Consumption | Comparative Statics 1
Income and Choice
Suppose the consumer’s income were to increase,Optimal choice changes,How?
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0 0
x2
x1
x2
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xb2
xb1
xb2
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The flrst diagram illustrates the case when both goods are normal,As income rises the demand for both goods rises.
The second diagram illustrates the case when good 1 is inferior,As income rises the demand for good 1 falls.
Good 1 is normal if ￠x1=￠m > 0 and inferior if ￠x1=￠m < 0,Both goods cannot be inferior,Why not?
Consumption | Comparative Statics 2
Income Ofier Curves and Engel Curves
When the optimal choices are connected up for each income level the resultant curve is called the income ofier curve.
The Engel curve is derived from this by plotting each level of income m against the resultant optimal quantity for
one of the goods | say good 1,x1,Both curves are drawn below.
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0 0
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m
x1
Income ofier curve Engel curve
Income ofier curve
Engel curves distinguish between luxury goods and necessary goods,If demand rises proportionally more than
income the good is a luxury | and necessary in the opposite case.
If the Engel curve is a straight line (through the origin),demand rises proportionally with income,The good is
neither luxury nor necessity,Homothetic preferences guarantee this,This is when the consumer only cares about
the ratio consumed of the two goods,That is,x ′ y implies tx ′ ty for all numbers t.
Income ofier curves are also straight lines in this case,Perfect complements,perfect substitutes and Cobb-Douglas
preferences are all homothetic.
Consumption | Comparative Statics 3
Price and Choice
Suppose the price of good 1 were to increase,Optimal choice changes,How?
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x2
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xb1xa1xb1 xa1
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The flrst diagram illustrates the case when good 1 is ordinary,As price rises the demand for good 1 falls.
The second diagram illustrates the case when good 1 is Gifien,As price rises the demand for good 1 rises.
Good 1 is ordinary if ￠x1=￠p1 < 0 and Gifien if ￠x1=￠p1 > 0,Both goods cannot be Gifien,Why not?
Consumption | Comparative Statics 4
Price Ofier Curves and Demand Curves
When the optimal choices are connected up for each price level the resultant curve is called the price ofier curve.
The demand curve is derived from this by plotting each level of price p1 against the resultant optimal quantity of
good 1,x1,Both curves are drawn below.
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0 0
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x1
p1
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Price ofier curve Demand curve
Price ofier curve
It is also possible to derive the demand curve for good 1 in terms of the price of good 2.
If optimal x1 increases with an increase in p2 (￠x1=￠p2 > 0) then the goods are (gross) substitutes,if optimal x1
decreases with an increase in p2 (￠x1=￠p2 < 0) then the goods are (gross) complements.
The inverse demand function is simply price in terms of quantity,p1 = p1(x1).
Consumption | Comparative Statics 5
The Price Efiect
When the price changes,demand changes,Economists decompose the overall price efiect into three parts.
1,The Substitution Efiect,The change in demand due to relative price changes.
2,The Income Efiect,The change in demand due to changes in purchasing power with money income constant.
3,The Endowment Efiect,The change in demand due to changes in the value of the endowment.
When added together these three efiects give the price efiect:
Price Efiect = Substitution Efiect +Income Efiect +Endowment Efiect
The rest of the lecture examines this simple equation | the Slutsky decomposition | in some depth.
Consumption | Comparative Statics 6
A Price Change
Initially suppose the consumer has flxed money income m,Throughout the lecture a decrease in p1 will be
examined,An increase in p1 or a change in p2 can be analysed in a similar way.
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xa xb
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A decrease in p1 pivots the budget line outward resulting in a new optimal choice xb.
This is illustrated in the above graph,The consumer alters consumption of good 1 from xa1 to xb1,Notice the second
diagram also shows this but without indifierence curves,Often,indifierence curves can be suppressed and only
optimal choices drawn in,This simplifles the diagrams a great deal.
Consumption | Comparative Statics 7
The Substitution Efiect 1
Suppose the consumer’s optimal choice changes from xa to xb due to a decrease in p1.
