Lecture Examples

Frank Cowell: Lecture Examples

LECTURE EXAMPLESEC202http://darp.lse.ac.uk/ec202

Additional examples provided during lectures in 2014

Frank Cowell

8 Dec 2014 1


Example – single technique

z1

z2

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1 z1·

3z1

z2

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3 z2·

1

8 Dec 2014 2


Example – two techniques

z1

z2

0

3

1z1

z2·

·

31

8 Dec 2014 3


Example – multiple techniques

z1

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z2·

z1·

8 Dec 2014 4


Example:

58 Dec 2014

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

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}


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


Example

78 Dec 2014

• Production function

• Keep input 2 constant

• Marginal product of good 1

Frank Cowell: Lecture Examples 88 Dec 2014


Example – cost-min, single technique

z1

z2

0

1 z1·

3z1

z2

0

3 z2·

1

8 Dec 2014 9


Example – cost-min, two techniques

z1

z2

0

3

1z1

z2·

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31

8 Dec 2014 10


Example

118 Dec 2014

z2

z1

• Isoquant (as before)• does not touch either axis

• Constraint set for given q• Cost minimisation must have interior solution


• Lagrangean for cost minimisation

• Necessary and sufficient for minimum:

• Evaluate first-order conditions

Example

128 Dec 2014

z2

z1

z*


• 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


• 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






Example

168 Dec 2014


• 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



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



Examplex2

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


• Lagrangean for cost minimisation

• For a minimum:


Examplex2

x1

x*

8 Dec 2014 22






Example

8 Dec 2014 23


• 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


Example

x2

x1

x*

• Lagrangean for utility maximisation

8 Dec 2014 25



Example

x2

x1

x*

• Optimal demands are

• So at the optimum

8 Dec 2014 26



• Results from cost minimisation:

• Differentiate to get compensated demand:

• Results from utility maximisation:

Example

8 Dec 2014 28


• 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


• 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



• 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



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



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



Example• Suppose property distribution is:• Reservation utility

• Incomes are• Demands by a and b (offer curves):

• Equilibrium where

388 Dec 2014


Example• Marginal Rate of Substitution:

• Assume that total endowment is (12,12)

• Contract curve is

• Which implies:

398 Dec 2014



Example• Suppose property distribution is:• Incomes are• Demands by a and b :

• Excess demands:

• Walras’ Law

• Equilibrium price:

• Equilibrium allocation

418 Dec 2014



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



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



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



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


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



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



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



Example• Suppose property distribution is:• Incomes are• Demands by a and b :

• Materials balance:

• Equilibrium price:

• Incomes in equilibrium allocation

568 Dec 2014


Example• Property distribution is:• Incomes in equilibrium allocation:• Extreme cases:

• Income-possibility set

578 Dec 2014

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