Transcript Ex 4.12
Frank Cowell: Microeconomics November 2006 Exercise 4.12 MICROECONOMICS Principles and Analysis Frank Cowell Ex 4.12(1) Question Frank Cowell: Microeconomics purpose: to derive solution and response functions for quasilinear preferences method: substitution of budget constraint into utility function and then simple maximisation Ex 4.12(1) Preliminary Frank Cowell: Microeconomics First steps are as follows: Sketch indifference curves Write down budget constraint Straightforward – parabolic contours Straightforward – fixed-income case Set out optimisation problem Ex 4.12(1) Indifference curves Frank Cowell: Microeconomics x2 Slope is vertical here Could have x2 = 0 x1 0 0 1 2 Ex 4.12(1) Budget constraint, FOC Frank Cowell: Microeconomics Budget constraint: Substitute this into the utility function: We get the objective function: FOC for an interior solution: Ex 4.12(1) Using the FOC Frank Cowell: Microeconomics Remember that person might consume zero of commodity 2 consider two cases Case 1: x2* > 0 From the FOC: But, to make sense this case requires: Case 2: x2* = 0 We get x1* from the budget constraint x1* = y / p1 Ex 4.12(1) Demand functions Frank Cowell: Microeconomics We can summarise the optimal demands for the two goods thus Ex 4.12(1) Indirect utility function Frank Cowell: Microeconomics Get maximised utility by substituting x* into the utility function V(p1, p2, y) = U(x1*, x2*) = U(D1(p1, p2, y), D2(p1, p2, y)) Case 1: p1 >`p1 Case 2: p1 ≤`p1 Ex 4.12(1) Cost function Frank Cowell: Microeconomics Get cost function (expenditure function) from the indirect utility function maximised utility is u = V(p1, p2, y) invert this to get y = C(p1, p2, u) Case 1: p1 >`p1 Case 2: p1 ≤`p1 Ex 4.12(2) Question Frank Cowell: Microeconomics purpose: to derive standard welfare concept method: use part 1 and manipulate the indirect utility function Ex 4.12(2) Compute CV Frank Cowell: Microeconomics Get compensating variation (1) from indirect utility function before price change: u = V(p1, p2, y) after price change: u = V(p1', p2, y − CV) Equivalently (2) could use cost function directly CV = C(p1, p2, u) − C(p1', p2, u) In Case 1 above we have Rearranging, we find: Equivalently Ex 4.12(3) Frank Cowell: Microeconomics In case 1 we have x1* = [½ a p2 / p1]2 So demand for good 1 has zero income effect Therefore, in this case CV = CS = EV Ex 4.12: Points to remember Frank Cowell: Microeconomics It’s always a good idea to sketch the indifference curves in this case the sketch is revealing… …because of the possible corner solution A corner solution can sometimes just be handled as two separate cases There’s often more than one way of getting to a solution in this case two equivalent derivations of CV