1. Lecture Examples EC202 Frank Cowell http://darp.lse.ac.uk/ec202
Additional examples provided during lectures in 2014
Frank Cowell
To understand the basics of welfare economics
Gisela: 6674
6300 (or 7437 or 6460). A31 Audio Visual Unit
DESMOND Mr George 6271
FLOOD Mr Ray 7694
GALE Mr Adam 6520
HEAD Mr Chris 6417
Microphone safe: 0000# Put the mic back in once finished. Ensure that you have pressed the 'on' button for the radio mic and also that the 'Mute' button isn't on.
8 Dec 2014

2. Example – single techniquez2 z2 z2 3
Isoquant1-4 z1 1 z1 z1 3 1
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3. Example – two techniquesz2 z2 3
Isoquant1-4 z1 1 z1 1 3
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4. Example – multiple techniquesz2 z2 3
Isoquant1-4 z1 1 z1 1 3
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5. Example:
Use spreadsheet to find (z1, z2) such that log 2 = 0.25 log z log z2)
Plot on graph
Z(2) = {z: f (z) ³ 2} z2
Find the input requirement set for q = 2 z1
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6. Example Isoquant q = 2 (as before) Isoquant q = 1 Isoquant q = 3z2
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
Extend the example from before z1
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7. Example Production function Keep input 2 constant
Marginal product of good 1
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9. Example – cost-min, single techniquez2 z2 z2 3
Isoquant1-4 z1 1 z1 z1 3 1
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10. Example – cost-min, two techniquesz2 z2 3
Isoquant1-4 z1 1 z1 1 3
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11. Example Isoquant (as before) does not touch either axis
Constraint set for given q
Cost minimisation must have interior solution z2 z1
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12. Example z* Lagrangean for cost minimisation
Necessary and sufficient for minimum:
Evaluate first-order conditions z2 z* z1
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13. 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
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14. 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
Cost Function: 5 things to remember (and check above)
Non-decreasing in every input price
Increasing in at least one input price
Increasing in output
Concave in prices – note second derivative negative
Homogeneous of degree 1 in prices
Shephard's Lemma (qick check now more next lecture)
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16. 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
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17. Example From last lecture, cost function is
Differentiate w.r.t. w1 and w2
Slope of conditional demand functions
10 Oct 2012
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19. Example indiff curve u = log 1 indiff curve u = log 2
From the equation
Equation of IC is x2
Check against the example in lecture 1
Transformed utility function x1
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21. Example Indifference curve (as before) does not touch either axis
Constraint set for given u
Cost minimisation must have interior solution x2 x1
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22. Example x* Lagrangean for cost minimisation For a minimum:
Evaluate first-order conditions x2 x* x1
8 Dec 2014

23. 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

24. 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

25. Example x* Lagrangean for utility maximisation
Evaluate first-order conditions x* x1
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26. Example
Optimal demands are
So at the optimum x2 x* x1
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28. Example Results from cost minimisation:
Differentiate to get compensated demand:
Results from utility maximisation:
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29. 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:
Features of demand functions
Homogeneous of degree zero
Satisfy the “adding-up” constraint
Symmetric substitution effects
Negative own-price substitution effects
Income effects could be positive or negative:
in fact they are nearly always a pain
8 Dec 2014

30. Example in fact they are nearly always a pain
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:
Features of demand functions
Homogeneous of degree zero
Satisfy the “adding-up” constraint
Symmetric substitution effects
Negative own-price substitution effects
Income effects could be positive or negative:
in fact they are nearly always a pain
8 Dec 2014

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32. Example in fact they are nearly always a pain 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:
Features of demand functions
Homogeneous of degree zero
Satisfy the “adding-up” constraint
Symmetric substitution effects
Negative own-price substitution effects
Income effects could be positive or negative:
in fact they are nearly always a pain
8 Dec 2014

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34. 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
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36. 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
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38. Example Suppose property distribution is: Reservation utility
Incomes are
Demands by a and b (offer curves):
Equilibrium where
8 Dec 2014

39. Example Marginal Rate of Substitution:
Assume that total endowment is (12,12)
Contract curve is
Which implies:
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41. Example Suppose property distribution is: Incomes are
Demands by a and b :
Excess demands:
Walras’ Law
Equilibrium price:
Equilibrium allocation
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43. Example P0 indifference curves Implied probabilities
Marginal rate of substitution
A prospect
The mean
Find the certainty equivalent
xBLUE
Check against the example in lecture 5
Explain slope of ray
Show that slope is the same all along the 45 degree line
Explain slope of line through P0 P0
xRED
21 Nov 2012
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45. Example P0 A prospect Certainty equivalent
Risk premium: 1.75 – = 0.346
Felicity function
xBLUE P0
xRED
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22 Nov 2012

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47. 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?
W has got to be at least as large as 3L to be interesting
read off what happens if W increases and L decreases to keep LW constant
Read off what happens if W and L increase to keep W-3L constant
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49. Example: Cycles and aggregation
Righthanders: Ordering is q ,q’’ ,q’
versus q’’ 18 for 17 against: q beats q’’
q’’ versus q’ 35 for 0 against: q’’ beats q’
versus q’ 35 for 0 against: q beats q’
Lefthanders: (1) Cycle! (2) Ignore smallest win: q ,q’ ,q’’
versus q’’ 16 for 18 against: q’’ beats q
q’’ versus q’ 12 for 22 against: q’ beats q’’
versus q’ 22 for 12 against: q beats q’
Combined Ordering is q’’ ,q ,q’
versus q’’ for against: q’’ beats q
q’’ versus q’ for against: q’’ beats q’
versus q’ for 6+6 against: q beats q’
What happens if Right-handers vote?
What happens if Left-handers vote?
What happens if there’s a combined vote?
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50. 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]
W has got to be at least as large as 3L to be interesting
read off what happens if W increases and L decreases to keep LW constant
Read off what happens if W and L increase to keep W-3L constant
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52. Example: envy Utility functions for a and b: Suppose the allocation is
Is this envy free?
Now suppose the allocation is
21 units of good 1 and 9 units of good 2
First allocation is not envy free – both would prefer b’s bundle
Second allocation is envy free
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54. 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:
Explain how a’s consumption is determined
Explain what happens for all efficient points.
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56. Example Suppose property distribution is: Incomes are
Demands by a and b :
Materials balance:
Equilibrium price:
Incomes in equilibrium allocation
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57. Example yb ya Property distribution is:
Incomes in equilibrium allocation:
Extreme cases:
Income-possibility set yb ya
8 Dec 2014