Frank Cow

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owell: M

icroeconomics

Microeconom

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Revision Lecture

EC202: Microeconomic Principles IIEC202: Microeconomic Principles IIFrank Cowell Frank Cowell

April 2007 April 2007

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Objectives of the lecture A look back at Term 1A look back at Term 1 Exam preparationExam preparation Reference materials used (1)Reference materials used (1)

Exam papers (and outline answers)Exam papers (and outline answers) 2003 1(c)2003 1(c) 2004 1(c)2004 1(c) 2005 1(a)2005 1(a) 2006 1(a)2006 1(a)

Reference materials used (2) Reference materials used (2) CfD presentations 2.9 9.6CfD presentations 2.9 9.6 Both related to past exam questionsBoth related to past exam questions

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Principles Scope of exam materialScope of exam material

what’s covered in the lectures…what’s covered in the lectures… … … is definitive for the examis definitive for the exam

ResitResit a separate paper for anyone doing this second time arounda separate paper for anyone doing this second time around

Structure and format of paperStructure and format of paper follows that of last two yearsfollows that of last two years check out the rubric from, say 2005 papercheck out the rubric from, say 2005 paper

Mark schemeMark scheme 40 marks for question 1 (8 marks for each of the five parts)40 marks for question 1 (8 marks for each of the five parts) 20 marks for each of the other three questions20 marks for each of the other three questions multipart questions: except where it’s obvious, roughly equal multipart questions: except where it’s obvious, roughly equal

marks across parts marks across parts

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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 part A (question 1) usually gets you to do both types 1 and 2part A (question 1) usually gets you to do both types 1 and 2 type 3 usually only in parts B and C of papertype 3 usually only in parts B and C of paper

Examples from past question 1

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2004 1(c) Straightforward Straightforward

“principles” question“principles” question Just say what you Just say what you

need to sayneed to say

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2005 1(a) Straight “principles”Straight “principles” Note the contrast Note the contrast

between firm and between firm and consumerconsumer

Be sure to Be sure to give your give your reasonsreasons

Frank Cow

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2006 1(a) Principles againPrinciples again But format of question But format of question

gives you a hint…gives you a hint… ……write out write out

decomposition formuladecomposition formula Then read off resultsThen read off results

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2003 1(c) A model-solving A model-solving

questionquestion (i) just set (i) just set EE(() = 0 ) = 0

and twiddleand twiddle (ii) check what (ii) check what

happens to happens to EE if you if you change change

(iii) draw diagram (iii) draw diagram and reasonand reason

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Planning Answers What’s the point?What’s the point?

take a moment or two..take a moment or two.. ……make notes to yourself make notes to yourself what is the what is the main pointmain point of the question? of the question? and the subpoints?and the subpoints?

See the big pictureSee the big picture balance out the answerbalance out the answer imagine that you’re drawing a pictureimagine that you’re drawing a picture if pressed for time, don’t rush to put in extra detail…if pressed for time, don’t rush to put in extra detail… ……you can go backyou can go back

Be an economist with your own timeBe an economist with your own time don’t solve things twice!don’t solve things twice! reuse resultsreuse results answer the right number of questions!!!answer the right number of questions!!!

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Tips Follow the leadsFollow the leads

examiners may be on your side!examiners may be on your side! so if you’re pointed in the right direction, follow it…so if you’re pointed in the right direction, follow it…

PixPix help you to see the solutionhelp you to see the solution help you to explain your solution to examinerhelp you to explain your solution to examiner

What What shouldshould the answer be? the answer be? take a moment before each part of the questiontake a moment before each part of the question check the “shape” of the problemcheck the “shape” of the problem use your intuitionuse your intuition

Does it make sense?Does it make sense? again take a moment to check after each partagain take a moment to check after each part we we allall make silly slips make silly slips

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Long questions Let’s look at two examples Let’s look at two examples

taken from exercises in the booktaken from exercises in the book but of “exam type” difficultybut of “exam type” difficulty covered in CfDcovered in CfD

Illustrates two types of questionIllustrates two types of question Ex 2.9 is mainly model solvingEx 2.9 is mainly model solving Ex 9.6 incorporates model buildingEx 9.6 incorporates model building

Look out for tipsLook out for tips Use of pictures in both questionsUse of pictures in both questions following hints in 9.6following hints in 9.6

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Ex 2.9(1): Question

purposepurpose: demonstrate relationship between short and long run: demonstrate relationship between short and long run methodmethod: Lagrangean approach to cost minimisation. First part can be : Lagrangean approach to cost minimisation. First part can be

solved by a “trick”solved by a “trick”

