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Lecture Examples

Lecture Examples. EC202 http://darp.lse.ac.uk/ec202 Additional examples provided during lectures in 2013 Frank Cowell. Example – single technique. z 2. z 2. z 2. 3. z 1. 1. z 1. z 1. 0. 0. 3. 1. Example – two techniques. z 2. z 2. 3. z 1. 1. z 1. 0. 3. 1.

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Lecture Examples

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  1. Lecture Examples EC202 http://darp.lse.ac.uk/ec202 Additional examples provided during lectures in 2013 Frank Cowell

  2. Example – single technique z2 z2 z2 3 z1 1 z1 z1 0 0 3 1

  3. Example – two techniques z2 z2 3 z1 1 z1 0 3 1

  4. Example – multiple techniques z2 z2 3 z1 1 z1 0 3 1

  5. Example: • Use spreadsheet to find (z1, z2) such that log 2 = 0.25 log z1+ 0.75log z2) • Plot on graph • Z(2) = {z: f (z) ³ 2} z2 z1

  6. Example z2 • Isoquant q = 2 (as before) • Isoquant q = 1 • Isoquant q = 3 • Equation of isoquant • Homotheticity • Check HD 1 from original equation • double inputs → double output z1

  7. Example • Production function • Keep input 2 constant • Marginal product of good 1

  8. Example – cost-min, single technique z2 z2 z2 3 z1 1 z1 z1 0 0 3 1

  9. Example – cost-min, two techniques z2 z2 3 z1 1 z1 0 3 1

  10. Example • Isoquant (as before) • does not touch either axis • Constraint set for given q • Cost minimisation must have interior solution z2 z1

  11. Example • Lagrangean for cost minimisation • Necessary and sufficient for minimum: • Evaluate first-order conditions z2 z* z1

  12. Example • First-order conditions for cost-min: • Rearrange the first two of these: • Substitute back into the third FOC: • Rearrange to get the optimised Lagrange multiplier

  13. Example • From first-order conditions: • Rearrange to get cost-min inputs: • By definition minimised cost is: • In this case the expression just becomes l* • So cost function is

  14. Example • First-order conditions for cost-min: • Rearrange the first two of these: • Substitute back into the third FOC: • Rearrange to get the optimised Lagrange multiplier

  15. Example • From last lecture, cost function is • Differentiate w.r.t. w1and w2 • Slope of conditional demand functions 10 Oct 2012 17

  16. Example • indiff curve u = log 1 • indiff curve u = log 2 • indiff curve u = log 3 • From the equation • Equation of IC is x2 • Transformed utility function x1

  17. Example • Indifference curve (as before) • does not touch either axis • Constraint set for given u • Cost minimisation must have interior solution x2 x1

  18. Example • Lagrangean for cost minimisation • For a minimum: • Evaluate first-order conditions x2 x* x1

  19. Example • First-order conditions for cost-min: • Rearrange the first two of these: • Substitute back into the third FOC: • Rearrange to get the optimised Lagrange multiplier

  20. Example • From first-order conditions: • Rearrange to get cost-min inputs: • By definition minimised cost is: • In this case the expression just becomes l* • So cost function is

  21. Example • Lagrangean for utility maximisation • Evaluate first-order conditions x2 x* x1

  22. Example • Optimal demands are • So at the optimum x2 x* x1

  23. Example • Results from cost minimisation: • Differentiate to get compensated demand: • Results from utility maximisation:

  24. Example • Ordinary and compensated demand for good 1: • Response to changes in y and p1: • Use cost function to write last term in y rather than u: • Slutsky equation: • In this case:

  25. Example • Take a case where income is endogenous: • Ordinary demand for good 1: • Response to changes in y and p1: • Modified Slutskyequation: • In this case:

  26. Example • Cost function: • Indirect utility function: • If p1falls to tp1 (where t < 1) then utility rises fromu to u′: • So CV of change is: • And the EV is:

  27. Example • Rearranged production function: • Three goods • goods 1 and 2 are outputs (+) • good 3 is an input () • If all of resource 3 used as input: • Attainable set high R3 q2 low R3 q1

  28. Example • Suppose property distribution is: • Incomes are • Given Cobb-Douglas preferences demands are • So, total demand for good 1 is • From materials-balance condition • Which can only hold if • So, equilibrium consumption of a is • Therefore equilibrium consumption of b is

  29. Example • Suppose property distribution is: • Reservation utility • Incomes are • Demands by a and b (offer curves): • Equilibrium where

  30. Example • Marginal Rate of Substitution: • Assume that total endowment is (12,12) • Contract curve is • Which implies:

  31. Example • Suppose property distribution is: • Incomes are • Demands by a and b : • Excess demands: • Walras’ Law • Equilibrium price: • Equilibrium allocation

  32. Example • indifference curves • Implied probabilities • Marginal rate of substitution • A prospect • The mean • Find the certainty equivalent xBLUE • P0 xRED 43 21 Nov 2012

  33. Example • A prospect • Certainty equivalent • Risk premium: 1.75 – 1.414 = 0.346 • Felicity function xBLUE • P0 xRED 45 22 Nov 2012

  34. Example • Suppose, if you win return is r = W, if you lose return is r = L • Expected rate of return is • If you invest b, then expected utility is • FOC • Optimal investment • Do rich people invest more? 47

  35. Example: Cycles and aggregation • What happens if Right-handers vote? • What happens if Left-handers vote? • What happens if there’s a combined vote? 49

  36. Example: IID • Suppose, Alf, Bill and Charlie have the following rankings • Everyone allocates 1 vote to the worst, 2 to the second worst,… • Votes over the four states are [8,7,7,8] • What if we exclude states 2 and 3? • If focus just on states 1 and 4 votes are [4,5] 50

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