The change in demand for good 1 is ￠x1 = xb1?xa1,This change can be decomposed into two parts | a
substitution efiect and an income efiect.
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x2
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xa1 xb1
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Suppose the consumer’s money income m was changed so that they could still just afiord to buy their initially
chosen bundle,xa,Their initial budget line was A which pivoted to B following the decrease in p1,Now some of
their income is \taken away" resulting in budget line C above.
At budget line C only the relative price change has taken place,Furthermore,given budget line C the consumer
might no longer choose bundle xa,They could prefer a bundle further down C,The next slide illustrates this.
Consumption | Comparative Statics 8
The Substitution Efiect 2
Suppose at the new budget line C,the consumer purchases bundle xc = (xc1; xc2) as in the graph below.
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0 0
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The substitution efiect is then the change in demand due to the relative price change alone,given by:
￠xs1 = xc1?xa1
Notice that xc is further down budget line C than xa,It could not go further up the budget line,Why not?
Because of this,the substitution efiect is always in the opposite direction to the price change.
Consumption | Comparative Statics 9
The Income Efiect
Returning to the consumer the income previously \taken away" results in another optimal choice | xb.
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0 0
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The income efiect is then the change in demand due to the change in purchasing power alone,given by:
￠xn1 = xb1?xc1
If good 1 is normal,xb will lie to the right of xc,If good 1 is inferior,xb will lie to the left of xc,The income efiect
can be in the opposite direction to the substitution efiect (an inferior good) or in the same direction (a normal
good),The diagram above illustrates a normal good.
Gifien goods arise when the good is inferior and the income efiect is large enough to outweigh the substitution
efiect,In this case xb lies to the left of xa | Gifien goods are always inferior.
Consumption | Comparative Statics 10
The Law of Demand
The Slutsky decomposition can be written as ￠x1 = ￠xs1 +￠xn1.
Although ￠x1 could be of either sign | the good could be ordinary or Gifien,something can be said about the
relationship between ￠x1 and ￠xn1 with the current assumptions in place.
The substitution efiect is always in the opposite direction to the price change so the following is true.
The Law of Demand,If demand for a good increases when income increases then demand for that good must
decrease when its price increases.
Consumption | Comparative Statics 11
The Endowment Efiect
Up until now the consumer’s income has been a flxed number m,When the consumer receives an endowment !,
there is a third efiect to take into consideration | the endowment efiect.
Recall a price change will pivot the budget line through the endowment point,The flrst graph shows a fall in p1,!
is the endowment,xa the initially chosen bundle and xd the bundle chosen after the price change.
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......0 0 0
x2
x1
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x1
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A D A DB C
xa xa
xb
xcxd xd
! ! !
xa1 xb1 xc1xd1
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The second graph shows the decomposition,Budget line A represents the initial prices,Budget line B corresponds
to the substitution efiect,C to the income efiect and D to the endowment efiect.
The third graph shows the three efiects,￠xs1 = xb1?xa1 is the substitution efiect,￠xn1 = xc1?xb1 is the income
efiect,￠x!1 = xd1?xc1 is the endowment efiect,So:
￠x1 = ￠xs1 +￠xn1 +￠x!1 = xd1?xa1
Consumption | Comparative Statics 12
Buying and Selling
Remember that the endowment efiect is a change in demand due to a shift in the budget line,therefore its sign is
determined by whether the good is normal or inferior.
On the last graph,xd lies to the right of xb and to the left of xc,This is because the good is normal,The income
and endowment efiects are related in this way,Take care | xd could not lie to the left of xb,for example.
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Given a fall in p1,budget line A moves to budget line D in both graphs,Optimal choice changes from xa to xd.
In the flrst graph,the consumer was a net buyer of good 1 | after p1 falls they always remain a net buyer.
In the second graph,the consumer was a net seller of good 1 | after p1 falls they need not be.
