Indifference Analysis

Shape of ‘Clive’s’ an indifference curve Notice that this curve is downward sloping Why is this and does it have any economic meaning? b c Pears d e f g Oranges

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 6

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… 6

The MRS of Y for X is: Pears 30 24 20 14 10 8 6 Oranges 7 13 15 Point b c d e f g

Deriving the marginal rate of substitution (MRS) 24 b Units of good Y 6 7 Units of good X

Deriving the marginal rate of substitution (MRS) MRS = DY/ DX= - 6 DY = - 6 24 b DX = 1 Units of good Y 6 7 Units of good X

Consider instead the MRS at 13 oranges MRS = DY/ DX= - 6 DY = - 6 24 b DX = 1 Pears 30 24 20 14 10 9 6 Oranges 7 8 13 Point a b c d e f g Units of good Y 6 7 Units of good X

Deriving the marginal rate of substitution (MRS) Pears 30 24 20 14 10 9 6 Oranges 7 8 13 Point a b c d e f g MRS = - 6 DY = - 6 24 b DX = 1 Units of good Y MRS = - 1 c DY = - 1 d 9 DX = 1 6 7 13 14 Units of good X

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 6

An indifference map Units of good Y I1 I1 Units of good X

An indifference map Units of good Y I2 I1 Units of good X

An indifference map Units of good Y I3 I2 I1 Units of good X

An indifference map Units of good Y I4 I3 I2 I1 Units of good X

An indifference map Units of good Y I5 I4 I3 I2 I1 Units of good X

This is basically a map of the happiness mountain An indifference map This is basically a map of the happiness mountain The further up we are, the happier we are Units of good Y I5 I4 I3 I2 I1 Units of good X

‘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:

Similarly, for a given X, the more Y we have the happier we are:

So the more X and Y we have the happier we are :

Of course the issue about the happiness mountain is that we never get there Y Y0 X

This is basically the indifference map we started with a while ago 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 X X0

An indifference map Units of good Y I5 I4 I3 I2 I1 Units of good X

An indifference map Note for a given quantity of x, a rise in y increases our utility, from I1 to I4 Units of good Y I5 I4 I3 I2 I1 Units of good X

An indifference map Similarly, for a given quantity of y, a rise in x increases our utility, from I2 to I5 Units of good Y I5 I4 I3 I2 I1 Units of good X

Down and left decreases it An indifference map So to labour the point, anything to the right an upwards increases our utility Down and left decreases it Units of good Y I5 I4 I3 I2 I1 Units of good X

Could two indifference curves ever cross? Units of good Y I1 Units of good X

Could two indifference curves ever cross? Could we get this? Units of good Y I1 Units of good X

Could two indifference curves ever cross? Could we get this? I2 Consider points a,b and c Point a is indifferent to b And point a is indifferent to c Units of good Y => b is indifferent to c a b I1 c Units of good X

Could two indifference curves ever cross? Could we get this? I2 Consider points a,b and c But point b has more X and Y than c so b is preferred to C Contradiction, so indifference curves can never cross Units of good Y a b I1 c Units of good X

An indifference map So just like the map of a mountain, our contours of happiness (the indifference curves) never cross. Units of good Y I5 I4 I3 I2 I1 Units of good X

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 6

Budget Line Our expenditure on X (PxX) and Y (PyY) must sum to our total income:

Budget Line Our expenditure on X and Y must sum to our total income: If Y =0, then 2X = 30 and Max Consumption of X = 15

Budget Line Our expenditure on X and Y must sum to our total income: If X= 0, then 1Y =30 and Max Consumption of Y = 30

A budget line a Units of good Y Units of good X Units of good X 15 15 good Y 30 Point on budget line Units of good X 15 Units of good Y 30 Point on budget line a b Units of good Y Assumptions PX = £2 PY = £1 Budget = £30 Units of good X

We join these two points to get the budget line a Units of good X 15 good Y 30 Point on budget line Units of good Y Assumptions PX = £2 PY = £1 Budget = £30 Units of good X

A budget line a Units of good Y Units of good X Units of good X 15 15 good Y 30 Point on budget line Point on budget line a Units of good Y Assumptions PX = £2 PY = £1 Budget = £30 Units of good X

A budget line a Units of good Y Units of good X Units of good X 5 10 5 10 15 Units of good Y 30 20 10 Point on budget line a Units of good Y Assumptions PX = £2 PY = £1 Budget = £30 Units of good X

