Frank Cowell: Lecture Examples
LECTURE EXAMPLESEC202http://darp.lse.ac.uk/ec202
Additional examples provided during lectures in 2014
Frank Cowell
8 Dec 2014 1
Frank Cowell: Lecture Examples
Example – single technique
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8 Dec 2014 2
Frank Cowell: Lecture Examples
Example – two techniques
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8 Dec 2014 3
Frank Cowell: Lecture Examples
Example – multiple techniques
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Frank Cowell: Lecture Examples
Example:
58 Dec 2014
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z2
z1
• 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}
Frank Cowell: Lecture Examples
Example
68 Dec 2014
z2
z1
• 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
Frank Cowell: Lecture Examples
Example
78 Dec 2014
• Production function
• Keep input 2 constant
• Marginal product of good 1
Frank Cowell: Lecture Examples 88 Dec 2014
Frank Cowell: Lecture Examples
Example – cost-min, single technique
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8 Dec 2014 9
Frank Cowell: Lecture Examples
Example – cost-min, two techniques
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8 Dec 2014 10
Frank Cowell: Lecture Examples
Example
118 Dec 2014
z2
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• Isoquant (as before)
• does not touch either axis
• Constraint set for given q
• Cost minimisation must have interior solution
Frank Cowell: Lecture Examples
• Lagrangean for cost minimisation
• Necessary and sufficient for minimum:
• Evaluate first-order conditions
Example
128 Dec 2014
z2
z1
z*
Frank Cowell: Lecture Examples
• 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
Example
138 Dec 2014
Frank Cowell: Lecture Examples
• 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
Example
148 Dec 2014
Frank Cowell: Lecture Examples8 Dec 2014 15
Frank Cowell: Lecture Examples
• 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
Example
168 Dec 2014
Frank Cowell: Lecture Examples
• From last lecture, cost function is
• Differentiate w.r.t. w1 and w2
• Slope of conditional demand functions
Example
1710 Oct 20128 Dec 2014 17
Frank Cowell: Lecture Examples 188 Dec 2014
Frank Cowell: Lecture Examples
Example
19
x2
x1
• 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
8 Dec 2014
Frank Cowell: Lecture Examples 208 Dec 2014
Frank Cowell: Lecture Examples
Example
x2
x1
• Indifference curve (as before)
• does not touch either axis
• Constraint set for given u• Cost minimisation must have interior solution
8 Dec 2014 21
Frank Cowell: Lecture Examples
• Lagrangean for cost minimisation
• For a minimum:
• Evaluate first-order conditions
Example
x2
x1
x*
8 Dec 2014 22
Frank Cowell: Lecture Examples
• 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
Example
8 Dec 2014 23
Frank Cowell: Lecture Examples
• 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
Example
8 Dec 2014 24
Frank Cowell: Lecture Examples
Example
x2
x1
x*
• Lagrangean for utility maximisation
8 Dec 2014 25
• Evaluate first-order conditions
Frank Cowell: Lecture Examples
Example
x2
x1
x*
• Optimal demands are
• So at the optimum
8 Dec 2014 26
Frank Cowell: Lecture Examples 278 Dec 2014
Frank Cowell: Lecture Examples
• Results from cost minimisation:
• Differentiate to get compensated demand:
• Results from utility maximisation:
Example
8 Dec 2014 28
Frank Cowell: Lecture Examples
• 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:
Example
8 Dec 2014 29
Frank Cowell: Lecture Examples
• 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:
Example
8 Dec 2014 30
Frank Cowell: Lecture Examples 318 Dec 2014
Frank Cowell: Lecture Examples
• 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:
Example
8 Dec 2014 32
Frank Cowell: Lecture Examples 338 Dec 2014
Frank Cowell: Lecture Examples
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
8 Dec 2014
q2
q1
high R3
low R3
34
Frank Cowell: Lecture Examples 358 Dec 2014
Frank Cowell: Lecture Examples
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
368 Dec 2014
Frank Cowell: Lecture Examples 378 Dec 2014
Frank Cowell: Lecture Examples
Example• Suppose property distribution is:
• Reservation utility
• Incomes are
• Demands by a and b (offer curves):
• Equilibrium where
388 Dec 2014
Frank Cowell: Lecture Examples
Example• Marginal Rate of Substitution:
• Assume that total endowment is (12,12)
• Contract curve is
• Which implies:
398 Dec 2014
Frank Cowell: Lecture Examples 408 Dec 2014
Frank Cowell: Lecture Examples
Example• Suppose property distribution is:
• Incomes are
• Demands by a and b :
• Excess demands:
• Walras’ Law
• Equilibrium price:
• Equilibrium allocation
418 Dec 2014
Frank Cowell: Lecture Examples 428 Dec 2014
Frank Cowell: Lecture Examples
Example
43
xBLUE
xRED
· P0
• indifference curves
• Implied probabilities
• Marginal rate of substitution
• A prospect
• The mean
• Find the certainty equivalent
21 Nov 20128 Dec 2014 43
Frank Cowell: Lecture Examples 448 Dec 2014
Frank Cowell: Lecture Examples
Example
45
xBLUE
xRED
· P0
• A prospect
• Certainty equivalent
• Risk premium: 1.75 – 1.414 = 0.346
• Felicity function
22 Nov 2012 458 Dec 2014
Frank Cowell: Lecture Examples 468 Dec 2014
Frank Cowell: Lecture Examples
Example
47
• 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?
478 Dec 2014
Frank Cowell: Lecture Examples 488 Dec 2014
Frank Cowell: Lecture Examples
Example: Cycles and aggregation
49
• What happens if Right-handers vote?
• What happens if Left-handers vote?
• What happens if there’s a combined vote?
498 Dec 2014
Frank Cowell: Lecture Examples
Example: IID
50
• 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]
508 Dec 2014
Frank Cowell: Lecture Examples 518 Dec 2014
Frank Cowell: Lecture Examples
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?
528 Dec 2014
Frank Cowell: Lecture Examples 538 Dec 2014
Frank Cowell: Lecture Examples
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:
548 Dec 2014
Frank Cowell: Lecture Examples 558 Dec 2014
Frank Cowell: Lecture Examples
Example• Suppose property distribution is:
• Incomes are
• Demands by a and b :
• Materials balance:
• Equilibrium price:
• Incomes in equilibrium allocation
568 Dec 2014
Frank Cowell: Lecture Examples
Example• Property distribution is:
• Incomes in equilibrium allocation:
• Extreme cases:
• Income-possibility set
578 Dec 2014
ya
yb