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

Frank Cowell: Microeconomics April 2007 Revision Lecture EC202: Microeconomic Principles II Frank Cowell Objectives of the lecture Frank Cowell: Microeconomics A look back at Term 1 Exam preparation Reference materials used (1) Exam papers (and outline answers) 2003 1(c) 2004 1(c) 2005 1(a) 2006 1(a) Reference materials used (2) CfD presentations 2.9 9.6 Both related to past exam questions Principles Frank Cowell: Microeconomics Scope of exam material Resit a separate paper for anyone doing this second time around Structure and format of paper what’s covered in the lectures… … is definitive for the exam follows that of last two years check out the rubric from, say 2005 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 usually only in parts B and C of paper 2004 1(c) 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 2003 1(c) Frank Cowell: Microeconomics A model-solving question (i) just set E(r) = 0 and twiddle (ii) check what happens to E if you change r (iii) draw diagram and reason 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 two examples Illustrates two types of question taken from exercises in the book but of “exam type” difficulty covered in CfD Ex 2.9 is mainly model solving Ex 9.6 incorporates model building Look out for tips Use of pictures in both questions following hints in 9.6 Ex 2.9(1): Question Frank Cowell: Microeconomics purpose: demonstrate relationship between short and long run method: Lagrangean approach to cost minimisation. First part can be solved by a “trick” Ex 2.9(1): Long-run costs Frank Cowell: Microeconomics Production function is homogeneous of degree 1 CRTS implies constant average cost increase all inputs by a factor t > 0 (i.e. z → tz)… …and output increases by the same factor (i.e. q → tq) constant returns to scale in the long run C(w, q) / q = A (a constant) so C(w, q) = Aq differentiating: Cq(w, q) = A So LRMC = LRAC = constant Their graphs will be an identical straight line Ex 2.9(2): Question Frank Cowell: Microeconomics method: Standard Lagrangean approach Ex 2.9(2): short-run Lagrangean Frank Cowell: Microeconomics In the short run amount of good 3 is fixed z3 = `z3 Could write the Lagrangean as But it is more convenient to transform the problem thus where Ex 2.9(2): Isoquants Frank Cowell: Microeconomics Sketch the isoquant map z2 z1 Isoquants do not touch the axes So maximum problem must have an interior solution Ex 2.9(2): short-run FOCs Frank Cowell: Microeconomics Differentiating Lagrangean, the FOCS are This implies To find conditional demand function must solve for l use the above equations… …and the production function Ex 2.9(2): short-run FOCs (more) Frank Cowell: Microeconomics Using FOCs and the production function: This implies where This will give us the short-run cost function Ex 2.9(2): short-run costs Frank Cowell: Microeconomics By definition, shortrun costs are: This becomes Substituting for k: From this we get SRAC: SRMC: Ex 2.9(2): short-run MC and AC Frank Cowell: Microeconomics marginal cost average cost q Ex 2.9(3): Question Frank Cowell: Microeconomics method: Draw the standard supply-curve diagram Manipulate the relationship p = MC Ex 2.9(3): short-run supply curve Frank Cowell: Microeconomics average cost curve marginal cost curve minimum average cost p supply curve p q q Ex 2.9(3): short-run supply elasticity Frank Cowell: Microeconomics Use the expression for marginal cost: Set p = MC for p ≥ p Rearrange to get supply curve Differentiate last line to get supply elasticity Ex 2.9: Points to remember Frank Cowell: Microeconomics Exploit CRTS to give you easy results Try transforming the Lagrangean to make it easier to manipulate Use MC curve to derive supply curve Ex 9.6(1): Question Frank Cowell: Microeconomics purpose: 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 method: find price-taking