A budget line a b Units of good Y Units of good X Units of good X 5 10 5 10 15 Units of good Y 30 20 10 Point on budget line a b b Units of good Y Assumptions PX = £2 PY = £1 Budget = £30 Units of good X

A budget line a b Units of good Y c Units of good X Units of good X 5 5 10 15 Units of good Y 30 20 10 Point on budget line a b c b Units of good Y c Assumptions PX = £2 PY = £1 Budget = £30 Units of good X

A budget line a b Units of good Y c d Units of good X Units of good X 5 10 15 Units of good Y 30 20 10 Point on budget line a b c d b Units of good Y c Assumptions PX = £2 PY = £1 Budget = £30 d Units of good X

Budget Line So all these points satisfy the equation:

Note the Maximum x we can consume is 30/Px A budget line a Units of good X 15 Units of good Y 30 Point on budget line a d b Units of good Y Note the Maximum x we can consume is 30/Px c d Units of good X =30/Px=30/2

And the maximum Y we can consume is 30/PY = 30/1 A budget line a Units of good X 15 Units of good Y 30 Point on budget line a d b Units of good Y And the maximum Y we can consume is 30/PY = 30/1 c d =30/Px=30/2

30/PY = A budget line So Height is 30/Py And length is 30/Px a b Units of good Y And length is 30/Px c d =30/Px=30/2

So (minus) the slope of the curve is Height / Length A budget line 30/PY = a So (minus) the slope of the curve is Height / Length b Units of good Y c d =30/Px=30/2

So (minus) the slope of the curve is Height / Length A budget line 30/PY = a So (minus) the slope of the curve is Height / Length b Units of good Y c d =30/Px=30/2

So (minus) the slope of the curve is Height / Length 30/PY = A budget line a So (minus) the slope of the curve is Height / Length b X Units of good Y X c d =30/Px=30/2

So (minus) the slope of the curve is Height / Length A budget line 30/PY = a So (minus) the slope of the curve is Height / Length b Units of good Y c Which is the relative price of x in terms of good y d =30/Px=30/2

A budget line 30/PY = a Actually the slope is a negative number since we must GIVE UP X to get Y. b Units of good Y c d =30/Px=30/2

A budget line 30/PY = a So the slope of the budget constraint is the relative price of X in terms of Y b Units of good Y c d =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 7

Finding the optimum consumption Units of good Y I5 I4 I3 I2 I1 O Units of good X

Finding the optimum consumption Units of good Y Budget line I5 I4 I3 I2 I1 O Units of good X

Finding the optimum consumption Consider the Points r and v What indifference curve are they on? Could we do better? r Units of good Y I5 I4 v I3 I2 I1 O Units of good X

Finding the optimum consumption Clearly s and u are better. They are on I2 r s Units of good Y u I5 I4 v I3 I2 I1 O Units of good X

Finding the optimum consumption At point t we have reached the highest point possible and this is our optimum consumption point r s Units of good Y t u I5 I4 v I3 I2 I1 O Units of good X

Finding the optimum consumption Utility is maximised at the point of tangency between the indifference map and the budget constraint. r s Units of good Y t Y1 u I5 I4 v I3 I2 I1 O X1 Units of good X

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 7

Same slope at t of indifference curve and budget line So when the consumer chooses optimally: r s Units of good Y t Y1 u I5 I4 v I3 I2 I1 O X1 Units of good X

Same slope at t of indifference curve and budget line 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. Units of good Y t Y1 u I5 I4 v I3 I2 I1 O X1 Units of good X

Same slope at t of indifference curve and budget line 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. Units of good Y t Y1 u I5 I4 v I3 I2 I1 O X1 Units of good X

Same slope at t of indifference curve and budget line The optimum must therefore satisfy: r s Units of good Y t Y1 u I5 I4 v I3 I2 I1 O X1 Units of 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? 7

We have found the demand for X at particular set of prices and Income Good Y Y1 I3 I2 I1 B1 B2 B3 X1 Good X

We have found the demand for X at particular set of prices and Income Good Y Y1 I3 I2 I1 B1 B2 B3 X1 Good X Suppose we draw a diagram below with Price on one axis and quantity on the other Price of X Good X X1

We have found the demand for X at particular set of prices and Income Good Y Y1 I3 I2 I1 B1 B2 B3 X1 Good X And now we project the demand for x downwards at the Price, Px=2 PX=2 Price of X Good X X1

We have found the demand for X at particular set of prices and Income Good Y Y1 I3 I2 I1 B1 B2 B3 X1 Good X What does this point represent? Ans: A point on the demand curve! PX=2 Price of X Good X X1

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. 7