[QuantEcon]MATLAB混编FORTRAN语言

Hicks-McFadden elasticity of substitution
$$
\sigma_{i,j} = \begin{cases}
        \dfrac{\dfrac{1}{\theta_i} + \dfrac{1}{\theta_j}}
        {(1-\rho_1) \left( \dfrac{1}{\theta_i} - \dfrac{1}{\theta^*} \right)
        + (1-\rho_2) \left( \dfrac{1}{\theta_j} - \dfrac{1}{\theta} \right)
        + (1-\rho) \left( \dfrac{1}{\theta^*} - \dfrac{1}{\theta} \right) }
        & \text{for } i=1,2; \; j=3 \\
        & \\
        (1-\rho_1)^{-1}
        & \mbox{text } i=1; \; j=2
   \end{cases}
$$
with
$$
\begin{align}
& \theta^* = \delta
            B_1^{\frac{\rho}{\rho_1}} \cdot y^\rho \\
& \theta = (1-\delta) x_3^{-\rho} \cdot y^{\rho} \\
& \theta_1 = \delta \delta_1 x_1^{-\rho_1}
            B_1^{-\frac{\rho_1 - \rho}{\rho_1}}
            \cdot y^{\rho} \\
& \theta_2 = \delta (1 - \delta_1) x_2^{-\rho_1}
            B_1^{-\frac{\rho_1 - \rho}{\rho_1}}
            \cdot y^{\rho} \\
& \theta_3 = \theta
\end{align}
$$

$$f_{\delta}\left(\rho\right)=h_{\delta}\left(\rho\right)\,\exp\left(-g_{\delta}\left(\rho\right)\right)
$$

$$
\frac{\partial y}{\partial\nu}=-\gamma \, e^{\lambda \, t} \, \frac{1}{\rho}\,\ln\left(\delta\, x_{1}^{-\rho}+\left(1-\delta\right)\, x_{2}^{-\rho}\right)\,\left(\delta\, x_{1}^{-\rho}+\left(1-\delta\right)\, x_{2}^{-\rho}\right)^{-\frac{\nu}{\rho}}$$
Now we define the function $f_{\nu}\left(\rho\right)$

$$
\begin{eqnarray}
f_{\nu}\left(\rho\right) & = & \frac{1}{\rho}\,\ln\left(\delta\, x_{1}^{-\rho}+\left(1-\delta\right)\, x_{2}^{-\rho}\right)\,\left(\delta\, x_{1}^{-\rho}+\left(1-\delta\right)\, x_{2}^{-\rho}\right)^{-\frac{\nu}{\rho}}\\
& = & \frac{1}{\rho}\,\ln\left(\delta\, x_{1}^{-\rho}+\left(1-\delta\right)\, x_{2}^{-\rho}\right)\,\exp\left(-\frac{\nu}{\rho}\,\ln\left(\delta\, x_{1}^{-\rho}+\left(1-\delta\right)\, x_{2}^{-\rho}\right)\right)\end{eqnarray}
$$

so that we can approximate $\partial y/\partial\nu$ by using the
first-order Taylor series approximation of $f_{\nu}\left(\rho\right)$:
$$
\begin{eqnarray}
\frac{\partial y}{\partial\nu} & = & -\gamma \, e^{\lambda \, t} \, f_{\nu}\left(\rho\right)\\
& \approx & -\gamma \, e^{\lambda \, t} \, \left(f_{\nu}\left(0\right)+\rho f_{\nu}'\left(0\right)\right)\end{eqnarray}$$

