Frank Cowell: Microeconomics Exercise 4.12 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

Frank Cowell: Microeconomics Ex 4.12(1) Question purpose: to derive solution and response functions for quasilinear preferences purpose: to derive solution and response functions for quasilinear preferences method: substitution of budget constraint into utility function and then simple maximisation method: substitution of budget constraint into utility function and then simple maximisation

Frank Cowell: Microeconomics Ex 4.12(1) Preliminary First steps are as follows: First steps are as follows: Sketch indifference curves Sketch indifference curves  Straightforward – parabolic contours Write down budget constraint Write down budget constraint  Straightforward – fixed-income case Set out optimisation problem Set out optimisation problem

Frank Cowell: Microeconomics Ex 4.12(1) Indifference curves x1x1 x2x2 Could have x 2 = 0 Slope is vertical here

Frank Cowell: Microeconomics Ex 4.12(1) Budget constraint, FOC Budget constraint: Budget constraint: Substitute this into the utility function: Substitute this into the utility function: We get the objective function: We get the objective function: FOC for an interior solution: FOC for an interior solution:

Frank Cowell: Microeconomics Ex 4.12(1) Using the FOC Remember that person might consume zero of commodity 2 Remember that person might consume zero of commodity 2  consider two cases Case 1: x 2 * > 0 Case 1: x 2 * > 0 From the FOC: From the FOC: But, to make sense this case requires: But, to make sense this case requires: Case 2: x 2 * = 0 Case 2: x 2 * = 0 We get x 1 * from the budget constraint We get x 1 * from the budget constraint  x 1 * = y / p 1

Frank Cowell: Microeconomics Ex 4.12(1) Demand functions We can summarise the optimal demands for the two goods thus We can summarise the optimal demands for the two goods thus

Frank Cowell: Microeconomics Ex 4.12(1) Indirect utility function Get maximised utility by substituting x * into the utility function Get maximised utility by substituting x * into the utility function  V(p 1, p 2, y) = U(x 1 *, x 2 * )  = U(D 1 (p 1, p 2, y), D 2 (p 1, p 2, y)) Case 1: p 1 >  p 1 Case 1: p 1 >  p 1 Case 2: p 1 ≤  p 1 Case 2: p 1 ≤  p 1

Frank Cowell: Microeconomics Ex 4.12(1) Cost function Get cost function (expenditure function) from the indirect utility function Get cost function (expenditure function) from the indirect utility function  maximised utility is  = V(p 1, p 2, y)  invert this to get y = C(p 1, p 2,  ) Case 1: p 1 >  p 1 Case 1: p 1 >  p 1 Case 2: p 1 ≤  p 1 Case 2: p 1 ≤  p 1

Frank Cowell: Microeconomics Ex 4.12(2) Question purpose: to derive standard welfare concept purpose: to derive standard welfare concept method: use part 1 and manipulate the indirect utility function method: use part 1 and manipulate the indirect utility function

Frank Cowell: Microeconomics Ex 4.12(2) Compute CV Get compensating variation (1) from indirect utility function Get compensating variation (1) from indirect utility function  before price change:  = V(,, )  before price change:  = V(p 1, p 2, y)  after price change:  = V(,, )  after price change:  = V(p 1 ', p 2, y − CV) Equivalently (2) could use cost function directly Equivalently (2) could use cost function directly  = C(,,  )C(,,  )  CV = C(p 1, p 2,  ) − C(p 1 ', p 2,  ) In Case 1 above we have In Case 1 above we have Rearranging, we find: Rearranging, we find: Equivalently Equivalently

Frank Cowell: Microeconomics Ex 4.12(3) In case 1 we have x 1 * = [½  p 2 / p 1 ] 2 In case 1 we have x 1 * = [½  p 2 / p 1 ] 2 So demand for good 1 has zero income effect So demand for good 1 has zero income effect Therefore, in this case CV = CS = EV Therefore, in this case CV = CS = EV

Frank Cowell: Microeconomics Ex 4.12: Points to remember It’s always a good idea to sketch the indifference curves 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 A corner solution can sometimes just be handled as two separate cases There’s often more than one way of getting to a solution There’s often more than one way of getting to a solution  in this case two equivalent derivations of CV