Definition (Homogeneity of degree \\(k\\)). A utility function \\(u:\\mathbb{r}^n\\rightarrow \\mathbb{R}\\) is homogeneous of degree \\(k\\) if and only if for all \\(x \\in \\mathbb{R}^n\\) and all \\(\\lambda>0\\), \\(u(\\lambda x)=\\lambda^ku(x)\\).<\/p>\n\n\n\n
$$f(\\lambda x_1,…,\\lambda x_n)=\\lambda^kf(x_1,…,x_n)$$<\/p>\n\n\n\n
Property<\/strong><\/p>\n\n\n\n $$CRTS:\\quad F(aK,aL)=aF(K,L) \\quad a>0$$<\/p>\n\n\n\n $$IRTS:\\quad F(aK,aL)>aF(K,L) \\quad a>1$$<\/p>\n\n\n\n $$DRTS:\\quad F(aK,aL)<aF(K,L) \\quad a>1$$<\/p>\n\n\n\n Theorem (Euler’s Theorem) Let \\(f(x_1,…,x_n)\\) be a function that is homogeneous of degree k. Then,<\/em><\/p>\n\n\n\n $$ x_1\\frac{\\partial f(x)}{\\partial x_1}+…+ x_n\\frac{\\partial f(x)}{\\partial x_n} =kf(x) $$<\/p>\n\n\n\n or, in gradient notation,<\/p>\n\n\n\n $$ x\\cdot \\nabla f(x)=kf(x) $$<\/p>\n\n\n\n Proof:<\/strong> Differentiate \\(f(tx_1,…,tx_n)=t^k f(x_1,…,x_n)\\) w.r.t \\(t\\) and then set \\(t=1\\).<\/p>\n\n\n\n P.S. We use Euler’s Theorem in the proof of the Solow Model.<\/p>\n\n\n\n Motivation:<\/strong><\/p>\n\n\n\n Given \\(y=ax^2+bx+c, a>0, b,c \\in \\mathbb{R}\\), we need to know how does a change in the parameter \\(a\\) affect the maximum value of \\(y\\), \\(y^*\\)?<\/p>\n\n\n\n We first define \\(y^*=\\max_{x} y= \\max_{x} ax^2+bx+c \\). The solution is \\(x^*=-\\frac{b}{2a}\\), and plug it back into \\(y\\), we get \\(y^*=f(x^*)=\\frac{4ac-b^2}{4a}\\). Now, we take derivative w.r.t. \\(a\\). \\(\\frac{\\partial y^*}{\\partial a}=\\frac{b^2}{4a^2}\\). We would find that,<\/p>\n\n\n\n $$\\frac{\\partial y^*}{\\partial a}= {\\frac{\\partial y}{\\partial a}}|_{x=x^*} $$<\/p>\n\n\n\n A Simple Envelop Theorem<\/strong><\/p>\n\n\n\n $$v(q)=\\max_{x} f(x,q)$$<\/p>\n\n\n\n $$=f(x^*(q),q)$$<\/p>\n\n\n\n $$ \\frac{d}{dq}v(q)=\\underbrace{\\frac{\\partial}{\\partial x}f(x^*(q),q)}_{=0\\ by\\ f.o.c.}\\frac{\\partial}{\\partial q}x^*(q)+\\frac{\\partial}{\\partial q} f(x^*(q),q) $$<\/p>\n\n\n\n $$ \\frac{d}{dq}v(q) =\\frac{\\partial}{\\partial q}f(x^*(q),q) $$<\/p>\n\n\n\n Think of the ET as an application of the chain rule and then F.O.C., our goal is to find how does parameter affect the already maximised function \\(v(q)=f(x^*(q),q)\\).<\/p>\n\n\n\n A formal expression<\/strong><\/p>\n\n\n\n Theorem (Envelope Theorem). Consider a constrained optimisation problem \\(v(\\theta)=\\max_x f(x,\\theta)\\) such that \\(g_1(x,\\theta)\\geq0,…,g_K(x,\\theta)\\geq0\\).<\/p>\n\n\n\n Comparative statics on the value function are given by: (\\(v(\\theta)=f(x,\\theta)|_{x=x^*(\\theta)}=f(x^*(\\theta),\\theta)\\))<\/p>\n\n\n\n $$ \\frac{\\partial v}{\\partial \\theta_i}=\\sum_{k=1}^{K}\\lambda_k \\frac{\\partial g_k}{\\partial \\theta_i}|_{x^*}+{\\frac{\\partial f}{\\partial \\theta_i}}|_{x^*}=\\frac{\\partial \\mathcal{L}}{\\partial \\theta_i}|_{x^*} $$<\/p>\n\n\n\n (for Lagrangian \\(\\mathcal{L}(x,\\theta,\\lambda)\\equiv f(x,\\theta)+\\sum_{k}\\lambda_k g_k(x,\\theta)\\)) for all \\(\\theta\\) such that the set of binding constraints does not change in an open neighborhood.<\/p>\n\n\n\n Roughly, the derivative of the value function is the derivative of the Lagrangian w.r.t. parameters, \\(\\theta\\), while argmax those unknows (\\(x=x^*\\)).<\/p>\n\n\n\n To be continued.<\/p>\n\n\n\n2. Euler’s Theorem<\/h4>\n\n\n\n
3. Envelop Theorem<\/h4>\n\n\n\n
4. Hicksian and Marshallian demand + Shepherd’s Lemma<\/h4>\n\n\n\n