#### Transcript The Firm: Optimisation - London School of Economics

LECTURE EXAMPLES EC202 http://darp.lse.ac.uk/ec202 Additional examples provided during lectures in 2014 Frank Cowell 8 Dec 2014 Frank Cowell: Lecture Examples 1 Example – single technique z2 z2 2 z 3 1 z 1 0 8 Dec 2014 3 z1 0 z1 1 Frank Cowell: Lecture Examples 2 Example – two techniques z2 3 1 0 8 Dec 2014 z2 1 3 z1 z1 Frank Cowell: Lecture Examples 3 Example – multiple techniques z2 3 1 0 8 Dec 2014 z2 1 3 z1 z1 Frank Cowell: Lecture Examples 4 Example: • Use spreadsheet to find (z1, z2) such that log 2 = 0.25 log z1+ 0.75log z2) 7 z2 6 • Plot on graph 5 • Z(2) = {z: f (z) 2} 4 3 2 1 0 0 8 Dec 2014 1 2 3 4 5 6 z1 7 Frank Cowell: Lecture Examples 5 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 8 Dec 2014 Frank Cowell: Lecture Examples 6 Example • Production function • Keep input 2 constant • Marginal product of good 1 8 Dec 2014 Frank Cowell: Lecture Examples 7 8 Dec 2014 Frank Cowell: Lecture Examples 8 Example – cost-min, single technique z2 z2 2 z 3 1 z 1 0 8 Dec 2014 3 z1 0 z1 1 Frank Cowell: Lecture Examples 9 Example – cost-min, two techniques z2 3 1 0 8 Dec 2014 z2 1 3 z1 z1 Frank Cowell: Lecture Examples 10 Example z2 • Isoquant (as before) • does not touch either axis • Constraint set for given q • Cost minimisation must have interior solution z1 8 Dec 2014 Frank Cowell: Lecture Examples 11 Example • Lagrangean for cost minimisation z2 • Necessary and sufficient for minimum: • Evaluate first-order conditions z* z1 8 Dec 2014 Frank Cowell: Lecture Examples 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 8 Dec 2014 Frank Cowell: Lecture Examples 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 8 Dec 2014 Frank Cowell: Lecture Examples 14 8 Dec 2014 Frank Cowell: Lecture Examples 15 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 8 Dec 2014 Frank Cowell: Lecture Examples 16 Example • From last lecture, cost function is • Differentiate w.r.t. w1 and w2 • Slope of conditional demand functions 810Dec Oct2014 2012 17 Frank Cowell: Lecture Examples 17 8 Dec 2014 Frank Cowell: Lecture Examples 18 Example x2 • indiff curve u = log 1 • indiff curve u = log 2 • indiff curve u = log 3 • From the equation • Equation of IC is • Transformed utility function x1 8 Dec 2014 Frank Cowell: Lecture Examples 19 8 Dec 2014 Frank Cowell: Lecture Examples 20 Example x2 • Indifference curve (as before) • does not touch either axis • Constraint set for given u • Cost minimisation must have interior solution x1 8 Dec 2014 Frank Cowell: Lecture Examples 21 Example • Lagrangean for cost minimisation x2 • For a minimum: • Evaluate first-order conditions x* x1 8 Dec 2014 Frank Cowell: Lecture Examples 22 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 8 Dec 2014 Frank Cowell: Lecture Examples 23 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 8 Dec 2014 Frank Cowell: Lecture Examples 24 Example • Lagrangean for utility maximisation x2 • Evaluate first-order conditions x* x1 8 Dec 2014 Frank Cowell: Lecture Examples 25 Example • Optimal demands are x2 • So at the optimum x* x1 8 Dec 2014 Frank Cowell: Lecture Examples 26 8 Dec 2014 Frank Cowell: Lecture Examples 27 Example • Results from cost minimisation: • Differentiate to get compensated demand: • Results from utility maximisation: 8 Dec 2014 Frank Cowell: Lecture Examples 28 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: 8 Dec 2014 Frank Cowell: Lecture Examples 29 Example • Take a case where income is endogenous: • Ordinary demand for good 1: • Response to changes in y and p1: • Modified Slutsky equation: • In this case: 8 Dec 2014 Frank Cowell: Lecture Examples 30 8 Dec 2014 Frank Cowell: Lecture Examples 31 Example • Cost function: • Indirect utility function: • If p1 falls to tp1 (where t < 1) then utility rises from u to u′: • So CV of change is: • And the EV is: 8 Dec 2014 Frank Cowell: Lecture Examples 32 8 Dec 2014 Frank Cowell: Lecture Examples 33 Example • Rearranged production function: q2 high R3 • Three goods • goods 1 and 2 are outputs (+) • good 3 is an input () • If all of resource 3 used as input: • Attainable set low R3 q1 8 Dec 2014 Frank Cowell: Lecture Examples 34 8 Dec 2014 Frank Cowell: Lecture Examples 35 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 8 Dec 2014 Frank Cowell: Lecture Examples 36 8 Dec 2014 Frank Cowell: Lecture Examples 37 Example • Suppose property distribution is: • Reservation utility • Incomes are • Demands by a and b (offer curves): • Equilibrium where 8 Dec 2014 Frank Cowell: Lecture Examples 38 Example • Marginal Rate of Substitution: • Assume that total endowment is (12,12) • Contract curve is • Which implies: 8 Dec 2014 Frank Cowell: Lecture Examples 39 8 Dec 2014 Frank Cowell: Lecture Examples 40 Example • Suppose property distribution is: • Incomes are • Demands by a and b : • Excess demands: • Walras’ Law • Equilibrium price: • Equilibrium allocation 8 Dec 2014 Frank Cowell: Lecture Examples 41 8 Dec 2014 Frank Cowell: Lecture Examples 42 Example xBLUE • indifference curves • Implied probabilities • Marginal rate of substitution • A prospect • The mean • Find the certainty equivalent P0 xRED 218Nov Dec 2012 2014 Frank Cowell: Lecture Examples 43 8 Dec 2014 Frank Cowell: Lecture Examples 44 Example • A prospect • Certainty equivalent • Risk premium: 1.75 – 1.414 = 0.346 • Felicity function xBLUE P0 xRED Dec 2012 2014 228Nov Frank Cowell: Lecture Examples 45 8 Dec 2014 Frank Cowell: Lecture Examples 46 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? 8 Dec 2014 Frank Cowell: Lecture Examples 47 8 Dec 2014 Frank Cowell: Lecture Examples 48 Example: Cycles and aggregation • What happens if Right-handers vote? • What happens if Left-handers vote? • What happens if there’s a combined vote? 8 Dec 2014 Frank Cowell: Lecture Examples 49 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] 8 Dec 2014 Frank Cowell: Lecture Examples 50 8 Dec 2014 Frank Cowell: Lecture Examples 51 Example: envy • Utility functions for a and b: • Suppose the allocation is • Is this envy free? • Now suppose the allocation is • Is this envy free? 8 Dec 2014 Frank Cowell: Lecture Examples 52 8 Dec 2014 Frank Cowell: Lecture Examples 53 Example • Suppose we have an exchange economy where stocks of the goods are (12, 12). • To find efficient points, max b’s utility keeping a’s utility constant • Lagrangean is • First-order conditions are: • Rearranging: • So efficient points are characterised by: 8 Dec 2014 Frank Cowell: Lecture Examples 54 8 Dec 2014 Frank Cowell: Lecture Examples 55 Example • Suppose property distribution is: • Incomes are • Demands by a and b : • Materials balance: • Equilibrium price: • Incomes in equilibrium allocation 8 Dec 2014 Frank Cowell: Lecture Examples 56 Example • Property distribution is: • Incomes in equilibrium allocation: • Extreme cases: • Income-possibility set yb ya 8 Dec 2014 Frank Cowell: Lecture Examples 57