Frank Cowell: Microeconomics Revision Lecture EC th April 2009 Frank Cowell

Frank Cowell: Microeconomics Overview... Styles of question Doing short questions Planning answers Doing long questions Revision lecture How to see what you need to do

Frank Cowell: Microeconomics Objectives of the lecture A look back at Term 1 A look back at Term 1 Exam preparation Exam preparation Reference materials used (1) Reference materials used (1)  Exam papers (and outline answers)  (a)  (a)  (d)  (d)  Reference materials used (2) Reference materials used (2)  CfD presentations 3.3, 4.12, 7.8  All related to past exam questions

Frank Cowell: Microeconomics The exam paper Scope of exam material Scope of exam material  what’s covered in the lectures…  … is definitive for the exam Structure and format of paper Structure and format of paper  follows that of last four years  check out the rubric from, last year’s paper Mark scheme Mark scheme  40 marks for question 1 (8 marks for each of the five parts)  20 marks for each of the other three questions  multipart questions: except where it’s obvious, roughly equal marks across parts

Frank Cowell: Microeconomics Question style – three types 1 Principles 1 Principles  reason on standard results and arguments  can use verbal and/or mathematical reasoning 2 Model solving 2 Model solving  a standard framework  you just turn the wheels 3 Model building 3 Model building  usually get guidance in the question  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 2  type 3 is usually only in parts B and C of paper Examples from past question 1

Frank Cowell: Microeconomics Overview... Styles of question Doing short questions Planning answers Doing long questions Revision lecture How to tackle the main types of question

Frank Cowell: Microeconomics (d) Straightforward “principles” question Straightforward “principles” question Just say what you need to say Just say what you need to say

Frank Cowell: Microeconomics (a) Straight “principles” Straight “principles” Note the contrast between firm and consumer Note the contrast between firm and consumer Be sure to give your reasons Be sure to give your reasons

Frank Cowell: Microeconomics (a) Principles again Principles again But format of question gives you a hint… But format of question gives you a hint… …write out decomposition formula …write out decomposition formula Then read off results Then read off results

Frank Cowell: Microeconomics (d) Principles and model- solving Principles and model- solving Jot down the principle (as note for self) Jot down the principle (as note for self) Write in the elements of the model Write in the elements of the model o Principle: vNM is of the form  0 u(y 0 )+  1 u(y 1 ) (or transform of this) o Model:  0 is  ;  1 is  : just need  =  to get the above form o Principle: risk premium defined by Ey −  o Model: plug  values into answer to part (ii) o [Check the online solutions for full detail on this one] (i) (iii) o Principle: c-e income  implicitly defined by u(  ) =  0 u(y 0 )+  1 u(y 1 ) o Model: plug y values into −  −  = −  y 0 −  −  y 1 −  (ii)

Frank Cowell: Microeconomics Straight principles can come up in long questions Straight principles can come up in long questions Don’t ignore them in a rush to get to the model! Don’t ignore them in a rush to get to the model! Could be some easy marks… Could be some easy marks… (a) (b) (c) State concept of PE State concept of PE In absence of externalities PE requires MRS=MRT In absence of externalities PE requires MRS=MRT But monopolist forces p>MC But monopolist forces p>MC  Four Marks (Yay!!) See CfD 4.12 (below) See CfD 4.12 (below) See on-line answers See on-line answers

Frank Cowell: Microeconomics Overview... Styles of question Doing short questions Planning answers Doing long questions Revision lecture How to do well in exams

Frank Cowell: Microeconomics Planning Answers What’s the point? What’s the point?  take a moment or two..  …make notes to yourself  what is the main point of the question?  and the subpoints? See the big picture See the big picture  balance out the answer  imagine that you’re drawing a picture  if pressed for time, don’t rush to put in extra detail…  …you can go back Be an economist with your own time Be an economist with your own time  don’t solve things twice!  reuse results  answer the right number of questions!!!

