CES: Fungsi produksi: Elastisitas substitusi

Saya harus membuktikan bahwa untuk fungsi produksi CES:$\sigma = 1/(1 + \rho)$

$$\begin{align} q = (l^\rho + k^\rho)^\frac{1}{\rho} \end{align}$$

Saya menemukan bahwa saya perlu menyelesaikan persamaan berikut:

$$\begin{align} \sigma = \frac{\frac{d(k/l)}{k/l}}{\frac{dRTS}{RTS}} = \frac{d(k/l)}{dRTS}\frac{RTS}{k/l} = \frac{d(k/l)}{d((k/l)^{1-\rho})}\frac{(k/l)^{1-\rho}}{k/l} \end{align}$$

Tapi saya tidak tahu bagaimana menulis ulang ungkapan ini menjadi $\sigma = 1/(1 + \rho)$

Fungsi produksi adalah: MPL dan MPK masing-masing: ql=∂

$$q = (l^\rho + k^\rho)^\frac{1}{\rho}$$ qk=∂ $$q_l = \frac{\partial q}{\partial l} = \frac{1}{\rho} \cdot (l^\rho + k^\rho)^{\frac{1}{\rho}-1} \cdot \rho\cdot l^{\rho-1}$$ $$q_k = \frac{\partial q}{\partial k} = \frac{1}{\rho} \cdot (l^\rho + k^\rho)^{\frac{1}{\rho}-1} \cdot \rho\cdot k^{\rho-1}$$

$f$

$$\sigma(x) = \frac{x f'(x)}{f(x)}\equiv \frac{\frac{df(x)}{f(x)}}{\frac{dx}{x}}$$

1. $u = ln(x)$$\rightarrow x = e^u$$v=ln(f(x))$$\rightarrow f(x) = e^v$
2. $v' = f'(x) / f(x)$$u'=\frac{1}{x}$ $$\frac{v'}{u'}=\frac{\frac{f'(x)}{f(x)}}{\frac{1}{x}} = \sigma(x)$$
3. $\frac{d ln f(x)}{d ln(x)}$$\frac{d ln f(x)}{d ln(x)} = \frac{d v}{d u}$ $$\frac{d v}{d u} = \frac{d v}{d x} \cdot \frac{d x}{d u} = \frac{f'(x)}{f(x)} \cdot x$$ $\sigma(x)$

Sekarang mari kita atasi masalah elastisitas Anda.

$$ln(\frac{q_k}{q_l})= log(\frac{\frac{1}{\rho} \cdot (l^\rho + k^\rho)^{\frac{1}{\rho}-1} \cdot \rho\cdot l^{\rho-1}}{\frac{1}{\rho} \cdot (l^\rho + k^\rho)^{\frac{1}{\rho}-1} \cdot \rho\cdot k^{\rho-1}}) = ln (\frac{l}{k})^{\rho-1} = (\rho-1) ln (l/k) = (1 - \rho) ln (k/l)$$ $$\Rightarrow ln (k/l) = \frac{1}{1-\rho} \cdot ln(\frac{q_k}{q_l})$$

$\sigma = \frac{1}{1-\rho}$