Frank Cowell: Lecture Examples LECTURE EXAMPLES EC202 Additional examples provided during lectures...

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

0

1 z1·

3z1

z2

0

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

z2

0

3

1

31

z2·

z1

·

8 Dec 2014 4


Example:

58 Dec 2014

0 1 2 3 4 5 6 7

0

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·

·

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



• From the equation

• Equation of IC is

• Transformed utility function

8 Dec 2014


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


• Lagrangean for cost minimisation

• For a minimum:


Example

x2

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



• 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



• 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



• 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

ya

yb

Frank Cowell: Lecture Examples LECTURE EXAMPLES EC202 Additional examples provided during lectures...

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