Frank Cowell: EC202 Microeconomics Revision Lecture 2 EC202: Microeconomic Principles II Frank Cowell May 2008

Frank Cowell: EC202 Microeconomics Objectives of the lecture Think more about method for long questions Think more about method for long questions Look at a few CfD Look at a few CfD  4.12, 4.13  5.1  (7.8)  9.6 See how they illustrate method See how they illustrate method Connect these to past exam questions Connect these to past exam questions

Frank Cowell: EC202 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  focus here on types 2 and 3  examples throughout the “long” questions of recent exam papers A simple type 2 question – Ex 5.1

Frank Cowell: EC202 Microeconomics Ex 5.1(1) Question purpose: construct a simple model of household supply and examine how it works method: build model up step-by-step through the question parts

Frank Cowell: EC202 Microeconomics 0 Ex 5.1(1) Preference map x2x2 x1x1 00 11 k  Shift the origin to (0, k)  Draw ICs homothetic to the shifted origin  indifference curves are “shifted” Cobb-Douglas  k is min requirement of other goods.   is share of budget of rice after an amount has been set aside to buy the min requirement rice other goods

Frank Cowell: EC202 Microeconomics Ex 5.1(2) Question method: Work out the budget constraint. Use the utility function to set out the Lagrangean Find the FOCs for an interior solution Find the demand functions Use these to get household supply function

Frank Cowell: EC202 Microeconomics Ex 5.1(2) Budget constraint Use good 2 as unit of value   price of rice (good 1) is p   price of all other goods (good 2) is 1 The consumer’s income is therefore: y := pR 1 + R 2 The budget constraint is px 1 + x 2  y where y is given by the above

Frank Cowell: EC202 Microeconomics Ex 5.1(2) Lagrangean method The Lagrangean is  log(x 1 ) + [1–  ]  log(x 2 –k) +  [ y – px 1 – x 2 ] The FOC for an interior maximum are  — – p  x 1 * 1–1– —— –  x 2 * –k y – px 1 * – x 2 * 

Frank Cowell: EC202 Microeconomics Ex 5.1(2) Demand functions From the FOC:  px 1 *  —  1–  x 2 *  k +  ——  Adding these and using the budget constraint, we have y  k + 1/ Eliminating  in the above:  x 1 *  — [y – k] p x 2 *  k + [1–  ] y

Frank Cowell: EC202 Microeconomics Ex 5.1(2) Supply function Supply of good 1 is given by S(p) := R 1 – x 1 *  Substituting in for y, we have  S(p)  [1–  ]R 1 –  — [R 2 – k] p Supply increases with price if R 2 > k

Frank Cowell: EC202 Microeconomics 0 Ex 5.1(2) Solution x2x2 x1x1 k  Endowment  Budget constraint rice other goods x* x2x2 p S R  Optimal consumption  Supply of rice

Frank Cowell: EC202 Microeconomics Ex 5.1(3) Let c be the amount of the ration If  k + [1–  ] y  c nothing changes from previous case Otherwise px 1 + c = y so that R 2 – c x 1 = R 1 +  ——— p c – R 2 S(p)  ——— p

Frank Cowell: EC202 Microeconomics 0 Ex 5.1(3) Modified solution x2x2 x1x1 k  Original solution rice other goods x* x2x2 S R  Generous ration  Severe ration x**  Generous ration has no effect  Severe ration on other goods affects supply of rice c c

Frank Cowell: EC202 Microeconomics Ex 5.1: Points to remember Use diagram to understand features of utility function Model supply as mirror image of demand Use diagram to see effect of ration Connection to exams   see 2005 q3   …it’s the same model! A “meatier” type 2 question – Ex 4.12, 4.13

Frank Cowell: EC202 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: EC202 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: EC202 Microeconomics Ex 4.12(1) Indifference curves x1x1 x2x2 Could have x 2 = 0 Slope is vertical here

Frank Cowell: EC202 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: EC202 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: EC202 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: EC202 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: EC202 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: EC202 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: EC202 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: EC202 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 Now look at the follow-up question

Frank Cowell: EC202 Microeconomics Ex 4.13(2) Question Method: Find monopolist’s AR from consumer demand using answer to Ex Find monopolist’s AR from consumer demand using answer to Ex Then use standard optimisation procedure Then use standard optimisation procedure

Frank Cowell: EC202 Microeconomics Ex 4.13(2) Monopoly profits Aggregate demand over N consumers using Exercise 4.12 Aggregate demand over N consumers using Exercise 4.12 Rearrange to get AR curve: Rearrange to get AR curve: Total Revenue is: Total Revenue is: Profits are therefore: Profits are therefore:

Frank Cowell: EC202 Microeconomics Ex 4.13(2) Maximising profits FOC (MC = MR) yields: FOC (MC = MR) yields: So monopolist’s optimal output is: So monopolist’s optimal output is: From AR curve, price at optimum is: From AR curve, price at optimum is: Simplify this to: Simplify this to:  (clearly price > MC)

Frank Cowell: EC202 Microeconomics Ex 4.13(3) Question Method: Aggregate the CV for each consumer to define L. Aggregate the CV for each consumer to define L. Use marginal cost and monopolist’s equilibrium price to evaluate L Use marginal cost and monopolist’s equilibrium price to evaluate L

Frank Cowell: EC202 Microeconomics Ex 4.13(3) Evaluating loss Use definition of CV with p 1 ' = c: Evaluate L at p 1 = 2c: Firm’s profits are: Clearly L > profits

Frank Cowell: EC202 Microeconomics Ex 4.13(4) Question Method: Add bonus B into the expression for profits Add bonus B into the expression for profits Again use standard optimisation procedure Again use standard optimisation procedure

Frank Cowell: EC202 Microeconomics Ex 4.13(4) Evaluating profits (again) Profits including bonus are: Value of bonus is: Use demand curve to express this in terms of q: So profits can now be expressed as:

Frank Cowell: EC202 Microeconomics Ex 4.13(4) Evaluating profits (again) Take the expression for profits including bonus FOC for a maximum is again MR = MC: Rearranging we get the value of optimal output for the regulated monopolist: Use demand curve to find: Clearly the regulated price = MC:

Frank Cowell: EC202 Microeconomics Ex 4.12 & 4.13: Points to note 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 Aggregate welfare loss is found from individual CV Unregulated monopoly makes profits smaller than losses to consumer Regulation causes monopoly to behave like competitive firm Now for a type 3 “model building” q

Frank Cowell: EC202 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: EC202 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: EC202 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: EC202 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: EC202 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: EC202 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: EC202 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: EC202 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: EC202 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: EC202 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: EC202 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: EC202 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: EC202 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: EC202 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 Link to exam   very similar to 2007 q5.   but also check out 2005 q4…   …same model, but you are asked to different tricks with it More practice with GE – see CfD 7.8