Frank Cowell: Microeconomics Tips Follow the leads Follow the leads  examiners may be on your side!  so if you’re pointed in the right direction, follow it… Pix Pix  help you to see the solution  help you to explain your solution to examiner What should the answer be? What should the answer be?  take a moment before each part of the question  check the “shape” of the problem  use your intuition Does it make sense? Does it make sense?  again take a moment to check after each part  we all make silly slips

Frank Cowell: Microeconomics Long questions Let’s look at three examples Let’s look at three examples  taken from exercises in the book  but of “exam type” difficulty  covered in CfD Illustrates two types of question Illustrates two types of question  Ex 3.3, 4.12 straight model solving  Ex 7.8 incorporates a little bit of model building Look out for tips Look out for tips  In all three questions, use pictures to clarify solution  following hints in 3.3 [The “Explain carefully…” bits]

Frank Cowell: Microeconomics Overview... Styles of question Doing short questions Planning answers Doing long questions Revision lecture A problem with discontinuous supply… CfD 3.3 CfD 4.12 CfD 7.8

Frank Cowell: Microeconomics Ex 3.3(1) Question purpose: to derive competitive supply function purpose: to derive competitive supply function method: derive AC, MC method: derive AC, MC

Frank Cowell: Microeconomics Ex 3.3(1) Costs Total cost is: F 0 + ½ aq i 2 Marginal cost: aq i Average cost: F 0 /q i + ½ aq i Therefore MC intersects AC where: This is at output level q where: At this point AC is at a minimum p where: For q below q there is IRTS and vice versa

Frank Cowell: Microeconomics Ex 3.3(1) Supply If p > p the firm supplies an amount of output such that   p = MC If p < p the firm supplies zero output   otherwise the firm would make a loss If p = p the firm is indifferent between supplying 0 or q   in either case firm makes zero profits To summarise the supply curve consists of :

Frank Cowell: Microeconomics Ex 3.3(1): Supply by a single firm qiqi p   Average cost   Marginal cost   Supply of output q

Frank Cowell: Microeconomics Ex 3.3(2) Question purpose: to demonstrate possible absence of equilibrium purpose: to demonstrate possible absence of equilibrium method: examine discontinuity in supply relationship method: examine discontinuity in supply relationship

Frank Cowell: Microeconomics Ex 3.3(2): Equilibrium? AC MC  qiqi p   AC,MC and supply of firm   Demand, low value of b   Demand, high value of b Supply (one firm)   Solution for high value of b is where Supply = Demand   Demand, med value of b

Frank Cowell: Microeconomics Ex 3.3(2) Equilibrium Outcome for supply by a single price-taking firm High demand: unique equilibrium on upper part of supply curve Low demand: equilibrium with zero output In between: no equilibrium Given case 1 “Supply = Demand” implies This implies: But for case 1 we need p ≥ p   from the above this implies

Frank Cowell: Microeconomics Ex 3.3(3) Question purpose: to demonstrate effect of averaging purpose: to demonstrate effect of averaging method: appeal to a continuity argument method: appeal to a continuity argument

Frank Cowell: Microeconomics Ex 3.3(3) Average supply, N firms Define average output Set of possible values for average output: Therefore the average supply function is

Frank Cowell: Microeconomics Ex 3.3(3) Average supply, limit case As N  the set J(q) becomes dense in [0, q] So, in the limit, if p = p average output can take any value in [0, q] Therefore the average supply function is

Frank Cowell: Microeconomics Ex 3.3(3): Average supply by N firms p   Average cost (for each firm)   Marginal cost (for each firm)   Supply of output for averaged firms q q 

Frank Cowell: Microeconomics Ex 3.3(4) Question purpose: to find equilibrium in large-numbers case purpose: to find equilibrium in large-numbers case method: re-examine small-numbers case method: re-examine small-numbers case

Frank Cowell: Microeconomics Ex 3.3(4) Equilibrium Equilibrium depends on where demand curve is located   characterise in terms of (price, average output) High demand   equilibrium is at (p, p/a) where p = aA / [a+b] Medium demand   equilibrium is at (p, [A – p]/b)   equivalent to (p,  q) where  := a[A – p] / [bp]   Achieve this with a proportion  at q and 1–  at 0 Low demand   equilibrium is at (p, 0)

