#### Transcript Extensive Form - London School of Economics

Frank Cowell: Microeconomics Revision Lecture EC202 http://darp.lse.ac.uk/ec202 30th April 2009 Frank Cowell Overview... Revision lecture Frank Cowell: Microeconomics Styles of question How to see what you need to do Doing short questions Planning answers Doing long questions Objectives of the lecture Frank Cowell: Microeconomics A look back at Term 1 Exam preparation Reference materials used (1) Exam papers (and outline answers) 2005 1(a) 2006 1(a) 2006 1(d) 2007 1(d) 2008 5 Reference materials used (2) CfD presentations 3.3, 4.12, 7.8 All related to past exam questions The exam paper Frank Cowell: Microeconomics Scope of exam material Structure and format of paper what’s covered in the lectures… … is definitive for the exam follows that of last four years check out the rubric from, last year’s paper 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 Question style – three types Frank Cowell: Microeconomics 1 Principles 2 Model solving a standard framework you just turn the wheels 3 Model building reason on standard results and arguments can use verbal and/or mathematical reasoning usually get guidance in the question longer question sometimes easier? Examples from past question 1 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 Overview... Revision lecture Frank Cowell: Microeconomics Styles of question How to tackle the main types of question Doing short questions Planning answers Doing long questions 2006 1(d) Frank Cowell: Microeconomics Straightforward “principles” question Just say what you need to say 2005 1(a) Frank Cowell: Microeconomics Straight “principles” Note the contrast between firm and consumer Be sure to give your reasons 2006 1(a) Frank Cowell: Microeconomics Principles again But format of question gives you a hint… …write out decomposition formula Then read off results 2007 1(d) Frank Cowell: Microeconomics (i) (ii) o o o o (iii) o o o Principles and modelsolving Jot down the principle (as note for self) Write in the elements of the model Principle: vNM is of the form p0u(y0)+p1u(y1) (or transform of this) Model: p0 is a/[a+b]; p1 is b/[a+b]: just need g=d to get the above form Principle: c-e income x implicitly defined by u(x) = p0u(y0)+p1u(y1) Model: plug y values into −x−g = −a y0−g −b y1−g Principle: risk premium defined by Ey − x Model: plug a,b,g,d values into answer to part (ii) [Check the online solutions for full detail on this one] 2008 5 Frank Cowell: Microeconomics (a) Straight principles can come up in long questions Don’t ignore them in a rush to get to the model! Could be some easy marks… State concept of PE In absence of externalities PE requires MRS=MRT But monopolist forces p>MC Four Marks (Yay!!) (b) See CfD 4.12 (below) (c) See on-line answers Overview... Revision lecture Frank Cowell: Microeconomics Styles of question How to do well in exams Doing short questions Planning answers Doing long questions Planning Answers Frank Cowell: Microeconomics What’s the point? See the big picture take a moment or two.. …make notes to yourself what is the main point of the question? and the subpoints? 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 don’t solve things twice! reuse results answer the right number of questions!!! Frank Cowell: Microeconomics Tips Follow the leads Pix help you to see the solution help you to explain your solution to examiner What should the answer be? examiners may be on your side! so if you’re pointed in the right direction, follow it… take a moment before each part of the question check the “shape” of the problem use your intuition 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 Illustrates two types of question taken from exercises in the book but of “exam type” difficulty covered in CfD Ex 3.3, 4.12 straight model solving Ex 7.8 incorporates a little bit of model building Look out for tips In all three questions, use pictures to clarify solution following hints in 3.3 [The “Explain carefully…” bits] Overview... Revision lecture