Frank Cowell: Microeconomics Revision Lecture EC202: Microeconomic Principles II Frank Cowell April 2007

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)  (c)  (c)  (a)  (a) Reference materials used (2) Reference materials used (2)  CfD presentations  Both related to past exam questions

Frank Cowell: Microeconomics Principles Scope of exam material Scope of exam material  what’s covered in the lectures…  … is definitive for the exam Resit Resit  a separate paper for anyone doing this second time around Structure and format of paper Structure and format of paper  follows that of last two years  check out the rubric from, say 2005 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 usually only in parts B and C of paper Examples from past question 1

Frank Cowell: Microeconomics (c) 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 (c) A model-solving question A model-solving question (i) just set E(  ) = 0 and twiddle (i) just set E(  ) = 0 and twiddle (ii) check what happens to E if you change  (ii) check what happens to E if you change  (iii) draw diagram and reason (iii) draw diagram and reason

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 two examples Let’s look at two 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 2.9 is mainly model solving  Ex 9.6 incorporates model building Look out for tips Look out for tips  Use of pictures in both questions  following hints in 9.6

Frank Cowell: Microeconomics Ex 2.9(1): Question purpose: demonstrate relationship between short and long run purpose: demonstrate relationship between short and long run method: Lagrangean approach to cost minimisation. First part can be solved by a “trick” method: Lagrangean approach to cost minimisation. First part can be solved by a “trick”

Frank Cowell: Microeconomics Ex 2.9(1): Long-run costs Production function is homogeneous of degree 1 Production function is homogeneous of degree 1  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 CRTS implies constant average cost CRTS implies constant average cost  C(w, q) / q = A (a constant)  so C(w, q) = Aq  differentiating: C q (w, q) = A So LRMC = LRAC = constant So LRMC = LRAC = constant  Their graphs will be an identical straight line

Frank Cowell: Microeconomics Ex 2.9(2): Question method: Standard Lagrangean approach Standard Lagrangean approach

Frank Cowell: Microeconomics Ex 2.9(2): short-run Lagrangean In the short run amount of good 3 is fixed In the short run amount of good 3 is fixed  z 3 =  z 3 Could write the Lagrangean as Could write the Lagrangean as But it is more convenient to transform the problem thus But it is more convenient to transform the problem thus where where

Frank Cowell: Microeconomics z2z2 z1z1 Ex 2.9(2): Isoquants Sketch the isoquant map Sketch the isoquant map Isoquants do not touch the axes Isoquants do not touch the axes So maximum problem must have an interior solution So maximum problem must have an interior solution

Frank Cowell: Microeconomics Ex 2.9(2): short-run FOCs Differentiating Lagrangean, the FOCS are Differentiating Lagrangean, the FOCS are This implies This implies To find conditional demand function must solve for To find conditional demand function must solve for  use the above equations…  …and the production function

Frank Cowell: Microeconomics Ex 2.9(2): short-run FOCs (more) Using FOCs and the production function: Using FOCs and the production function: This implies This implies  where This will give us the short-run cost function This will give us the short-run cost function

Frank Cowell: Microeconomics Ex 2.9(2): short-run costs By definition, short- run costs are: By definition, short- run costs are: This becomes This becomes Substituting for k: Substituting for k: From this we get From this we get  SRAC:  SRMC:

Frank Cowell: Microeconomics q Ex 2.9(2): short-run MC and AC marginal cost average cost

Frank Cowell: Microeconomics Ex 2.9(3): Question method: Draw the standard supply-curve diagram Draw the standard supply-curve diagram Manipulate the relationship p = MC Manipulate the relationship p = MC

Frank Cowell: Microeconomics Ex 2.9(3): short-run supply curve   average cost curve   marginal cost curve   supply curve q p q p   minimum average cost

Frank Cowell: Microeconomics Use the expression for marginal cost: Use the expression for marginal cost: Set p = MC for p ≥ p Set p = MC for p ≥ p Rearrange to get supply curve Rearrange to get supply curve Differentiate last line to get supply elasticity Differentiate last line to get supply elasticity Ex 2.9(3): short-run supply elasticity

Frank Cowell: Microeconomics Ex 2.9: Points to remember Exploit CRTS to give you easy results Exploit CRTS to give you easy results Try transforming the Lagrangean to make it easier to manipulate Try transforming the Lagrangean to make it easier to manipulate Use MC curve to derive supply curve Use MC curve to derive supply curve