Frank Cowell: Microeconomics Ex 3.3(4): Eqm (medium demand) p   AC and MC (for each firm)   Supply of output (averaged) q   Demand   Equilibrium q*   Equilibrium achieved by mixing firms at 0 and at q  here 1  here q 

Frank Cowell: Microeconomics Ex 3.4: Points to remember Model discontinuity carefully Model discontinuity carefully Averaging may eliminate discontinuity problem in a large economy Averaging may eliminate discontinuity problem in a large economy  depends whether individual agents are small. Equilibrium in averaged model may involve identical firms doing different things Equilibrium in averaged model may involve identical firms doing different things  equilibrium depends on the right mixture

Frank Cowell: Microeconomics Overview... Styles of question Doing short questions Planning answers Doing long questions Revision lecture Close to CfD 3.3 CfD 4.12 CfD 7.8

Frank Cowell: Microeconomics Ex 4.12(1) Question purpose: to derive solution and response functions for quasilinear preferences purpose: to derive solution and response functions for quasilinear preferences method: substitution of budget constraint into utility function and then simple maximisation method: substitution of budget constraint into utility function and then simple maximisation

Frank Cowell: Microeconomics Ex 4.12(1) Preliminary First steps are as follows: First steps are as follows: Sketch indifference curves Sketch indifference curves  Straightforward – parabolic contours Write down budget constraint Write down budget constraint  Straightforward – fixed-income case Set out optimisation problem Set out optimisation problem

Frank Cowell: Microeconomics Ex 4.12(1) Indifference curves x1x1 x2x2 Could have x 2 = 0 Slope is vertical here

Frank Cowell: Microeconomics Ex 4.12(1) Budget constraint, FOC Budget constraint: Budget constraint: Substitute this into the utility function: Substitute this into the utility function: We get the objective function: We get the objective function: FOC for an interior solution: FOC for an interior solution:

Frank Cowell: Microeconomics Ex 4.12(1) Using the FOC Remember that person might consume zero of commodity 2 Remember that person might consume zero of commodity 2  consider two cases Case 1: x 2 * > 0 Case 1: x 2 * > 0 From the FOC: From the FOC: But, to make sense this case requires: But, to make sense this case requires: Case 2: x 2 * = 0 Case 2: x 2 * = 0 We get x 1 * from the budget constraint We get x 1 * from the budget constraint  x 1 * = y / p 1

Frank Cowell: Microeconomics Ex 4.12(1) Demand functions We can summarise the optimal demands for the two goods thus We can summarise the optimal demands for the two goods thus

Frank Cowell: Microeconomics Ex 4.12(1) Indirect utility function Get maximised utility by substituting x * into the utility function Get maximised utility by substituting x * into the utility function  V(p 1, p 2, y) = U(x 1 *, x 2 * )  = U(D 1 (p 1, p 2, y), D 2 (p 1, p 2, y)) Case 1: p 1 >  p 1 Case 1: p 1 >  p 1 Case 2: p 1 ≤  p 1 Case 2: p 1 ≤  p 1

Frank Cowell: Microeconomics Ex 4.12(1) Cost function Get cost function (expenditure function) from the indirect utility function Get cost function (expenditure function) from the indirect utility function  maximised utility is  = V(p 1, p 2, y)  invert this to get y = C(p 1, p 2,  ) Case 1: p 1 >  p 1 Case 1: p 1 >  p 1 Case 2: p 1 ≤  p 1 Case 2: p 1 ≤  p 1

Frank Cowell: Microeconomics Ex 4.12(2) Question purpose: to derive standard welfare concept purpose: to derive standard welfare concept method: use part 1 and manipulate the indirect utility function method: use part 1 and manipulate the indirect utility function

Frank Cowell: Microeconomics Ex 4.12(2) Compute CV Get compensating variation (1) from indirect utility function Get compensating variation (1) from indirect utility function  before price change:  = V(,, )  before price change:  = V(p 1, p 2, y)  after price change:  = V(,, )  after price change:  = V(p 1 ', p 2, y − CV) Equivalently (2) could use cost function directly Equivalently (2) could use cost function directly  = C(,,  )C(,,  )  CV = C(p 1, p 2,  ) − C(p 1 ', p 2,  ) In Case 1 above we have In Case 1 above we have Rearranging, we find: Rearranging, we find: Equivalently Equivalently