Frank Cowell: Microeconomics Styles of question A problem with discontinuous supply… Doing short questions Planning answers Doing long questions •CfD 3.3 •CfD 4.12 •CfD 7.8 Ex 3.3(1) Question Frank Cowell: Microeconomics purpose: to derive competitive supply function method: derive AC, MC Ex 3.3(1) Costs Frank Cowell: Microeconomics Total cost is: F0 + ½ aqi2 Marginal cost: aqi Average cost: F0/qi + ½ aqi 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 Ex 3.3(1) Supply Frank Cowell: Microeconomics If p > p the firm supplies an amount of output such that 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 p = MC in either case firm makes zero profits To summarise the supply curve consists of : Ex 3.3(1): Supply by a single firm Frank Cowell: Microeconomics Average cost p Marginal cost Supply of output q qi Ex 3.3(2) Question Frank Cowell: Microeconomics purpose: to demonstrate possible absence of equilibrium method: examine discontinuity in supply relationship Ex 3.3(2): Equilibrium? Frank Cowell: Microeconomics AC,MC and supply of firm p Demand, low value of b Demand, med value of b Demand, high value of b Solution for high value of b is where Supply = Demand AC MC qi Ex 3.3(2) Equilibrium Frank Cowell: Microeconomics Outcome for supply by a single price-taking firm High demand: unique equilibrium on upper part of supply curve 2. Low demand: equilibrium with zero output 3. In between: no equilibrium 1. Given case 1 “Supply = Demand” implies This implies: But for case 1 we need p ≥ p from the above this implies Ex 3.3(3) Question Frank Cowell: Microeconomics purpose: to demonstrate effect of averaging method: appeal to a continuity argument Ex 3.3(3) Average supply, N firms Frank Cowell: Microeconomics Define average output Set of possible values for average output: Therefore the average supply function is Ex 3.3(3) Average supply, limit case Frank Cowell: Microeconomics 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 Ex 3.3(3): Average supply by N firms Frank Cowell: Microeconomics Average cost (for each firm) Marginal cost (for each firm) p Supply of output for averaged firms q q Ex 3.3(4) Question Frank Cowell: Microeconomics purpose: to find equilibrium in large-numbers case method: re-examine small-numbers case Ex 3.3(4) Equilibrium Frank Cowell: Microeconomics Equilibrium depends on where demand curve is located High demand characterise in terms of (price, average output) equilibrium is at (p, p/a) where p = aA / [a+b] Medium demand equilibrium is at (p, [A – p]/b) equivalent to (p, bq) where b := a[A – p] / [bp] Achieve this with a proportion b at q and 1–b at 0 Low demand equilibrium is at (p, 0) Ex 3.3(4): Eqm (medium demand) Frank Cowell: Microeconomics AC and MC (for each firm) Supply of output (averaged) Demand p Equilibrium Equilibrium achieved by mixing firms at 0 and at q b here 1b here q* q q Ex 3.4: Points to remember Frank Cowell: Microeconomics Model discontinuity carefully 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 depends on the right mixture Overview... Revision lecture Frank Cowell: Microeconomics Styles of question Close to 2008 5 Doing short questions Planning answers Doing long questions •CfD 3.3 •CfD 4.12 •CfD 7.8 Ex 4.12(1) Question Frank Cowell: Microeconomics purpose: to derive solution and response functions for quasilinear preferences method: substitution of budget constraint into utility function and then simple maximisation Ex 4.12(1) Preliminary Frank Cowell: Microeconomics First steps are as follows: Sketch indifference curves Write down budget constraint Straightforward – parabolic contours Straightforward – fixed-income case Set out optimisation problem Ex 4.12(1) Indifference curves Frank Cowell: Microeconomics x2 Slope is vertical here Could have x2 = 0 x1 0 0 1 2 Ex 4.12(1) Budget constraint, FOC Frank Cowell: Microeconomics Budget constraint: Substitute this into the utility function: We get the objective function: FOC for an interior solution: Ex 4.12(1) Using the FOC Frank Cowell: Microeconomics Remember