frank cowell: ec202 microeconomics revision lecture 2 ec202: microeconomic principles ii frank...
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Frank C
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Frank C
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C202 M
icroeconomics
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Revision Lecture 2
EC202: Microeconomic Principles IIEC202: Microeconomic Principles IIFrank Cowell Frank Cowell
May 2008May 2008
Frank C
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C202 M
icroeconomics
EC
202 Microeconom
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Objectives of the lecture
Think more about method for long questionsThink more about method for long questions Look at a few CfDLook at a few CfD
4.12, 4.134.12, 4.13 5.15.1 (7.8)(7.8) 9.69.6
See how they illustrate methodSee how they illustrate method Connect these to past exam questionsConnect these to past exam questions
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Question Style – three types
1 Principles1 Principles reason on standard results and argumentsreason on standard results and arguments can use verbal and/or mathematical reasoningcan use verbal and/or mathematical reasoning
2 Model solving2 Model solving a standard frameworka standard framework you just turn the wheelsyou just turn the wheels
3 Model building3 Model building usually get guidance in the questionusually get guidance in the question longer question sometimes easier?longer question sometimes easier?
One type not necessarily “easier” or “harder” than another One type not necessarily “easier” or “harder” than another focus here on types 2 and 3focus here on types 2 and 3 examples throughout the “long” questions of recent exam papersexamples throughout the “long” questions of recent exam papers
A simple type 2 question –
Ex 5.1
A simple type 2 question –
Ex 5.1
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Ex 5.1(1) Question
purpose: construct a simple model of household supply and examine how it works
method: build model up step-by-step through the question parts
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Ex 5.1(1) Preference mapx2
x1
•0
•
1•
•
k
Shift the origin to (0, k)
Draw ICs homothetic to the shifted origin
indifference curves are “shifted” Cobb-Douglas k is min requirement of other goods. is share of budget of rice after an amount has been set aside to buy the min requirementrice
othe
r go
ods
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Ex 5.1(2) Question
method: Work out the budget constraint. Use the utility function to set out the Lagrangean Find the FOCs for an interior solution Find the demand functions Use these to get household supply function
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Ex 5.1(2) Budget constraint
Use good 2 as unit of value price of rice (good 1) is p price of all other goods (good 2) is 1
The consumer’s income is therefore:y := pR1 + R2
The budget constraint is
px1 + x2 y where y is given by the above
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Ex 5.1(2) Lagrangean method
The Lagrangean is
log(x1) + [1– ]log(x2–k) + [ y – px1 – x2 ]
The FOC for an interior maximum are — – p
x1*
1–—— –
x2*–k
y – px1* – x2
*
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Ex 5.1(2) Demand functions
From the FOC: px1
*— 1–
x2*k + ——
Adding these and using the budget constraint, we have
y k + 1/ Eliminating in the above:
x1
*— [y – k] p
x2*k + [1– ] y
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Ex 5.1(2) Supply function
Supply of good 1 is given by
S(p) := R1 – x1*
Substituting in for y, we have S(p) [1– ]R1 – — [R2 – k]
p
Supply increases with price if R2 > k
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Ex 5.1(2) Solutionx2
x1
k
Endowment
Budget constraint
rice
othe
r go
ods
x*
x2
p
S
R•
•
Optimal consumption
Supply of rice
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Ex 5.1(3)
Let c be the amount of the ration If k + [1– ] y c nothing changes from previous case Otherwise
px1 + c = y so that R2 – c
x1 = R1 +——— p
c – R2 S(p) ———
p
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Ex 5.1(3) Modified solutionx2
x1
k
Original solution
rice
othe
r go
ods
x*
x2
S
R•
•
Generous ration
Severe ration
x**•
Generous ration has no effect
Severe ration on other goods affects supply of rice
c
c
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Ex 5.1: Points to remember Use diagram to understand features of utility function Model supply as mirror image of demand Use diagram to see effect of ration Connection to exams
see 2005 q3 …it’s the same model!