Frank Cowell: Microeconomics Ex 4.12(3) In case 1 we have x 1 * = [½  p 2 / p 1 ] 2 In case 1 we have x 1 * = [½  p 2 / p 1 ] 2 So demand for good 1 has zero income effect So demand for good 1 has zero income effect Therefore, in this case CV = CS = EV Therefore, in this case CV = CS = EV

Frank Cowell: Microeconomics Ex 4.12: Points to remember It’s always a good idea to sketch the indifference curves It’s always a good idea to sketch the indifference curves  in this case the sketch is revealing…  …because of the possible corner solution A corner solution can sometimes just be handled as two separate cases A corner solution can sometimes just be handled as two separate cases There’s often more than one way of getting to a solution There’s often more than one way of getting to a solution  in this case two equivalent derivations of CV

Frank Cowell: Microeconomics Overview... Styles of question Doing short questions Planning answers Doing long questions Revision lecture A standard GE exercise… with a little bit of a twist CfD 3.3 CfD 4.12 CfD 7.8

Frank Cowell: Microeconomics Ex 7.8: Question purpose: to show how to find equilibrium allocation in a GE model purpose: to show how to find equilibrium allocation in a GE model method: standard construction and solution of excess demand functions. method: standard construction and solution of excess demand functions.

Frank Cowell: Microeconomics Ex 7.8: approach Step 1: model behaviour of each type as a price taker   Write down budget constraint for the unknown price p   Set up Lagrangean for each type   Find the FOCs   Get demand functions from the FOCs Step 2: get excess demand function for one of the goods   Use the demand functions for each type from step 1   Other EDF follows by Walras’ law Step 3: find equilibrium price(s) as root(s) of EDF

Frank Cowell: Microeconomics Ex 7.8: type-a problem The endowment for type a is R 1 Let price of good 1 in terms of good 2 be p The income of type a is then pR 1 The utility function is: So the Lagrangean of type a is:

Frank Cowell: Microeconomics Ex 7.8: type-a demand Given the Lagrangean for a: FOCs for interior maximum: Rearrange and use the budget constraint: Demand by a for good 2:

Frank Cowell: Microeconomics Ex 7.8: type-b problem The endowment for type b is R 2 Recall that values are measured in terms of good 2 So the income of type b is just R 2 The utility function is: So the Lagrangean of type b is:

Frank Cowell: Microeconomics Ex 7.8: type-b demand Given the Lagrangean for b: FOCs for interior maximum: Rearrange and use the budget constraint: Demand by b for good 2:

Frank Cowell: Microeconomics Ex 7.8: excess demand Demand by the two types for good 2: Excess demand for good 2 is defined as x x 2 2  R 2 So the excess demand function for good 2 is: Letting  := 2R 1 /R 2 excess demand is zero where

Frank Cowell: Microeconomics Ex 7.8: how many equilibria?   Graph of p 2/3   Graph of p   1   Equilibrium p p 2/3 p  p*p*   Excess demand is zero where p 2/3 = p    There is clearly only one equilibrium p *

Frank Cowell: Microeconomics Ex 7.8: the equilibrium To find the equilibrium we need the resource values   R 1 = 5   R 2 = 16   So  := 2R 1 /R 2 = 5/8 Equilibrium price must satisfy p 2/3 =  5/8  p  Use trial and error to find solution   check whether there is excess demand/supply at certain prices   try easy numbers that have integer cube roots: p = 1? 8? 27? … Clearly p = 1 is too low and p = 27 is too high Try p = 8   LHS: p 2/3 = 4   RHS: (5/8) p  1 = 4   so this is the equilibrium

Frank Cowell: Microeconomics Ex 7.8: Points to note Step by step approach gets you very close to the solution   work out individual demands   set excess demand to zero   get a condition to determine equilibrium price Graphical intuition helps you get the form of the solution Don’t get fazed by awkward numbers   trial-and-error method quickly gives you the answer