Frank Cowell Microeconomics November 2006 Exercise 4 12
Frank Cowell: Microeconomics November 2006 Exercise 4. 12 MICROECONOMICS Principles and Analysis Frank Cowell
Ex 4. 12(1) Question Frank Cowell: Microeconomics n n 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 n First steps are as follows: n Sketch indifference curves u n Write down budget constraint u n Straightforward – parabolic contours Straightforward – fixed-income case Set out optimisation problem
Ex 4. 12(1) Indifference curves Frank Cowell: Microeconomics x 2 Slope is vertical here Could have x 2 = 0 x 1 0 0 1 2
Ex 4. 12(1) Budget constraint, FOC Frank Cowell: Microeconomics n Budget constraint: Substitute this into the utility function: We get the objective function: n FOC for an interior solution: n n
Ex 4. 12(1) Using the FOC Frank Cowell: Microeconomics n Remember that person might consume zero of commodity 2 u consider two cases n Case 1: x 2* > 0 From the FOC: n But, to make sense this case requires: n Case 2: x 2* = 0 We get x 1* from the budget constraint n n u x 1* = y / p 1
Ex 4. 12(1) Demand functions Frank Cowell: Microeconomics n We can summarise the optimal demands for the two goods thus
Ex 4. 12(1) Indirect utility function Frank Cowell: Microeconomics n Get maximised utility by substituting x* into the utility function u u 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)) n Case 1: p 1 >`p 1 n Case 2: p 1 ≤`p 1
Ex 4. 12(1) Cost function Frank Cowell: Microeconomics n Get cost function (expenditure function) from the indirect utility function u u maximised utility is u = V(p 1, p 2, y) invert this to get y = C(p 1 , p 2 , u) n Case 1: p 1 >`p 1 n Case 2: p 1 ≤ `p 1
Ex 4. 12(2) Question Frank Cowell: Microeconomics n n 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 n Get compensating variation (1) from indirect utility function u u n before price change: u = V(p 1, p 2, y) after price change: u = V(p 1', p 2, y − CV) Equivalently (2) could use cost function directly u CV = C(p 1, p 2, u) − C(p 1', p 2, u) n In Case 1 above we have n Rearranging, we find: n Equivalently
Ex 4. 12(3) Frank Cowell: Microeconomics n n n In case 1 we have x 1* = [½ a p 2 / p 1]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 n It’s always a good idea to sketch the indifference curves u u n n 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 u in this case two equivalent derivations of CV
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