Frank Cowell: Microeconomics Ex 9.6(1): Question 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 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  method: find price-taking optimising demands for each of the two types, use these to compute the excess demand function and solve for 

Frank Cowell: Microeconomics Ex 9.6(1): budget constraints Use commodity 2 as numéraire Use commodity 2 as numéraire  price of good 1 is   price of good 2 is 1 Evaluate incomes for the two types, given their resources: Evaluate incomes for the two types, given their resources:  type a has endowment (30, k)  therefore y a = 30  + k  type b has endowment (60, 210  k)  therefore y b = 60  + [210  k] Budget constraints for the two types are therefore: Budget constraints for the two types are therefore:   x 1 a + x 2 a ≤ 30  + k   x 1 b + x 2 b ≤ 60  + [210  k]

Frank Cowell: Microeconomics Ex 9.6(1): optimisation We could jump straight to a solution We could jump straight to a solution  utility functions are simple…  …so we can draw on known results Cobb-Douglas preferences imply Cobb-Douglas preferences imply  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) 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… Jump to “equilibrium price”

Frank Cowell: Microeconomics Ex 9.6(1): optimisation, type a The Lagrangean is: The Lagrangean is:  2log x 1 a + log x 2 a + a [y a   x 1 a  x 2 a ]  where a is the Lagrange multiplier  and y a is 30  + k FOC for an interior solution FOC for an interior solution  2/x 1 a  a  = 0  1/x 2 a  a  = 0  y a   x 1 a  x 2 a = 0 Eliminating a from these three equations, demands are Eliminating a from these three equations, demands are  x 1 a  = ⅔ y a /   x 2 a  = ⅓ y a

Frank Cowell: Microeconomics Ex 9.6(1): optimisation, type b The Lagrangean is: The Lagrangean is:  log x 1 b + 2log x 2 b + b [y b   x 1 b  x 2 b ]  where b is the Lagrange multiplier  and y b is 60   k FOC for an interior solution FOC for an interior solution  1/x 1 b  b  = 0  2/x 2 b  b  = 0  y b   x 1 b  x 2 b = 0 Eliminating b from these three equations, demands are Eliminating b from these three equations, demands are  x 1 b  = ⅓ y b /   x 2 b  = ⅔y b

Frank Cowell: Microeconomics Ex 9.6(1): equilibrium price Take demand equations for the two types 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: Excess demand for commodity 2:  [10  + ⅓k]+[40  +140 − ⅔k] − 210  which simplifies to 50  − ⅓k − 70 Set excess demand to 0 for equilibrium: Set excess demand to 0 for equilibrium:  equilibrium price must be:   = [210 + k] / 150

Frank Cowell: Microeconomics Ex 9.6(2): Question and solution Incomes for the two types are resources: Incomes for the two types are resources:  y a = 30  + k  y b = 60  + [210  k] The equilibrium price is: The equilibrium price is:   = [210 + k] / 150 So we can solve for incomes as: So we can solve for incomes as:  y a = [ k] / 5  y b = [1470  3k] / 5 Equivalently we can write y a and y b in terms of  as Equivalently we can write y a and y b in terms of  as  y a = 180   210  y b = 420  90 

Frank Cowell: Microeconomics Ex 9.6(3): Question purpose: to use the outcome of the GE model to plot the “income- possibility” set purpose: to use the outcome of the GE model to plot the “income- possibility” set method: plot incomes corresponding to extremes of allocating commodity 2, namely k = 0 and k = 210. Then fill in the gaps. method: plot incomes corresponding to extremes of allocating commodity 2, namely k = 0 and k = 210. Then fill in the gaps.

Frank Cowell: Microeconomics Income possibility set yaya ybyb (42, 294) (294, 168)   incomes for k = 0   incomes for k = 210   incomes for intermediate values of k   attainable set if income can be thrown away   y b = 315  ½y a

Frank Cowell: Microeconomics Ex 9.6(4): Question purpose: find a welfare optimum subject to the “income-possibility” set purpose: find a welfare optimum subject to the “income-possibility” set method: plot contours for the function W on the previous diagram. method: plot contours for the function W on the previous diagram.

Frank Cowell: Microeconomics Welfare optimum: first case yaya ybyb   income possibility set   Contours of W = log y a + log y b   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

Frank Cowell: Microeconomics Ex 9.6(5): Question purpose: as in part 4 purpose: as in part 4 method: as in part 4 method: as in part 4

Frank Cowell: Microeconomics Welfare optimum: second case yaya ybyb   income possibility set   Contours of W = y a + y b   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

Frank Cowell: Microeconomics 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