Utility functions
\[
\begin{aligned}
x_{njt,1} &= \text{carbon sequestration} \\
x_{njt,2} &= \text{above ground biodiversity} \\
x_{njt,3} &= \text{water pollution (N leaching)} \\
x_{njt,4} &= \text{below ground biodiversity} \\
x_{njt,5} &= \text{indirect landuse effects} \\
x_{njt,6} &= \text{tax}
\end{aligned}
\]
Interaction
\[
U_{njt} = \beta_0
+ \sum_{k=1}^{6} \beta_k x_{njt,k}
+ \sum_{k=1}^{4} \beta_k x_{njt,k} SQ_{n,k}
+ \beta_{11} x_{njt,1} x_{njt,3}
+ \beta_{12} x_{njt,1} x_{njt,5}
+ \varepsilon_{njt}
\]
Quadratic
\[
U_{njt} = \beta_0
+ \sum_{k=1}^{4} \left( \beta_k x_{njt,k} + \beta_k x_{njt,k}^2 \right)
+ \beta_5 x_{njt,5}
+ \beta_6 x_{njt,6}
+ \beta_{11} x_{njt,1} x_{njt,3}
+ \beta_{12} x_{njt,1} x_{njt,5}
+ \varepsilon_{njt}
\]
Linear
\[
U_{njt} = \beta_0
+ \sum_{k=1}^{6} \beta_k x_{njt,k}
+ \beta_{11} x_{njt,1} x_{njt,3}
+ \beta_{12} x_{njt,1} x_{njt,5}
+ \varepsilon_{njt}
\]
Logarithmic
\[
U_{njt} = \beta_0
+ \sum_{k=1}^{4} \beta_k \log(x_{njt,k})
+ \beta_5 x_{njt,5}
+ \beta_6 x_{njt,6}
+ \beta_{11} x_{njt,1} x_{njt,3}
+ \beta_{12} x_{njt,1} x_{njt,5}
+ \varepsilon_{njt}
\]
Flexible specifications (Conditional logit)
Examining the model fit characteristics and the coefficients of the
CLM suggests the need for more flexible model specifications. So far, we
specified each attribute for which we hypothesize diminishing marginal
returns as either logarithmic or quadratic. However, the model outputs
above suggest that the effect of changes in above-ground biodiversity on
utility is better captured with a logarithmic specification, while the
below-ground biodiversity attribute and its effect on utility might
follow a quadratic relationship. Using the same model selection strategy
as before, we re-estimate the quadratic model, but use a logarithmic
transformation of the above-ground biodiversity attribute (Flex 1).
Conversely, we re-estimate the logarithmic model but allow for a
quadratic effect of below-ground biodiversity on utility (Flex 2). The
interacted variables are also in log terms. Further, given the small
sample size, we estimated a main effects model, excluding the
interactions, to gauge MWTP values of the main attributes. The main
effects model uses the either Flex 1 or Flex 2, depending on which shows
the better fit characteristics.
Model fit characteristics
| Flex 1 |
2234.26 |
2300.76 |
-1105.13 |
| Flex 2 |
2229.31 |
2284.73 |
-1104.66 |
| Main effects |
2226.38 |
2270.71 |
-1105.19 |
Flex 1
Conditional logit model output
| asc_1 |
-0.1346255 |
0.2000046 |
-0.6731121 |
0.5008760 |
| asc_2 |
0.0000000 |
NA |
NA |
NA |
| b_climate |
0.6300469 |
0.7069646 |
0.8912000 |
0.3728219 |
| log_biodiv |
0.8054338 |
0.3197592 |
2.5188764 |
0.0117730 |
| b_bgbio |
0.4112114 |
0.1324175 |
3.1054152 |
0.0019001 |
| b_water |
-0.0499653 |
0.0241724 |
-2.0670413 |
0.0387303 |
| b_iLUC |
-0.0000062 |
0.0000031 |
-1.9948446 |
0.0460598 |
| b_tax |
-0.0071416 |
0.0007509 |
-9.5104222 |
0.0000000 |
| b_climate_2 |
-0.0465187 |
0.5642994 |
-0.0824361 |
0.9342999 |
| b_bgbio_2 |
-0.0242469 |
0.0083169 |
-2.9153702 |
0.0035527 |
| b_water_2 |
0.0004909 |
0.0005551 |
0.8844297 |
0.3764643 |
| b_cli_water |
-0.0159618 |
0.0145143 |
-1.0997288 |
0.2714503 |
| b_cli_iLUC |
-0.0000021 |
0.0000052 |
-0.4078813 |
0.6833608 |
Flex 2
Conditional logit model output
| asc_1 |
-0.1782785 |
0.1971978 |
-0.9040593 |
0.3659640 |
| asc_2 |
0.0000000 |
NA |
NA |
NA |
| log_climate |
0.7867420 |
0.7429937 |
1.0588812 |
0.2896539 |
| log_biodiv |
0.7603117 |
0.3128048 |
2.4306264 |
0.0150727 |
| b_bgbio |
0.4071531 |
0.1360935 |
2.9917151 |
0.0027742 |
| b_bgbio_2 |
-0.0241298 |
0.0085491 |
-2.8224859 |
0.0047653 |
| log_water |
-0.4042547 |
0.1315474 |
-3.0730734 |
0.0021187 |
| b_iLUC |
-0.0000077 |
0.0000034 |
-2.2797834 |
0.0226205 |
| b_tax |
-0.0072117 |
0.0007478 |
-9.6443831 |
0.0000000 |
| b_cli_water |
-0.1590021 |
0.2503875 |
-0.6350240 |
0.5254128 |
| b_cli_iLUC |
0.0000014 |
0.0000077 |
0.1864911 |
0.8520597 |
Main effects model
Conditional logit model output
| asc_1 |
-0.1635949 |
0.1874886 |
-0.8725589 |
0.3829036 |
| asc_2 |
0.0000000 |
NA |
NA |
NA |
| log_climate |
0.4408386 |
0.2091042 |
2.1082244 |
0.0350116 |
| log_biodiv |
0.7296738 |
0.3163015 |
2.3068935 |
0.0210608 |
| b_bgbio |
0.4099737 |
0.1358654 |
3.0175002 |
0.0025487 |
| b_bgbio_2 |
-0.0242406 |
0.0085499 |
-2.8352064 |
0.0045796 |
| log_water |
-0.4411707 |
0.0990269 |
-4.4550589 |
0.0000084 |
| b_iLUC |
-0.0000073 |
0.0000019 |
-3.9386991 |
0.0000819 |
| b_tax |
-0.0071825 |
0.0007430 |
-9.6673185 |
0.0000000 |
| b_cli_water |
0.0000000 |
NA |
NA |
NA |
| b_cli_iLUC |
0.0000000 |
NA |
NA |
NA |