that person might consume zero of commodity 2 consider two cases Case 1: x2* > 0 From the FOC: But, to make sense this case requires: Case 2: x2* = 0 We get x1* from the budget constraint x1* = y / p1 Ex 4.12(1) Demand functions Frank Cowell: Microeconomics We can summarise the optimal demands for the two goods thus Ex 4.12(1) Indirect utility function Frank Cowell: Microeconomics Get maximised utility by substituting x* into the utility function V(p1, p2, y) = U(x1*, x2*) = U(D1(p1, p2, y), D2(p1, p2, y)) Case 1: p1 >`p1 Case 2: p1 ≤`p1 Ex 4.12(1) Cost function Frank Cowell: Microeconomics Get cost function (expenditure function) from the indirect utility function maximised utility is u = V(p1, p2, y) invert this to get y = C(p1, p2, u) Case 1: p1 >`p1 Case 2: p1 ≤`p1 Ex 4.12(2) Question Frank Cowell: Microeconomics purpose: to derive standard welfare concept method: use part 1 and manipulate the indirect utility function Ex 4.12(2) Compute CV Frank Cowell: Microeconomics Get compensating variation (1) from indirect utility function before price change: u = V(p1, p2, y) after price change: u = V(p1', p2, y − CV) Equivalently (2) could use cost function directly CV = C(p1, p2, u) − C(p1', p2, u) In Case 1 above we have Rearranging, we find: Equivalently Ex 4.12(3) Frank Cowell: Microeconomics In case 1 we have x1* = [½ a p2 / p1]2 So demand for good 1 has zero income effect Therefore, in this case CV = CS = EV Ex 4.12: Points to remember Frank Cowell: Microeconomics 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 There’s often more than one way of getting to a solution in this case two equivalent derivations of CV Overview... Revision lecture Frank Cowell: Microeconomics Styles of question A standard GE exercise… with a little bit of a twist Doing short questions Planning answers Doing long questions •CfD 3.3 •CfD 4.12 •CfD 7.8 Ex 7.8: Question Frank Cowell: Microeconomics purpose: to show how to find equilibrium allocation in a GE model method: standard construction and solution of excess demand functions. Ex 7.8: approach Frank Cowell: Microeconomics Step 1: model behaviour of each type as a price taker Step 2: get excess demand function for one of the goods Write down budget constraint for the unknown price p Set up Lagrangean for each type Find the FOCs Get demand functions from the FOCs 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 Ex 7.8: type-a problem Frank Cowell: Microeconomics The endowment for type a is R1 Let price of good 1 in terms of good 2 be p The income of type a is then pR1 The utility function is: So the Lagrangean of type a is: Ex 7.8: type-a demand Frank Cowell: Microeconomics Given the Lagrangean for a: FOCs for interior maximum: Rearrange and use the budget constraint: Demand by a for good 2: Ex 7.8: type-b problem Frank Cowell: Microeconomics The endowment for type b is R2 Recall that values are measured in terms of good 2 So the income of type b is just R2 The utility function is: So the Lagrangean of type b is: Ex 7.8: type-b demand Frank Cowell: Microeconomics Given the Lagrangean for b: FOCs for interior maximum: Rearrange and use the budget constraint: Demand by b for good 2: Ex 7.8: excess demand Frank Cowell: Microeconomics Demand by the two types for good 2: Excess demand for good 2 is defined as x21 + x22 R2 So the excess demand function for good 2 is: Letting q:= 2R1/R2 excess demand is zero where Ex 7.8: how many equilibria? Frank Cowell: Microeconomics Graph of p2/3 pq 1 p2/3 Graph of pq 1 Equilibrium Excess demand is zero where p2/3 = pq 1 p* p There is clearly only one equilibrium p* Ex 7.8: the equilibrium Frank Cowell: Microeconomics To find the equilibrium we need the resource values Equilibrium price must satisfy p2/3 = (5/8) p 1 Use trial and error to find solution R1 = 5 R2 = 16 So q := 2R1/R2 = 5/8 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: p2/3 = 4 RHS: (5/8) p 1 = 4 so this is the equilibrium Ex 7.8: Points to note Frank Cowell: Microeconomics 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