0.208)
IDEO	—	0.200*** (0.064)
GRISK	—	–0.222*** (0.045)
Latent class probability	0.71	0.29
Log-likelihood	–3,571.8
AIC	7,195.7
BIC	7,358.6
Observations	11,655

Note: Standard errors are in parentheses. An average respondent has a yearly income of $49,991, a vocational/technical degree, knew what an aquifer was but did not know about the MRVA and its uses, and are mostly unaware of the pressure on the MRVA. Half are employed full- or part-time, have a middle-of-the-road ideology, and believe that the decline of the MRVA holds a medium risk to people. AIC = Akaike information criterion; BIC = Bayesian information criterion; C/T = cap and trade; SWI = surface water infrastructure.
*, **, *** Significance at the 10%, 5%, and 1% levels, respectively.

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Table 4

Results for Quasi-hyperbolic Discounted Latent Class Model in Willingness-to-Pay Space

	A (Reference)	B	C
Variable/Class	No Preferences	Policy Motivated	Business as Usual
Utility function
ASC1 (C/T)	3.20 (4.51)	19.21*** (5.84)	–4.49*** (1.68)
ASC2 (SWI)	5.16 (4.77)	21.02*** (5.82)	–4.11*** (1.61)
Buffer	1.46 (2.74)	18.89*** (7.66)	0.547 (0.879)
Quality	7.62 (10.11)	18.96 (12.74)	5.83* (3.39)
Jobs	2.62 (4.39)	15.26* (8.09)	1.68 (1.58)
Infrastructure	–3.53 (6.64)	13.50 (12.13)	6.80* (3.51)
Wildlife	0.033 (5.15)	14.14 (11.66)	–1.15 (2.36)
Scale (λ)	0.111 (0.077)	0.659 (0.093)	1.25 (0.27)
r	0.159 (0.157)	0.656*** (0.195)	0.185*** (0.05)
β	0.97 (0.961)	0.96*** (0.194)	0.98*** (0.168)
Class membership function
Constant	—	–4.96*** (1.04)	–2.49*** (0.683)
INC	—	0.114*** (0.044)	0.049 (0.041)
EDU	—	0.243*** (0.089)	0.329*** (0.081)
AQU	—	0.316* (0.161)	–0.030 (0.160)
PRESS	—	–0.663* (0.368)	–0.828** (0.349)
EMP	—	–0.656** (0.265)	–0.543** (0.262)
IDEO	—	0.014 (0.087)	0.248*** (0.088)
GRISK	—	0.442*** (0.086)	–0.029 (0.062)
Latent class probability	0.22	0.44	0.34
Log-likelihood	–3,489.96
AIC	7,071.93
BIC	7,360.12
Observations	11,655

Note: Standard errors are in parentheses. An average respondent has a yearly income of $49,991, a vocational/technical degree, knew what an aquifer was but did not know about the MRVA and its uses, and are mostly unaware of the pressure on the MRVA. Half are employed full- or part-time, have a middle-of-the-road ideology, and believe that the decline of the MRVA holds a medium risk to people. AIC = Akaike information criterion; BIC = Bayesian information criterion; C/T = cap and trade; SWI = surface water infrastructure.
*, **, *** Significance at the 10%, 5%, and 1% levels, respectively.

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Table 5

Results of Discounted Conditional Logit (D-CL), Discounted Mixed Logit (D-MXL), and Discounted Logit-Mixed Logit (D-LML) Models in a Willingness-to-Pay Space

