Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Indifference Analysis Shape of ‘Clive’s’ an indifference curve 30 a Notice that this curve is downward sloping 28 26 b 24 22 c Pears 20 18 16 d 14 Why is this and does it have any economic meaning? 12 e 10 f 8 g 6 4 2 0 0 2 4 6 8 10 12 Oranges 14 16 18 20 22 INDIFFERENCE ANALYSIS • To see the special significance of its shape we need to look at a concept known as the – diminishing marginal rate of substitution INDIFFERENCE ANALYSIS • The rate of Substitution is the rate we are prepared to exchange Pears (Y) for Oranges (X). • Or: If I give ‘Clive’ one more orange how many pears must I take away to leave him just as happy… MRSYX Y1 Y0 Y X 1 X 0 X The MRS of Y for X is: • Pears Oranges MRSYX Y1 Y0 Y X 1 X 0 X MRSYX 24 30 6 6 76 1 30 24 20 14 10 8 6 6 7 8 10 13 15 20 Point a b c d e f g Deriving the marginal rate of substitution (MRS) 30 a b Units of good Y 24 20 10 0 0 67 10 Units of good X 20 Deriving the marginal rate of substitution (MRS) a 30 MRS = Y/ X= - 6 b Y = - 6 24 Units of good Y X = 1 20 10 0 0 67 10 Units of good X 20 a 30 MRS = Y/ X= - 6 b Y = - 6 24 Units of good Y X = 1 Consider instead the MRS at 13 oranges Pears Oranges 20 30 24 20 14 10 9 6 10 0 0 67 10 Units of good X 6 7 8 10 13 14 20 20 Point a b c d e f g Deriving the marginal rate of substitution (MRS) 30 a Pears Oranges MRS = - 6 b Y = - 6 24 30 24 20 14 10 9 6 Units of good Y X = 1 20 10 9 Y = - 1 c 6 7 8 10 13 14 20 MRS = - 1 d X = 1 0 0 67 10 13 14 Units of good X 20 Point a b c d e f g INDIFFERENCE ANALYSIS • From an Indifference curves to an indifference Map • An indifference curve tells us the bundles which give Clive the same happiness as 13 pears and 10 oranges. • An indifference map shows us a whole series of different curves showing which bundles give different levels of happiness An indifference map Units of good Y 30 20 10 I1 I1 0 0 10 Units of good X 20 An indifference map Units of good Y 30 20 10 I2 0 0 10 Units of good X I1 20 An indifference map Units of good Y 30 20 10 I1 0 0 10 Units of good X I2 20 I3 An indifference map Units of good Y 30 20 10 I4 I1 0 0 10 Units of good X I2 20 I3 An indifference map Units of good Y 30 20 10 I5 I4 I1 0 0 10 Units of good X I2 20 I3 An indifference map Units of good Y 30 This is basically a map of the happiness mountain 20 The further up we are, the happier we are 10 I5 I4 I1 0 0 10 Units of good X I2 20 I3 ‘We’re Climbing up the Sunshine Mountain’ • We can think of picking bundles of goods as an attempt to climb the happiness mountain. ‘We’re Climbing up the Sunshine Mountain’ • A mountain of course is three dimensional and it is difficult to view on a screen.This might be a side view. • Putting in contours the mountain looks like this: • Putting in contours the mountain looks like this: • Gradually rotating and looking down from above it looks like this: • Gradually rotating and looking down from above it looks like this: • Gradually rotating and looking down from above it looks like this: • Gradually rotating and looking down from above it looks like this: • Gradually rotating and looking down from above it looks like this: • Looking down from above the mountasin looks like this: • Gradually rotating and looking down from above it looks like this: So from above the happiness mountain looks like this Putting two axes in: Y X For a given Y, the more X we have the happier we are: Y Y0 X Similarly, for a given X, the more Y we have the happier we are: Y Y0 X X0 So the more X and Y we have the happier we are : Y Y0 X Of course the issue about the happiness mountain is that we never get there Y Y0 X Of course the issue about the happiness mountain is that we never get there. So we are interested in just one corner of the mountain Y0 This is basically the indifference map we started with a while ago X0 X An indifference map Units of good Y 30 20 10 I5 I4 I1 0 0 10 Units of good X I2 20 I3 An indifference map Units of good Y 30 Note for a given quantity of x, a rise in y increases our utility, from I1 to I4 20 10 I5 I4 I1 0 0 10 Units of good X I2 20 I3 An indifference map Units of good Y 30 Similarly, for a given quantity of y, a rise in x increases our utility, from I2 to I5 20 10 I5 I4 I1 0 0 10 Units of good X I2 20 I3 An indifference map Units of good Y 30 So to labour the point, anything to the right an upwards increases our utility 20 Down and left decreases it 10 I5 I4 I1 0 0 10 Units of good X I2 20 I3 Could two indifference curves ever cross? Units of good Y 30 20 10 I1 0 0 10 Units of good X 20 Could two indifference curves ever cross? 