If they continue to be a net seller after a price fall they are worse ofi,If they are a net buyer and there is a price fall
they are better ofi,What about a price rise? Net sellers continue to sell and are better ofi,Buyers need not
continue to buy,but if they do they are worse ofi,Why?
Income and Choice
Suppose the consumer’s income were to increase,Optimal choice changes,How?
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The flrst diagram illustrates the case when both goods are normal,As income rises the demand for both goods rises.
The second diagram illustrates the case when good 1 is inferior,As income rises the demand for good 1 falls.
Good 1 is normal if ￠x1=￠m > 0 and inferior if ￠x1=￠m < 0,Both goods cannot be inferior,Why not?
Consumption | Comparative Statics 2
Income Ofier Curves and Engel Curves
When the optimal choices are connected up for each income level the resultant curve is called the income ofier curve.
The Engel curve is derived from this by plotting each level of income m against the resultant optimal quantity for
one of the goods | say good 1,x1,Both curves are drawn below.
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Income ofier curve Engel curve
Income ofier curve
Engel curves distinguish between luxury goods and necessary goods,If demand rises proportionally more than
income the good is a luxury | and necessary in the opposite case.
If the Engel curve is a straight line (through the origin),demand rises proportionally with income,The good is
neither luxury nor necessity,Homothetic preferences guarantee this,This is when the consumer only cares about
the ratio consumed of the two goods,That is,x ′ y implies tx ′ ty for all numbers t.
Income ofier curves are also straight lines in this case,Perfect complements,perfect substitutes and Cobb-Douglas
preferences are all homothetic.
Consumption | Comparative Statics 3
Price and Choice
Suppose the price of good 1 were to increase,Optimal choice changes,How?
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The flrst diagram illustrates the case when good 1 is ordinary,As price rises the demand for good 1 falls.
The second diagram illustrates the case when good 1 is Gifien,As price rises the demand for good 1 rises.
Good 1 is ordinary if ￠x1=￠p1 < 0 and Gifien if ￠x1=￠p1 > 0,Both goods cannot be Gifien,Why not?
Consumption | Comparative Statics 4
Price Ofier Curves and Demand Curves
When the optimal choices are connected up for each price level the resultant curve is called the price ofier curve.
The demand curve is derived from this by plotting each level of price p1 against the resultant optimal quantity of
good 1,x1,Both curves are drawn below.
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Price ofier curve Demand curve
Price ofier curve
It is also possible to derive the demand curve for good 1 in terms of the price of good 2.
If optimal x1 increases with an increase in p2 (￠x1=￠p2 > 0) then the goods are (gross) substitutes,if optimal x1
decreases with an increase in p2 (￠x1=￠p2 < 0) then the goods are (gross) complements.
The inverse demand function is simply price in terms of quantity,p1 = p1(x1).
Consumption | Comparative Statics 5
The Price Efiect
When the price changes,demand changes,Economists decompose the overall price efiect into three parts.
1,The Substitution Efiect,The change in demand due to relative price changes.
2,The Income Efiect,The change in demand due to changes in purchasing power with money income constant.
3,The Endowment Efiect,The change in demand due to changes in the value of the endowment.
When added together these three efiects give the price efiect:
Price Efiect = Substitution Efiect +Income Efiect +Endowment Efiect
The rest of the lecture examines this simple equation | the Slutsky decomposition | in some depth.
Consumption | Comparative Statics 6
A Price Change
Initially suppose the consumer has flxed money income m,Throughout the lecture a decrease in p1 will be
examined,An increase in p1 or a change in p2 can be analysed in a similar way.
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A decrease in p1 pivots the budget line outward resulting in a new optimal choice xb.
This is illustrated in the above graph,The consumer alters consumption of good 1 from xa1 to xb1,Notice the second
diagram also shows this but without indifierence curves,Often,indifierence curves can be suppressed and only
optimal choices drawn in,This simplifles the diagrams a great deal.
Consumption | Comparative Statics 7
The Substitution Efiect 1
Suppose the consumer’s optimal choice changes from xa to xb due to a decrease in p1.