Parameter	D-CL (Exponential)	D-MXL (Exponential)	D-LML (Quasi-hyperbolic)
ASC1 (C/T)	–2.44** (1.16)	–0.451*** (0.737)	13.12* (8.51)
ASC2 (SWI)	–0.789 (1.16)	16.471*** (0.801)	29.56*** (7.43)
Buffer	7.05*** (2.05)	6.327*** (0.192)	5.22* (3.85)
Quality	12.58** (4.91)	9.968*** (0.383)	9.32* (6.59)
Jobs	7.63*** (2.87)	4.315*** (0.220)	6.64*** (0.07)
Infrastructure	4.133 (3.99)	–8.254*** (0.187)	6.05 (11.19)
Wildlife	3.661 (3.95)	–1.556*** (0.198)	8.92* (4.20)
Scale (λ)	0.371*** (0.04)	1.639*** (0.241)	0.700* (0.124)
R	0.362*** (0.05)	0.491*** (0.0054)	0.626*** (0.047)
Β			0.613* (0.42)
SD (ASC1, C/T)		51.952*** (0.186)	74.40*** (4.88)
SD (ASC2, SWI)	—	44.645*** (0.151)	88.64*** (6.68)
SD (buffer)	—	16.237*** (0.180)	23.96*** (2.65)
SD (quality)	—	64.774*** (0.726)	34.83*** (3.72)
SD (jobs)	—	0.073 (0.058)	0.36*** (0.03)
SD (infrastructure)	—	53.070*** (0.706)	44.44*** (6.16)
SD (wildlife)	—	83.207*** (1.005)	22.06*** (2.04)
SD (scale (λ))	—	4.882*** (0.850)	0.458* (0.08)
SD (r)	—	0.005 (0.014)	2.65*** (0.24)
SD (β)	—		1.76*** (0.22)
Log-likelihood	–4,185.3	–3,531.7	–3,377.9
AICBBIC	8,388.6	7,099.4	7,005.9
BIC	8,445.0	7,212.2	7,789.0
Obs	11,655	11,655	11,655

Note: D-MIXL and D-LML have bootstrap standard errors given in parentheses obtained using 250 bootstrap samples. Results for all the discounting models for each estimation method are in Appendix Tables A1 and A2 (tests for β are against 1; all others are against 0). AIC = Akaike information criterion; BIC = Bayesian information criterion; C/T = cap and trade; SD = standard deviation; SWI = surface water infrastructure.
*, **, *** Significance at the 10%, 5%, and 1% levels, respectively.

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Table 6

Heterogeneity in Willingness to Pay (WTP) through Estimation Methods in Discounted Mixed Logit (D-MXL), Discounted Logit-Mixed Logit (D-LML), and Discounted Latent Class (D-LCM) Models in a WTP Space

Estimation Method	Scenario 1 (Status Quo Levels in 2050 and No Policy)	Scenario 2 (Current Levels in 2050 and C/T Policy)	Scenario 3 (Current Levels in 2050 and SWI Policy)
D-MXL (r = 0.491)
Mean	116.4***	115.5***	149.9***
Mean	(1.38)	(1.73)	(1.79)
Low	–233.3***	–210.1***	–186.0***
Low	(2.72)	(2.82)	(2.83)
High	400.6***	396.0***	392.53 ***
High	(4.02)	(4.09)	(4.10)
D-LML (r = 0.626; β = 0.613)
Mean	133.3***	160.0***	193.5***
Mean	(26.07)	(28.76)	(28.14)
Low	18.58	–3.12	12.37
Low	(28.21)	(29.86)	(29.73)
High	236.37***	238.89***	286.07***
High	(30.08)	(31.63)	(31.64)
D-LCM (r = 0.430 [class A]; r = 0.188 [class B])
Mean	426.43***	430.00***	433.8***
Mean	(34.92)	(35.12)	(35.10)
Class A	363.4***	383.16***	387.7***
Class A	(35.32)	(35.48)	(35.50)
Class B	580.7***	544.67***	546.7 ***
Class B	(33.93)	(34.21)	(34.20)

Note: Standard errors are in parentheses. Scenario 1 supposes no change from current levels by 2050 without a policy; scenarios 2 and 3 suppose no change from current levels by 2050 with the cap-and-trade (C/T) policy and surface water infrastructure (SWI) policy, respectively. Low = bottom 33rd percentile of WTP from 1,000 random draws of WTP; high = upper 33rd percentile of WTP from 1,000 random draws of WTP.
*, **, *** Significance at the 10%, 5%, and 1% levels, respectively.