optimising demands for each of the two types, use these to compute the excess demand function and solve for r Ex 9.6(1): budget constraints Frank Cowell: Microeconomics Use commodity 2 as numéraire Evaluate incomes for the two types, given their resources: price of good 1 is r price of good 2 is 1 type a has endowment (30, k) therefore ya = 30r + k type b has endowment (60, 210 k) therefore yb = 60r + [210 k] Budget constraints for the two types are therefore: rx1a + x2a ≤ 30r + k rx1b + x2b ≤ 60r + [210 k] Ex 9.6(1): optimisation Frank Cowell: Microeconomics Jump to “equilibrium price” We could jump straight to a solution Cobb-Douglas preferences imply utility functions are simple… …so we can draw on known results indifference curves do not touch the origin… …so we need consider only interior solutions also demand functions for the two commodities exhibit constant expenditure shares In this case (for type a) coefficients of Cobb-Douglas are 2 and 1 so expenditure shares are ⅔ and ⅓ (and for b they will be ⅓ and ⅔ ) gives the optimal demands immediately… Ex 9.6(1): optimisation, type a Frank Cowell: Microeconomics The Lagrangean is: FOC for an interior solution 2log x1a + log x2a + na[ya rx1a x2a ] where na is the Lagrange multiplier and ya is 30r + k 2/x1a nar = 0 1/x2a na = 0 ya rx1a x2a= 0 Eliminating na from these three equations, demands are x1a = ⅔ ya / r x2a = ⅓ ya Ex 9.6(1): optimisation, type b Frank Cowell: Microeconomics The Lagrangean is: FOC for an interior solution log x1b + 2log x2b + nb[yb rx1b x2b ] where nb is the Lagrange multiplier and yb is 60r + 210 k 1/x1b nbr = 0 2/x2b nb = 0 yb rx1b x2b= 0 Eliminating nb from these three equations, demands are x1b = ⅓ yb / r x2b = ⅔yb Ex 9.6(1): equilibrium price Frank Cowell: Microeconomics Take demand equations for the two types substitute in the values for income type-a demand becomes type-b demand becomes Excess demand for commodity 2: [10r + ⅓k]+[40r +140 − ⅔k] − 210 which simplifies to 50r − ⅓k − 70 Set excess demand to 0 for equilibrium: equilibrium price must be: r = [210 + k] / 150 Ex 9.6(2): Question and solution Frank Cowell: Microeconomics Incomes for the two types are resources: The equilibrium price is: r = [210 + k] / 150 So we can solve for incomes as: ya = 30r + k yb = 60r + [210 k] ya = [210 + 6k] / 5 yb = [1470 3k] / 5 Equivalently we can write ya and yb in terms of r as ya = 180r 210 yb = 420 90r Ex 9.6(3): Question Frank Cowell: Microeconomics purpose: to use the outcome of the GE model to plot the “incomepossibility” set method: plot incomes corresponding to extremes of allocating commodity 2, namely k = 0 and k = 210. Then fill in the gaps. Income possibility set Frank Cowell: Microeconomics incomes for k = 0 yb incomes for k = 210 incomes for intermediate values of k 300 • attainable set if income can be thrown away (42, 294) yb = 315 ½ya 200 • (294, 168) 100 ya 0 100 200 300 Ex 9.6(4): Question Frank Cowell: Microeconomics purpose: find a welfare optimum subject to the “income-possibility” set method: plot contours for the function W on the previous diagram. Welfare optimum: first case Frank Cowell: Microeconomics income possibility set yb Contours of W = log ya + log yb Maximisation of W over incomepossibility set 300 W is maximised at 200 corner • incomes are (294, 168) here k = 210 100 so optimum is where all of resource 2 is allocated to type a ya 0 100 200 300 Ex 9.6(5): Question Frank Cowell: Microeconomics purpose: as in part 4 method: as in part 4 Welfare optimum: second case Frank Cowell: Microeconomics income possibility set yb Contours of W = ya + yb Maximisation of W over incomepossibility set 300 again W is maximised 200 at corner • …where k = 210 so optimum is where all 100 of resource 2 is allocated to type a ya 0 100 200 300 Ex 9.6: Points to note Frank Cowell: Microeconomics 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