The change in demand for good 1 is ￠x1 = xb1?xa1,This change can be decomposed into two parts | a
substitution efiect and an income efiect.
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Suppose the consumer’s money income m was changed so that they could still just afiord to buy their initially
chosen bundle,xa,Their initial budget line was A which pivoted to B following the decrease in p1,Now some of
their income is \taken away" resulting in budget line C above.
At budget line C only the relative price change has taken place,Furthermore,given budget line C the consumer
might no longer choose bundle xa,They could prefer a bundle further down C,The next slide illustrates this.
Consumption | Comparative Statics 8
The Substitution Efiect 2
Suppose at the new budget line C,the consumer purchases bundle xc = (xc1; xc2) as in the graph below.
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0 0
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The substitution efiect is then the change in demand due to the relative price change alone,given by:
￠xs1 = xc1?xa1
Notice that xc is further down budget line C than xa,It could not go further up the budget line,Why not?
Because of this,the substitution efiect is always in the opposite direction to the price change.
Consumption | Comparative Statics 9
The Income Efiect
Returning to the consumer the income previously \taken away" results in another optimal choice | xb.
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The income efiect is then the change in demand due to the change in purchasing power alone,given by:
￠xn1 = xb1?xc1
If good 1 is normal,xb will lie to the right of xc,If good 1 is inferior,xb will lie to the left of xc,The income efiect
can be in the opposite direction to the substitution efiect (an inferior good) or in the same direction (a normal
good),The diagram above illustrates a normal good.
Gifien goods arise when the good is inferior and the income efiect is large enough to outweigh the substitution
efiect,In this case xb lies to the left of xa | Gifien goods are always inferior.
Consumption | Comparative Statics 10
The Law of Demand
The Slutsky decomposition can be written as ￠x1 = ￠xs1 +￠xn1.
Although ￠x1 could be of either sign | the good could be ordinary or Gifien,something can be said about the
relationship between ￠x1 and ￠xn1 with the current assumptions in place.
The substitution efiect is always in the opposite direction to the price change so the following is true.
The Law of Demand,If demand for a good increases when income increases then demand for that good must
decrease when its price increases.
Consumption | Comparative Statics 11
The Endowment Efiect
Up until now the consumer’s income has been a flxed number m,When the consumer receives an endowment !,
there is a third efiect to take into consideration | the endowment efiect.
Recall a price change will pivot the budget line through the endowment point,The flrst graph shows a fall in p1,!
is the endowment,xa the initially chosen bundle and xd the bundle chosen after the price change.
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The second graph shows the decomposition,Budget line A represents the initial prices,Budget line B corresponds
to the substitution efiect,C to the income efiect and D to the endowment efiect.
The third graph shows the three efiects,￠xs1 = xb1?xa1 is the substitution efiect,￠xn1 = xc1?xb1 is the income
efiect,￠x!1 = xd1?xc1 is the endowment efiect,So:
￠x1 = ￠xs1 +￠xn1 +￠x!1 = xd1?xa1
Consumption | Comparative Statics 12
Buying and Selling
Remember that the endowment efiect is a change in demand due to a shift in the budget line,therefore its sign is
determined by whether the good is normal or inferior.
On the last graph,xd lies to the right of xb and to the left of xc,This is because the good is normal,The income
and endowment efiects are related in this way,Take care | xd could not lie to the left of xb,for example.
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0
x2
x1 0
x2
x1
A D
A
Dxa
xd
!
xd
!
xa
Given a fall in p1,budget line A moves to budget line D in both graphs,Optimal choice changes from xa to xd.
In the flrst graph,the consumer was a net buyer of good 1 | after p1 falls they always remain a net buyer.
In the second graph,the consumer was a net seller of good 1 | after p1 falls they need not be.
If they continue to be a net seller after a price fall they are worse ofi,If they are a net buyer and there is a price fall
they are better ofi,What about a price rise? Net sellers continue to sell and are better ofi,Buyers need not
continue to buy,but if they do they are worse ofi,Why?
