Problem 1
Use the Lagrange-multiplier method to find the stationary values of \(z\) : \((a) z=x y,\) subject to \(x+2 y=2\) (b) \(z=x(y+4),\) subject to \(x+y=8\) (c) \(z=x-3 y-x y,\) subject to \(x+y=6\) \((d) z=7-y+x^{2},\) subject to \(x+y=0\)
Problem 1
Given \(U=(x+2)(y+1)\) and \(P_{x}=4, P_{y}=6,\) and \(B=130\) (a) Write the Lagrangian function. (b) Find the optimal levels of purchase \(x^{*}\) and \(y^{*}\) (c) ts the second-order sufficient condition for maximum satisfied? (d) Does the answer in (b) give any comparative-static information?
Problem 3
Write the Lagrangian function and the first-order condition for stationary values (without solving the equations) for each of the following: (a) \(z=x+2 y+3 w+x y-y w,\) subject to \(x+y+2 w=10\) (b) \(z=x^{2}+2 x y+y w^{2},\) subject to \(2 x+y+w^{2}=24\) and \(x+w=8\)
Problem 3
(a) Let \(z=f(x)\) plot as a negatively sloped curve shaped like the right half of a bell in the first quadrant, passing through the points \((0,5),(2,4),(3,2),\) and \((5,1) .\) Let \(z=g(x)\) plot as a positively sloped 45 . line. Are \(f(x)\) and \(g(x)\) quasiconcave?
Problem 4
Write out the bordered Hessian for a constrained optimization problem with four choice variables and two constraints. Then state specifically the second- order sufficient condition for a maximum and for a minimum of \(z,\) respectively.
Problem 4
If, instead of \(g(x, y)=c,\) the constraint is written in the form of \(C(x, y)=0,\) how should the Lagrangian function and the first-order condition be modified as a consequence?
Problem 5
(a) Verify that a cubic function \(z=a x^{3}+b x^{2}+c x+d\) is in general neither quasiconcave nor quasiconvex. (b) is it possible to impose restrictions on the parameters such that the function becomes both quasiconcave and quasiconvex for \(x \geq 0 ?\)
Problem 5
Comment on the validity of the statement: "If the derivative \(\left(\partial x^{*} / \partial P_{x}\right)\) is negative, then \(x\) cannot possibly represent an inferior good."
Problem 6
(a) The CES production function rules out \(\rho=-1 .\) If \(\rho=-1,\) however, what would be the general shape of the isoquants for positive \(K\) and \(l ?\) (b) is \(\sigma\) defined for \(\rho=-1 ?\) What is the limit of \(\sigma\) as \(\rho \rightarrow-1 ?\) (c) Interpret economically the results for parts \((a)\) and \((b)\)
Problem 7
(a) Does the assumption of diminishing marginal utility to goods \(x\) and \(y\) imply strictly convex indifference curves? (b) Does the assumption of strict convexity in the indifference curves imply diminishing marginal utility to goods \(x\) and \(y ?\)
