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Exercise 4.12

Exercise 4.12. MICROECONOMICS Principles and Analysis Frank Cowell. November 2006. Ex 4.12(1) Question. purpose : to derive solution and response functions for quasilinear preferences method : substitution of budget constraint into utility function and then simple maximisation.

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Exercise 4.12

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  1. Exercise 4.12 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

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

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

  4. Ex 4.12(1) Indifference curves x2 Slope is vertical here Could have x2 = 0 x1 0 0 1 2

  5. Ex 4.12(1) Budget constraint, FOC • Budget constraint: • Substitute this into the utility function: • We get the objective function: • FOC for an interior solution:

  6. Ex 4.12(1) Using the FOC • 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

  7. Ex 4.12(1) Demand functions • We can summarise the optimal demands for the two goods thus

  8. Ex 4.12(1) Indirect utility function • 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

  9. Ex 4.12(1) Cost function • 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

  10. Ex 4.12(2) Question • purpose: to derive standard welfare concept • method: use part 1 and manipulate the indirect utility function

  11. Ex 4.12(2) Compute CV • 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

  12. Ex 4.12(3) • In case 1 we have x1* = [½ap2 / p1]2 • So demand for good 1 has zero income effect • Therefore, in this case CV = CS = EV

  13. Ex 4.12: Points to remember • 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

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