30 I2 Units of good Y Could we get this? 20 10 I1 0 0 10 Units of good X 20 Could two indifference curves ever cross? 30 Could we get this? I2 Units of good Y Consider points a,b and c Point a is indifferent to b 20 And point a is indifferent to c => b is indifferent to c a 10 b c I1 0 0 10 Units of good X 20 Could two indifference curves ever cross? Could we get this? I2 30 Units of good Y Consider points a,b and c But point b has more X and Y than c so b is preferred to C 20 a Contradiction, so indifference curves can never cross 10 b c I1 0 0 10 Units of good X 20 An indifference map Units of good Y 30 So just like the map of a mountain, our contours of happiness (the indifference curves) never cross. 20 10 I5 I4 I1 0 0 10 Units of good X I2 20 I3 INDIFFERENCE ANALYSIS • We now have a map which represents peoples’ choices. What else do we need to make an economic decision about what to consume? • Answer: Information on prices and Income • The budget line: • Suppose the Price of X is Px • And the Price of Y is Py • And Income is represented by I Budget Line • Our expenditure on X (PxX) and Y (PyY) must sum to our total income: PX X PY Y I e.g . PX 2 PY 1 I 30 2 X 1Y 30 Budget Line • Our expenditure on X and Y must sum to our total income: e.g . PX 2 If Y =0, then PY 1 2X = 30 I 30 2 X 1Y 30 and Max Consumption of X = 15 Budget Line • Our expenditure on X and Y must sum to our total income: e.g . PX 2 If X= 0, then 1Y =30 PY 1 and I 30 2 X 1Y 30 Max Consumption of Y = 30 A budget line 30 a Units of good Y Units of Units of Point on Units of Units of Point on good X good Y budget line good X good Y budget line 0 0 15 15 20 10 30 30 0 0 a b Assumptions PX = £2 PY = £1 Budget = £30 0 0 5 10 Units of good X 15 20 We join these two points to get the budget line 30 a Units of good Y Units of good X Units of Point on good Y budget line 0 15 20 10 30 0 Assumptions PX = £2 PY = £1 Budget = £30 0 0 5 10 Units of good X 15 20 A budget line a 30 Units of good Y Units of good X Units of Point on good Y budget line 0 15 20 30 0 a Assumptions 10 PX = £2 PY = £1 Budget = £30 0 0 5 10 Units of good X 15 20 A budget line a 30 Units of good Y Units of good X Units of Point on good Y budget line 0 5 10 15 20 30 20 10 0 a Assumptions 10 PX = £2 PY = £1 Budget = £30 0 0 5 10 Units of good X 15 20 A budget line a 30 Units of good Y Units of good X 0 5 10 15 b 20 Units of Point on good Y budget line 30 20 10 0 a b Assumptions 10 PX = £2 PY = £1 Budget = £30 0 0 5 10 Units of good X 15 20 A budget line a 30 Units of good Y Units of good X 0 5 10 15 b 20 Units of Point on good Y budget line 30 20 10 0 c 10 a b c Assumptions PX = £2 PY = £1 Budget = £30 0 0 5 10 Units of good X 15 20 A budget line a 30 Units of good Y Units of good X 0 5 10 15 b 20 Units of Point on good Y budget line 30 20 10 0 c 10 a b c d Assumptions PX = £2 PY = £1 Budget = £30 d 0 0 5 10 Units of good X 15 20 Budget Line • So all these points satisfy the equation: PX X PY Y I becomes 2 X 1Y 30 when PX 2, PY 1, I 30 A budget line a 30 Units of good Y Units of good X 0 15 b 20 Units of Point on good Y budget line 30 0 Note the Maximum x we can consume is 30/Px c 10 a d d 0 0 5 10 Units of good X 15 20 =30/Px=30/2 A budget line 30/PY = a 30 Units of good Y Units of good X 0 15 b 20 Units of Point on good Y budget line 30 0 And the maximum Y we can consume is 30/PY = 30/1 c 10 a d d 0 0 5 10 15 20 =30/Px=30/2 A budget line 30/PY = a 30 Units of good Y So Height is 30/Py b 20 And length is 30/Px c 10 d 0 0 5 10 15 20 =30/Px=30/2 A budget line 30/PY = Units of good Y 30 a b 20 So (minus) the slope of the curve is Height / Length c 10 30 Py 30 Px d 0 0 5 10 15 20 =30/Px=30/2 