Envelope Theorem: How the factor of optimization can simplify
Langrange multipliers interpretation use.
Buchanan's theory of club goods.
Common sense tells us that if the price of a firm's input change, this will set a cascade of readjustments. Consider this familiar firm:
\(\pi=zL^\alpha -wL\) We know optimial \(L^*\) will depend on \(z, \alpha , w\), \(L^* = (\frac{w}{\alpha z})^\frac{2}{\alpha -1} =(\frac{\alpha z}{w})^\frac{1}{1-\alpha}\)
Consider \(\pi ^*=z (\frac{az}{w})^\frac{\alpha}{1-\alpha} -w(\frac{\alpha z}{w})^\frac{1}{1-\alpha} =zL^{*^{\alpha}} -wL^*\) where pi* is all choice variables "optimized out"
\(\frac{d\pi^*}{dw}=\alpha z L^{*^{\alpha -}} \frac{dL^*}{dw} -(L^* +w\frac{dL^*}{dw})=\) some unknown?
\(=(MPL-w)\frac{dL^*}{dw}-L^*=\frac{d\pi^*}{dw}\)  where MPL-w is zero as they are optimizing. Because the entreprenuer is optimizing, \(\frac{d\pi ^*}{dw}=-L^*\)
You're maximizing: \(y=f(x_1,x_2,\alpha)\)  so \(y^*=f(x_1 ^*,x_2 ^*, \alpha)\) since you optimized \(\frac{\delta f^*}{\delta x_2 ^*}=0\)\(\frac{\delta f^*}{\delta x_1 ^*}=0\)
What is \(\frac{d y^*}{d \alpha}\) ?               \(\frac{dy^*}{d \alpha}=\frac{df}{dx_1 ^x} \frac{dx_1 ^*}{d \alpha} +\frac{df}{dx_2 ^x} \frac{dx_2 ^*}{d \alpha} +\frac{df}{d\alpha}\)        Optimization simplified: \(\frac{df}{dx_i ^*} =0\)   so \(\frac{dy^y}{d\alpha}=\frac{df}{d\alpha}\)
The Cost Function:
Handy to study in itself, math is often simpler than w/ profit maximization, lessons widely applied for any firm output level, the firm always wants to minimize costs.
Elements of duality: F.O.C.s describing cost minimization, are very similar to profit maximization F.O.Cs.
Consider: Minimize \(wL+rK\) subject to \(2z \sqrt{K} \sqrt{L} =Q\)
\(\mathcal{L} =wL+rK-\lambda (Q-2z \sqrt{K} \sqrt{L})\)     \(\frac{\delta \mathcal{L}}{\delta L}=0=w-\lambda z (\frac{K}{L})^{\frac{1}{2}}\)   \(\frac{\delta \mathcal{L}}{\delta K}=0=r-\lambda z (\frac{K}{L})^{-\frac{1}{2}}\)
\(\frac{w}{r}=(\frac{K}{L})^*\)                 \(K=\frac{w}{r} L\)             \(Q=2z \sqrt{\frac{w}{r} L} \sqrt{L}\)         \(\frac{Q}{2z} \sqrt{\frac{r}{w}} =L^*\)       \(\epsilon_{L,w} =-\frac{1}{2}\)
\(w=\lambda z\frac{K}{L}^{\frac{-1}{2}}\)      \(\lambda=\frac{w}{z} \frac{w}{r}^{\frac{-1}{2}}=\frac{\sqrt{wr}}{z}\) which is marginal cost