A “meatier” type 2 question – Ex
4.12, 4.13
A “meatier” type 2 question – Ex
4.12, 4.13
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Ex 4.12(1) Question
purposepurpose: to derive solution and response functions for quasilinear : to derive solution and response functions for quasilinear preferencespreferences
methodmethod: substitution of budget constraint into utility function and then : substitution of budget constraint into utility function and then simple maximisationsimple maximisation
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Ex 4.12(1) Preliminary
First steps are as follows:First steps are as follows:
Sketch indifference curvesSketch indifference curves Straightforward – parabolic contoursStraightforward – parabolic contours
Write down budget constraintWrite down budget constraint Straightforward – fixed-income caseStraightforward – fixed-income case
Set out optimisation problemSet out optimisation problem
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0 1 2
Ex 4.12(1) Indifference curves
x1
x2
Could have x2 = 0
Could have x2 = 0
Slope is vertical here
Slope is vertical here
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Ex 4.12(1) Budget constraint, FOC
Budget constraint:Budget constraint: Substitute this into the utility Substitute this into the utility
function:function: We get the objective function:We get the objective function:
FOC for an interior solution:FOC for an interior solution:
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Ex 4.12(1) Using the FOC
Remember that person might consume zero of commodity 2Remember that person might consume zero of commodity 2 consider two cases consider two cases
Case 1: Case 1: xx22** > 0 > 0
From the FOC: From the FOC:
But, to make sense this case requires:But, to make sense this case requires:
Case 2: Case 2: xx22** = 0 = 0
We get We get xx11** from the budget constraint from the budget constraint
xx11** = = yy / / pp11
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Ex 4.12(1) Demand functions
We can summarise the optimal demands for We can summarise the optimal demands for the two goods thus the two goods thus
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Ex 4.12(1) Indirect utility function
Get maximised utility by substituting Get maximised utility by substituting xx** into the utility into the utility function function VV((pp11, , pp22, , yy) = ) = UU((xx11
**, , xx22**) )
= = UU((DD11((pp11, , pp22, , yy), ), DD22((pp11, , pp22, , yy))))
Case 1: Case 1: pp11 > >pp11
Case 2: Case 2: pp11 ≤ ≤pp11
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Ex 4.12(1) Cost function
Get cost function (expenditure function) from the indirect Get cost function (expenditure function) from the indirect utility functionutility function maximised utility is maximised utility is = = VV((pp11, , pp22, , yy))
invert this to get invert this to get yy = = CC((pp11, , pp22, , ))
Case 1: Case 1: pp11 > >pp11
Case 2: Case 2: pp11 ≤ ≤pp11
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Ex 4.12(2) Question
purposepurpose: to derive standard welfare concept: to derive standard welfare concept methodmethod: use part 1 and manipulate the indirect utility function : use part 1 and manipulate the indirect utility function
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Ex 4.12(2) Compute CV Get compensating variation (1) from indirect utility functionGet compensating variation (1) from indirect utility function
before price change: before price change: = = VV((p1, , p2, , y)) after price change: after price change: = = VV((p1', , p2, , y − CV))
Equivalently (2) could use cost function directlyEquivalently (2) could use cost function directly CV = = CC((p1, , p2, , )) − CC((p1', , p2, , ))
In Case 1 above we haveIn Case 1 above we have
Rearranging, we find:Rearranging, we find:
EquivalentlyEquivalently
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Ex 4.12(3)
In case 1 we have In case 1 we have xx11** = [ = [½½ pp2 2 / / pp11]]22
So demand for good 1 has zero income effectSo demand for good 1 has zero income effect Therefore, in this case CV = CS = EVTherefore, in this case CV = CS = EV
Now look at the follow-up
question
Now look at the follow-up
question
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Ex 4.13(2) Question
MethodMethod: : Find monopolist’s AR from consumer demand using Find monopolist’s AR from consumer demand using
answer to Ex 4.12. answer to Ex 4.12. Then use standard optimisation procedureThen use standard optimisation procedure
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Ex 4.13(2) Monopoly profits
Aggregate demand over Aggregate demand over NN consumers using Exercise 4.12consumers using Exercise 4.12
Rearrange to get AR curve:Rearrange to get AR curve:
Total Revenue is:Total Revenue is: Profits are therefore:Profits are therefore:
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Ex 4.13(2) Maximising profits