Now we define the helper function $g_{\nu}\left(\rho\right)$
$$\begin{eqnarray}
g_{\nu}\left(\rho\right) & = & \frac{1}{\rho}\,\ln\left(\delta\, x_{1}^{-\rho}+\left(1-\delta\right)\, x_{2}^{-\rho}\right)\\
& = & \frac{1}{\rho}\,\ln\left(g\left(\rho\right)\right)\end{eqnarray}$$

with first and second derivative$$\begin{eqnarray}
g_{\nu}'\left(\rho\right) & = & \frac{\rho\frac{g'\left(\rho\right)}{g\left(\rho\right)}-\ln\left(g\left(\rho\right)\right)}{\rho^{2}}\\
& = & \frac{1}{\rho}\,\frac{g'\left(\rho\right)}{g\left(\rho\right)}-\frac{\ln\left(g\left(\rho\right)\right)}{\rho^{2}}\\
g_{\nu}''\left(\rho\right) & = & -\frac{1}{\rho^{2}}\,\frac{g'\left(\rho\right)}{g\left(\rho\right)}+\frac{1}{\rho}\,\frac{g''\left(\rho\right)}{g\left(\rho\right)}-\frac{1}{\rho}\,\frac{\left(g'\left(\rho\right)\right)^{2}}{\left(g\left(\rho\right)\right)^{2}}+2\frac{\ln\left(g\left(\rho\right)\right)}{\rho^{3}}-\frac{1}{\rho^{2}}\frac{g'\left(\rho\right)}{g\left(\rho\right)}\\
& = & \frac{-2\rho\,\frac{g'\left(\rho\right)}{g\left(\rho\right)}+\rho^{2}\,\frac{g''\left(\rho\right)}{g\left(\rho\right)}-\rho^{2}\,\frac{\left(g'\left(\rho\right)\right)^{2}}{\left(g\left(\rho\right)\right)^{2}}+2\ln\left(g\left(\rho\right)\right)}{\rho^{3}}\end{eqnarray}$$

and use the function $f\left(\rho\right)$ defined above so that$$
f_{\nu}\left(\rho\right)=g_{\nu}\left(\rho\right)\,\exp\left(f\left(\rho\right)\right)$$
and$$\begin{eqnarray}
f_{\nu}'\left(\rho\right) & = & g_{\nu}'\left(\rho\right)\,\exp\left(f\left(\rho\right)\right)+g_{\nu}\left(\rho\right)\,\exp\left(f\left(\rho\right)\right)\, f'\left(\rho\right)\end{eqnarray}$$

Now we can calculate the limits of $g_{\nu}\left(\rho\right)$, $g_{\nu}'\left(\rho\right)$,
and $g_{\nu}''\left(\rho\right)$ for $\rho\rightarrow0$ by
$$\begin{eqnarray}
g_{\nu}\left(0\right) & = & \lim_{\rho\rightarrow0}g_{\nu}\left(\rho\right)\\
& = & \lim_{\rho\rightarrow0}\frac{\ln\left(g\left(\rho\right)\right)}{\rho}\\
& = & \lim_{\rho\rightarrow0}\frac{\frac{g'\left(\rho\right)}{g\left(\rho\right)}}{1}\\
& = & -\delta\,\ln x_{1}-\left(1-\delta\right)\,\ln x_{2}\end{eqnarray}$$