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Ex 2.9(1): Long-run costs Production function is homogeneous of degree 1Production function is homogeneous of degree 1

increase all inputs by a factor increase all inputs by a factor tt > 0 (i.e. > 0 (i.e. zz →→ ttzz)…)… ……and output increases by the same factor (i.e. and output increases by the same factor (i.e. qq →→ tqtq)) constant returns to scale in the long runconstant returns to scale in the long run

CRTS implies constant average costCRTS implies constant average cost CC((ww, , qq) / ) / qq = = AA (a constant) (a constant) so so CC((ww, , qq) = ) = AqAq differentiating: differentiating: CCqq((ww, , qq) = ) = AA

So LRMC = LRAC = constantSo LRMC = LRAC = constant Their graphs will be an identical straight lineTheir graphs will be an identical straight line

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Ex 2.9(2): Question

methodmethod: : Standard Lagrangean approachStandard Lagrangean approach

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Ex 2.9(2): short-run Lagrangean In the short run amount of good 3 is fixedIn the short run amount of good 3 is fixed

zz33 = = zz33 Could write the Lagrangean asCould write the Lagrangean as

But it is more convenient to transform the problem thusBut it is more convenient to transform the problem thus

wherewhere

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z2

z1

Ex 2.9(2): Isoquants Sketch the isoquant mapSketch the isoquant map

Isoquants do not touch the axesIsoquants do not touch the axes So maximum problem must have an interior solutionSo maximum problem must have an interior solution

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Ex 2.9(2): short-run FOCs Differentiating Lagrangean, the FOCS areDifferentiating Lagrangean, the FOCS are

This impliesThis implies

To find conditional demand function must solve for To find conditional demand function must solve for use the above equations… use the above equations… ……and the production functionand the production function

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Ex 2.9(2): short-run FOCs (more) Using FOCs and the production function:Using FOCs and the production function:

This impliesThis implies

wherewhere

This will give us the short-run cost functionThis will give us the short-run cost function

Frank Cow

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Ex 2.9(2): short-run costs By definition, short-By definition, short-

run costs are:run costs are:

This becomesThis becomes

Substituting for Substituting for kk::

From this we getFrom this we get

SRAC:SRAC:

SRMC:SRMC:

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q

Ex 2.9(2): short-run MC and AC

marginalcost

averagecost

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Ex 2.9(3): Question

methodmethod: : Draw the standard supply-curve diagramDraw the standard supply-curve diagram Manipulate the relationship Manipulate the relationship pp = MC = MC

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Ex 2.9(3): short-run supply curveaverage cost curvemarginal cost curve

supply curve

q

p

q

p

minimum average cost

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Use the expression for marginal cost:Use the expression for marginal cost: Set Set p p = MC for = MC for pp ≥≥ pp Rearrange to get supply curveRearrange to get supply curve

Differentiate last line to get supply elasticityDifferentiate last line to get supply elasticity

Ex 2.9(3): short-run supply elasticity

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Ex 2.9: Points to remember Exploit CRTS to give you easy resultsExploit CRTS to give you easy results Try transforming the Lagrangean to make it easier Try transforming the Lagrangean to make it easier

to manipulateto manipulate Use MC curve to derive supply curveUse MC curve to derive supply curve

Frank Cow

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icroeconomics

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Ex 9.6(1): Question

purposepurpose: to derive equilibrium prices and incomes as a function of : to derive equilibrium prices and incomes as a function of endowment. To show the limits to redistribution within the GE model for a endowment. To show the limits to redistribution within the GE model for a alternative SWFs 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 ≤ ≤ 3030 + + kk

xx11bb + + xx22

b b ≤ ≤ 6060 + [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 also demand functions for the two commodities exhibit constant

expenditure shares 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”

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Ex 9.6(1): optimisation, type a The Lagrangean is:The Lagrangean is:

2log 2log xx11aa + log + log xx22

a a + + aa[[yyaa xx11aa xx22

aa ] ] 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/xx22

aa aa = 0 = 0 yyaa xx11

aa xx22aa= 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/xx22

bb bb = 0 = 0 yybb xx11

bb xx22bb= 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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icroeconomics

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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

Frank Cow

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Ex 9.6(3): Question

purposepurpose: to use the outcome of the GE model to plot the “income-: to use the outcome of the GE model to plot the “income-possibility” setpossibility” set

methodmethod: plot incomes corresponding to extremes of allocating : plot incomes corresponding to extremes of allocating commodity 2, namely commodity 2, 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 = 210incomes 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