Exercícios- Teoria da Firma - Microeconomia 1

Questão 1 Calcule a Taxa Marginal de Substituição Técnica\ny(X1,X2) = [α1X1^ρ + α2X2^ρ]^(1/ρ)\ndln(X2|X1) = (1/X2 - 1/X1)\ndgn(1) = - (1/X1 dx1 - 1/X2 dx2)\n* Calculando a TMST\ndy/∂X1 = [α1X1^ρ + α2X2^ρ]^(ρ-1)α1X1^(ρ-1) = 0\ndy/∂X2 = 1/e[α1X1^ρ + α2X2^ρ]^(ρ-1)α2X2^(ρ-1)\nTMST = [α1X1^ρ]/[α2X2^ρ]\nσ1,2 = - (1/X1 dx1 - 1/X2 dx2)/(ρ-1)(1/X1 dx1 + 1/X2 dx2) => -1/(ρ-1)(-1) => σ1,2 = 1/(1-ρ)\n* A elasticidade de substituição é constante. Questão 2\ny = X1^β1 · X2^β2\na) Calcule a Taxa Marginal de Substituição Técnica\n- ln(X2/X1) = lnX2 - lnX1\n- dln(X2|X1) = - (1/X1 dx1 - 1/X2 dx2)\n* Calculando a TMST\ndy = β1X1^β1-1 X2^β2 = 0\n∂y/∂X2 = β2X1^β1 X2^β2-1 = 0\nTMST = β1X1^β1-1 X2^β2/β2X1^β1 X2^β2-1 = β1X2/β2X1\n* Derivando os ln da TMST\ndln(X1|X2) = ln(β1/β2) + ln(X2/X1) => lnX2 - lnX1\ndln(X2|X1) = 1/X2 dx2 = - (1/X1 dx1 - 1/X2 dx2)\nσ1,2 = - (1/X1 dx1 - 1/X2 dx2)/(1/X1 dx1 - 1/X2 dx2) => σ1,2 = 1\nb) Encontre os demandantes por insumo e a função custo\nmin w1X1 + w2X2 p. a. X1^β1X2^β2 ≥ y\n∂(X1,X2,λ)\n∂X1 = w1X1^β1-1 X2^β2 = 0 (1)\n∂(X2)\n∂X2 = w2X1^β1 X2^β2-1 = 0 (2)\n∂(X3)\n∂λ = X1^β1 + X2^β2 - y (3) Dividindo (1 por 2)\nw1 = λβ1X1^β1-1 X2^β2\nw2 = λβ2X1^β1 X2^β2-1 = β1X2^β2 - β2X1^β1 = w1/w2 × β3X2/β2X1\nX1 = w1β2/w3β1 (4)\nSubst. (4) em (3)\ny = X1^β1 · (w1β2/w2)^(1-β1) => y = X1^(β1+β2)w2^β2/(w1β1)^(β2) => y(w2β1)^(w1β2)\nX3^β1/w2^β2 =>\n(w2B1)*(X1)^(β1)\nX2 = w2b1/w1b2 =>\nX2 = (y^β1)(β1)\n(y^β2)(β2) • FUNÇÃO CUSTO\nc(w1,g) = w1x1 + w2x2\nc(w1,g) = w1.g(y1) + w2.g(y3)\nc(w1,g) = w1.g[(B1/B2)(B2/B2) + (B3/B2)(B1/B2)] + w2.g[(B1/(w2))^B1/(B2)]\n\nc(w1,g) = (w1,w2)(B1/B2 + (B2/B2)(B2^1/B1))(y) + w2.g[(B1/(w1))(w2/(B1 + B2))]\n+\n\nc(w1,g) = (w1^g)(w2^2)(B1/B2 + (B2/B2)(B2^1)/(B1^g))(y) + w1.