$$\begin{eqnarray}
g_{\nu}'\left(0\right) & = & \lim_{\rho\rightarrow0}g_{\nu}'\left(\rho\right)\\
& = & \lim_{\rho\rightarrow0}\left(\frac{1}{\rho}\,\frac{g'\left(\rho\right)}{g\left(\rho\right)}-\frac{\ln\left(g\left(\rho\right)\right)}{\rho^{2}}\right)\\
& = & \lim_{\rho\rightarrow0}\frac{\rho\,\frac{g'\left(\rho\right)}{g\left(\rho\right)}-\ln\left(g\left(\rho\right)\right)}{\rho^{2}}\\
& = & \lim_{\rho\rightarrow0}\frac{\frac{g'\left(\rho\right)}{g\left(\rho\right)}+\rho\,\frac{g''\left(\rho\right)g\left(\rho\right)-\left(g'\left(\rho\right)\right)^{2}}{\left(g\left(\rho\right)\right)^{2}}-\frac{g'\left(\rho\right)}{g\left(\rho\right)}}{2\rho}\\
& = & \lim_{\rho\rightarrow0}\frac{g''\left(\rho\right)g\left(\rho\right)-\left(g'\left(\rho\right)\right)^{2}}{2\left(g\left(\rho\right)\right)^{2}}\\
& = & \frac{g''\left(0\right)g\left(0\right)-\left(g'\left(0\right)\right)^{2}}{2\left(g\left(0\right)\right)^{2}}\\
& = & \frac{1}{2}\left(\delta\left(\ln x_{1}\right)^{2}+\left(1-\delta\right)\left(\ln x_{2}\right)^{2}-\left(-\delta\,\ln x_{1}-\left(1-\delta\right)\,\ln x_{2}\right)^{2}\right)\\
& = & \frac{1}{2}\biggl(\delta\left(\ln x_{1}\right)^{2}+\left(1-\delta\right)\left(\ln x_{2}\right)^{2}-\delta^{2}\left(\ln x_{1}\right)^{2}\\
&  & -2\delta\left(1-\delta\right)\,\ln x_{1}\ln x_{2}-\left(1-\delta\right)^{2}\left(\ln x_{2}\right)^{2}\biggr)\nonumber \\
& = & \frac{1}{2}\biggl(\left(\delta-\delta^{2}\right)\left(\ln x_{1}\right)^{2}+\left(\left(1-\delta\right)-\left(1-\delta\right)^{2}\right)\left(\ln x_{2}\right)^{2}\\
&  & -2\delta\left(1-\delta\right)\,\ln x_{1}\ln x_{2}\biggr)\nonumber \\
& = & \frac{1}{2}\biggl(\delta\left(1-\delta\right)\left(\ln x_{1}\right)^{2}+\left(1-\delta\right)\left(1-\left(1-\delta\right)\right)\left(\ln x_{2}\right)^{2}\\
&  & -2\delta\left(1-\delta\right)\,\ln x_{1}\ln x_{2}\biggr)\nonumber \\
& = & \frac{\delta\left(1-\delta\right)}{2}\left(\left(\ln x_{1}\right)^{2}-2\,\ln x_{1}\ln x_{2}+\left(\ln x_{2}\right)^{2}\right)\\
& = & \frac{\delta\left(1-\delta\right)}{2}\left(\ln x_{1}-\ln x_{2}\right)^{2}\end{eqnarray}
$$

so that we can calculate the limit of $f_{\nu}\left(\rho\right)$and
$f_{\nu}'\left(\rho\right)$ for $\rho\rightarrow0$ by

$$
\begin{eqnarray}
f_{\nu}\left(0\right) & = & \lim_{\rho\rightarrow0}f_{\nu}\left(\rho\right)\\
& = & \lim_{\rho\rightarrow0}\left(g_{\nu}\left(\rho\right)\,\exp\left(f\left(\rho\right)\right)\right)\\
& = & \lim_{\rho\rightarrow0}g_{\nu}\left(\rho\right)\,\lim_{\rho\rightarrow0}\exp\left(f\left(\rho\right)\right)\\
& = & \lim_{\rho\rightarrow0}g_{\nu}\left(\rho\right)\,\exp\left(\lim_{\rho\rightarrow0}f\left(\rho\right)\right)\\
& = & g_{\nu}\left(0\right)\,\exp\left(f\left(0\right)\right)\\
& = & \left(-\delta\,\ln x_{1}-\left(1-\delta\right)\,\ln x_{2}\right)\exp\left(\nu\left(\delta\,\ln x_{1}+\left(1-\delta\right)\,\ln x_{2}\right)\right)\\
& = & -\left(\delta\,\ln x_{1}+\left(1-\delta\right)\,\ln x_{2}\right)x_{1}^{\nu\delta}x_{2}^{\nu\left(1-\delta\right)}\end{eqnarray}
$$