Table 1.

Mathematical results

Case	Optimization	Variable conductivity?	Carbon cost increases as ${\psi }_L$ decreases?	β(${\psi }_L$)=	MXTE
1	CMH	N	N	NA	NA
2	CMH	Y	N	NA	NA
3	CMH	N	Y	$\sqrt{\frac{K_{\max }}{-\mathrm{\Theta }\prime \left({\psi }_L\right)}}$	$-\mathrm{\Theta }\prime \left({\psi }_L\right)$
4	CMH	Y	Y	$\sqrt{\frac{K\left({\psi }_L\right)}{-\mathrm{\Theta }\prime \left({\psi }_L\right)}}$	$-\mathrm{\Theta }\prime \left({\psi }_L\right)$
5	WUEH	N	N	${\lambda }^{-\left(1/2\right)}$	$\lambda K_{\max }$
6	WUEH	Y	N	${\lambda }^{-\left(1/2\right)}$	$\lambda K\left({\psi }_L\right)$
7	WUEH	N	Y	$\sqrt{\frac{K_{\max }}{\lambda K_{\max }-\mathrm{\Theta }\prime \left({\psi }_L\right)}}$	$\lambda K_{\max }-\mathrm{\Theta }\prime \left({\psi }_L\right)$
8	WUEH	Y	Y	$\sqrt{\frac{K\left({\psi }_L\right)}{\lambda K\left({\psi }_L\right)-\mathrm{\Theta }\prime \left({\psi }_L\right)}}$	$\lambda K\left({\psi }_L\right)-\mathrm{\Theta }\prime \left({\psi }_L\right)$

Optimization column: CMH means net carbon gain maximization; WUEH means the constant marginal water use efficiency hypothesis. Variable conductivity column: Y means that K$\left({\psi }_L\right)$ decreases with ${\psi }_L$ (empirically as a Weibull function); N means that K$\left({\psi }_L\right)$ is equal to the constant K_max. Carbon cost Increases as ${\psi }_L$ decreases column: Y means that the derivative of the cost function $\mathrm{\Theta }\prime \left({\psi }_L\right)$ increases as ${\psi }_L$ decreases, which implies that the cost function itself is concave-up; N means that the cost function is constant (or zero) so that its derivative is zero. β$\left({\psi }_L\right)$ = column: The optimal stomatal conductance $\left(g_s^{\text{opt}}\right)$ in cases 3–8 is approximately proportional to $\frac{A_N\beta \left({\psi }_L\right)}{\sqrt{\text{saturation}\hspace{0.17em}\text{deficit}}\sqrt{C_a-\mathrm{\Gamma }}}$ in the special case in which T_L ∼ T_a and photosynthesis is carbon or light limited (see text). Recall that β$\left({\psi }_L\right)$ is the stomatal sensitivity to leaf water potential. NA means “not applicable” because there is no internal optimum in cases 1 and 2. Instead, stomates are predicted to be always wide open in these cases. MXTE column: The MXTE is defined as $\text{MXTE}=\frac{\mathrm{\partial }{A}_N}{\mathrm{\partial }{g}_s}/-\frac{\mathrm{\partial }{\psi }_L}{\mathrm{\partial }{g}_s}.$