(1/(B2 + B2))\ny\n\nc(w1,g) =\n(w2)(w1, (B@/B2 + (B2/B2)(B^2/((B/w2)) = y) c) y = x1B1x2B2\nM1,2(x1) = B1xB2^1/x2B2 \nb1 = B1/(x1)xB3^1/x2 => B1 \n= B3(b) => x1/x1 => B3\nM1,2(x2) = B2(x1/y1)/x1 = x2 => xB3\nM2,1(x2) = B2(x1/y1)/(x2/B2) => B2 \n\nd d(i) = B1 + B2\n\nB1 + B2 = 1 Retornos constantes\nB1 + B2 > 1 Retornos crescentes\nB1 + B2 < 1 Retornos decrescentes\n\nNote: Sem produtos com inversos < 0, portanto B1 + B2 não podem ser negativos. y(K,L) = 100L^1/2K^1/4\na) Encontre as funções demanda por insumos\nmin w1L + w2K s.a 100(L^1/2K^1/4) = y\nL(K,L,λ) = w1L+w2K−λ[100(L^1/2K^1/4)−y] = 0\n\n∂L/∂L = w1 − λ 100 * 1/2L^(-1/2)K^(1/4) = 0\n∂L/∂K = w2 − λ 100 * 1/4L^(1/2)K^(-3/4) = 0\n∂L/∂λ = 100 * L^1/2K^1/4 − y = 0\n\nDividindo (1) por (2)\n... = ... = L^(3/4)\nAplicando 4 em 3\ny = 100(∂ω2K) = (B(y)^{1/2}(K^y))y^3 => ...\n= K^{4/3} ω^3 = y/(100(C^3)^2 L^2*ω2^3)\n Aproximando 5 em 4\n\n2.w2.\n\ny^{4/3}\n\nw2^{2/3}\n\nL = --------------------- -> L = 2.w2.\n\ny^{4/3}\n\nw1^{1/3}\n\n100^{4/3}(2.w2)^{2/3}\n\n--------------\n\nw1\n\n=> L = 2.w2.\n\ny^{4/3}.\n\nw1^{1/3}\n\nL = 2 y^{4/3}.\n\nw2^{1/3}\n\n--------------------------------\n\n100^{1/3}(2.w2)^{2/3}\n\n=> L = ------ .\n\ny^{4/3}_2.w2^{2/3}\n\n100^{4/3}.\n\n(2.w2)^{2/3}\n\n=> L =\n\n= 2.y^{4/3}.\n\nw1^{1/3}\n\n=> L = 2.y^{4/3}.w2^{1/3}\n\n---------------b\n\n100^{2/3}.\n\nw14/3\n\n=> L = 1/w1\n\nb) CUSTO TOTAL\n\nC(w1,w2,y1,y2,x) = w1.2^{1/3}.\n\ny^{3/3}.\n\nw2^{1/3} + w2.x.y^{4/3}.\n\n100^{4/3}.\n\n2^{2/3}.\n\ny^{2/3}\n\n=> w1^{2/3}.w2^{2/3}.y^{3/3} +\n\nw1^{1/3}.y.w2^{1/3}.\n\ny\n\n100^{1/3}.y^{2/3}.\n\nw1^{2/3}.w2^{2/3}\n\n=> w1^{2/3}.w2^{2/3}.2y.\n\n=> w1^{2/3}.\n\nw2^{2/3}.2.y\n\n=> w1^{2/3}.w2^{2/3}.y\n\n=> w1.------ .w1^{1/3} +\n\nw2. .w2^{1/3}\n\n= 400^{1/3}\n\n=> w3.2^{1/3}.\n\n + w2.y^{3/3} + w1.w2.y. +\n\n+ .w1.y +\n\n=> w2^{4/3}.\n\ny^{1/3}\n\n100^{2/3})\n\n(w2^{2/3} -\n\n(w1^{2/3} -\n\n + w2(w1^{2/3} -\n\n 100)\n\n + w2^{2/3}\n\n+ w2^{11/3}) \n\n +\n\n + w2.w1^{1/2}.\n\nw2^{3/3})\n\n + w2.w2^{2/3} + w2.\n\nw1^{/3}