FOC (MC = MR) yields:FOC (MC = MR) yields:
So monopolist’s optimal output is:So monopolist’s optimal output is:
From AR curve, price at optimum is:From AR curve, price at optimum is:
Simplify this to:Simplify this to: (clearly price > MC)(clearly price > MC)
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Ex 4.13(3) Question
MethodMethod: : Aggregate the CV for each consumer to define Aggregate the CV for each consumer to define LL. . Use marginal cost and monopolist’s equilibrium price to evaluate Use marginal cost and monopolist’s equilibrium price to evaluate LL
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Ex 4.13(3) Evaluating loss Use definition of CV with
p1' = c:
Evaluate L at p1 = 2c:
Firm’s profits are:
Clearly L > profits
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Ex 4.13(4) Question
MethodMethod: : Add bonus Add bonus BB into the expression for profits into the expression for profits Again use standard optimisation procedureAgain use standard optimisation procedure
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Ex 4.13(4) Evaluating profits (again) Profits including bonus are:
Value of bonus is:
Use demand curve to express this in terms of q:
So profits can now be expressed as:
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Ex 4.13(4) Evaluating profits (again) Take the expression for profits including bonus FOC for a maximum is again MR = MC:
Rearranging we get the value of optimal output for the regulated monopolist:
Use demand curve to find:
Clearly the regulated price = MC:
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Ex 4.12 & 4.13: Points to note It’s always a good idea to sketch the indifference curvesIt’s always a good idea to sketch the indifference curves
in this case the sketch is revealing…in this case the sketch is revealing… ……because of the possible corner solutionbecause of the possible corner solution
A corner solution can sometimes just be handled as two A corner solution can sometimes just be handled as two separate casesseparate cases
There’s often more than one way of getting to a solutionThere’s often more than one way of getting to a solution in this case two equivalent derivations of CVin this case two equivalent derivations of CV
Aggregate welfare loss is found from individual CV Unregulated monopoly makes profits smaller than losses
to consumer Regulation causes monopoly to behave like competitive
firmNow for a
type 3 “model building” q
Now for a type 3 “model
building” q
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Ex 9.6(1): Question
purposepurpose: to derive equilibrium prices and incomes as a function of endowment. : to derive equilibrium prices and incomes as a function of endowment. To show the limits to redistribution within the GE model for a alternative SWFs To show the limits to redistribution within the GE model for a alternative SWFs
methodmethod: find price-taking optimising demands for each of the two types, use : find price-taking optimising demands for each of the two types, use these to compute the excess demand function and solve for these to compute the excess demand function and solve for
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Ex 9.6(1): budget constraints Use commodity 2 as numéraireUse commodity 2 as numéraire
price of good 1 is price of good 1 is price of good 2 is 1price of good 2 is 1
Evaluate incomes for the two types, given their resources:Evaluate incomes for the two types, given their resources: type type aa has endowment ( has endowment (30, 30, kk)) therefore therefore yyaa = 30= 30 + + kk type type bb has endowment ( has endowment (60, 210 60, 210 kk)) therefore therefore yybb = 60= 60 + [210 + [210 kk]]
Budget constraints for the two types are therefore:Budget constraints for the two types are therefore: xx11
aa + + xx22aa ≤ 30≤ 30 + + kk
xx11bb + + xx22
b b ≤ 60≤ 60 + [210 + [210 kk]]
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Ex 9.6(1): optimisation
We could jump straight to a solutionWe could jump straight to a solution utility functions are simple…utility functions are simple… ……so we can draw on known resultsso we can draw on known results
Cobb-Douglas preferences implyCobb-Douglas preferences imply indifference curves do not touch the origin…indifference curves do not touch the origin… ……so we need consider only interior solutionsso we need consider only interior solutions also demand functions for the two commodities exhibit constant expenditure shares also demand functions for the two commodities exhibit constant expenditure shares
In this case (for type In this case (for type aa) ) coefficients of Cobb-Douglas are 2 and 1coefficients of Cobb-Douglas are 2 and 1 so expenditure shares are so expenditure shares are ⅔ ⅔ and and ⅓⅓ (and for (and for bb they will be they will be ⅓ ⅓ and and ⅔⅔ ) ) gives the optimal demands immediately… gives the optimal demands immediately…