A budget line 30/PY = Units of good Y 30 a b 20 So (minus) the slope of the curve is Height / Length 30 Px . Py 30 c 10 d 0 0 5 10 15 20 =30/Px=30/2 A budget line 30/PY = Units of good Y 30 a b 20 So (minus) the slope of the curve is Height / Length Px 30 X . Py 30 X c 10 d 0 0 5 10 15 20 =30/Px=30/2 A budget line 30/PY = Units of good Y 30 a b 20 So (minus) the slope of the curve is Height / Length Px Py c 10 d 0 0 5 10 Which is the relative price of x in terms of good y 15 20 =30/Px=30/2 A budget line 30/PY = Units of good Y 30 a b 20 Actually the slope is a negative number since we must GIVE UP X to get Y. c 10 Px Py d 0 0 5 10 15 20 =30/Px=30/2 A budget line 30/PY = Units of good Y 30 a b 20 So the slope of the budget constraint is the relative price of X in terms of Y c 10 Px Py d 0 0 5 10 15 20 =30/Px=30/2 INDIFFERENCE ANALYSIS • We now combine the indifference curve we found before and the budget constraint to find: • The optimum consumption point Units of good Y Finding the optimum consumption I5 I4 I1 O Units of good X I2 I3 Units of good Y Finding the optimum consumption Budget line I5 I4 I1 O Units of good X I2 I3 Finding the optimum consumption r Consider the Points r and v Units of good Y What indifference curve are they on? Could we do better? I5 I4 v I1 O Units of good X I2 I3 Finding the optimum consumption Clearly s and u are better. r s Units of good Y They are on I2 u I5 I4 v I1 O Units of good X I2 I3 Finding the optimum consumption At point t we have reached the highest point possible and this is our optimum consumption point r Units of good Y s t u I5 I4 v I1 O Units of good X I2 I3 Finding the optimum consumption Utility is maximised at the point of tangency between the indifference map and the budget constraint. r Units of good Y s Y1 t u I5 I4 v I1 O X1 Units of good X I2 I3 INDIFFERENCE ANALYSIS • The optimum consumption point • Recall that the slope of the indifference curve is • the marginal rate of substitution • While the slope of the budget constraint is: • (Minus) the relative price of X and Y Same slope at t of indifference curve and budget line So when the consumer chooses optimally: r Units of good Y s Y1 Px MRS Py t u I5 I4 v I1 O X1 Units of good X I2 I3 Same slope at t of indifference curve and budget line If r Units of good Y s Y1 Px MRS Py At s MRS is above the market price. ‘Clive’ is prepared to give up more Y than market requires to get X – so he does. t u I5 I4 v I1 O X1 Units of good X I2 I3 Same slope at t of indifference curve and budget line If r Units of good Y s Y1 Px MRS Py At u MRS is below the market price. ‘Clive’ is prepared to give up more X than market requires to get Y – so he does. t u I5 I4 v I1 O X1 Units of good X I2 I3 Same slope at t of indifference curve and budget line The optimum must therefore satisfy: r Units of good Y s Y1 Px MRS Py t u I5 I4 v I1 O X1 Units of good X I2 I3 INDIFFERENCE ANALYSIS • So now we have established what Clive will choose to consume IF the price of X is 2, the price of Y is 1 and his income is 30. • So XD = f(px,py,I) = f(2,1,30) = x1 • What can we say about X1? Good Y We have found the demand for X at particular set of prices and Income Y1 I3 B1 X1 B2 I1 I2 B3 Good X Good Y We have found the demand for X at particular set of prices and Income Y1 I3 B1 X1 B2 I1 I2 B3 Good X Price of X Suppose we draw a diagram below with Price on one axis and quantity on the other X1 Good X Good Y We have found the demand for X at particular set of prices and Income Y1 I3 B1 Price of X X1 B2 I1 I2 B3 Good X And now we project the demand for x downwards at the Price, Px=2 PX=2 X1 Good X Good Y We have found the demand for X at particular set of prices and Income Y1 I3 B1 Price of X X1 B2 I1 I2 B3 Good X What does this point represent? PX=2 Ans: A point on the demand curve! X1 Good X INDIFFERENCE ANALYSIS • So now we have established what Clive will choose to consume IF the price of X is 2, the price of Y is 1 and his income is 30. • So XD = f(px,py,I) = f(2,1,30) = x1 • What can we say about X1? • What would happen if Px were to change? • or Py • Or I • It is to these issues we now turn.