Jump to “equilibrium
price”
Jump to “equilibrium
price”
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Ex 9.6(1): optimisation, type a
The Lagrangean is:The Lagrangean is: 2log 2log xx11
aa + log + log xx22a a + + aa[[yyaa xx11
aa xx22aa ] ]
where where aa is the Lagrange multiplieris the Lagrange multiplier and and yyaa is 30 is 30+ + k k
FOC for an interior solutionFOC for an interior solution 2/2/xx11
aa aa = 0 = 0
1/1/xx22aa aa = 0 = 0
yyaa xx11aa xx22
aa= 0= 0
Eliminating Eliminating aa from these three equations, demands are from these three equations, demands are xx11
aa = = ⅔ ⅔ yya a / / xx22
aa = = ⅓ ⅓ yya a
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Ex 9.6(1): optimisation, type b The Lagrangean is:The Lagrangean is:
log log xx11bb + 2log + 2log xx22
b b + + bb[[yybb xx11bb xx22
bb ] ] where where bb is the Lagrange multiplieris the Lagrange multiplier and and yybb is 60 is 60 + 210 + 210 kk
FOC for an interior solutionFOC for an interior solution 1/1/xx11
bb bb = 0 = 0
2/2/xx22bb bb = 0 = 0
yybb xx11bb xx22
bb= 0= 0
Eliminating Eliminating bb from these three equations, demands are from these three equations, demands are xx11
bb = = ⅓ ⅓ yyb b / / xx22
bb = = ⅔ ⅔yybb
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Ex 9.6(1): equilibrium price Take demand equations for the two typesTake demand equations for the two types
substitute in the values for incomesubstitute in the values for income type-type-a a demand becomes demand becomes
type-type-b b demand becomes demand becomes
Excess demand for commodity 2:Excess demand for commodity 2: [10[10 ++ ⅓⅓kk]+[40]+[40+140+140 −− ⅔ ⅔kk]] −− 210210 which simplifies to 5which simplifies to 500 −− ⅓ ⅓kk −− 7700
Set excess demand to 0 for equilibrium:Set excess demand to 0 for equilibrium: equilibrium price must be:equilibrium price must be: = [= [212100 + + kk] / 15] / 1500
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Ex 9.6(2): Question and solution
Incomes for the two types are resources:Incomes for the two types are resources: yyaa = 30= 30 + + kk yybb = 60= 60 + [210 + [210 kk]]
The The equilibrium price is:equilibrium price is: = [= [212100 + + kk] / 15] / 1500
So we can solve for incomes as:So we can solve for incomes as: yyaa = = [[212100 + 6 + 6kk] / 5] / 5 yybb = = [1[1474700 3 3kk] / 5 ] / 5
Equivalently we can write Equivalently we can write yyaa and and yybb in terms of in terms of as as yyaa = = 180180 210 210 yybb = = 420 420 90 90
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Ex 9.6(3): Question
purposepurpose: to use the outcome of the GE model to plot the “income-possibility” set: to use the outcome of the GE model to plot the “income-possibility” set methodmethod: plot incomes corresponding to extremes of allocating commodity 2, : plot incomes corresponding to extremes of allocating commodity 2,
namely namely k k = 0 and = 0 and k k = 210. Then fill in the gaps. = 210. Then fill in the gaps.
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Income possibility set
ya
yb
0
200
300
100 200 300
100
• (42, 294)
• (294, 168)
incomes for k = 0incomes for k = 210
incomes for intermediate values of k
attainable set if income can be thrown away
yb = 315 ½ya
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Ex 9.6(4): Question
purposepurpose: find a welfare optimum subject to the “income-possibility” set: find a welfare optimum subject to the “income-possibility” set methodmethod: plot contours for the function : plot contours for the function WW on the previous diagram. on the previous diagram.
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Welfare optimum: first case
ya
yb
0
200
300
100 200 300
100
income possibility setContours of W = log ya + log yb
Maximisation of W over income-possibility set
• W is maximised at
corner
incomes are (294, 168)
here k = 210
so optimum is where all of resource 2 is allocated to type a
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Ex 9.6(5): Question
purposepurpose: as in part 4: as in part 4 methodmethod: as in part 4: as in part 4
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Welfare optimum: second case
ya
yb
0
200
300
100 200 300
100
income possibility setContours of W = ya + yb
Maximisation of W over income-possibility set
• again W is maximised
at corner
…where k = 210
so optimum is where all of resource 2 is allocated to type a
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Ex 9.6: Points to note
Applying GE methods gives the feasible set Limits to redistribution
natural bounds on k asymmetric attainable set
Must take account of corners Get the same W-maximising solution
where society is averse to inequality where society is indifferent to inequality
Link to exam very similar to 2007 q5. but also check out 2005 q4… …same model, but you are asked to different tricks with it
More practice with GE – see CfD 7.8