Consumption | Applications 1
Hicks Decomposition
There is another way to decompose the price efiect,For simplicity consider a consumer with flxed income m.
The Hicks decomposition involves pivoting the budget line around the initial indifierence curve rather than the
initial bundle,The diagram below illustrates for a price decrease in good 1.
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The consumer now chooses (as before) xb1 on the outermost budget line,The Hicks substitution efiect is the change
in demand due to the relative price change given a constant level of utility rather than income.
Notice the consumer is indifierent between bundles xa and xc,On this new parallel budget line,the consumer can
no longer afiord xa | unlike before,The Hicks substitution efiect is then given by ￠xs1 = xc1?xa1.
The Hicks income efiect is ￠xn1 = xb1?xc1,These can be difierent in size from the Slutsky efiects,The price efiect is
the same of course,This means a Slutsky inferior good is not always a Hicks inferior good and vice-versa.
The substitution efiect is still always in the opposite direction to the price change,Why?
Consumption | Applications 2
Some Examples
Returning to Slutsky decompositions,consider the cases of perfect complements and perfect substitutes.
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0 0
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In both cases there is a decrease in p1,The consumer changes from xa1 to xb1.
The flrst graph shows the perfect complement case | it is all income efiect,The substitution efiect is zero.
The second graph shows the perfect substitute case | it is all substitution efiect,The income efiect is zero.
Note the Slutsky and Hicks decompositions are identical for both cases,Why?
Consumption | Applications 3
Rates of Change
Mathematically,the Slutsky decomposition can be written in the following way.
￠x1 = ￠xs1 + ￠xn1 +￠x!1
What is the rate of change of demand with respect to price,￠x1=￠p1?
Suppose price changes from pa1 to pb1,The old and new amounts of money income required to purchase x are
ma = pa1x1 + p2x2 and mb = pb1x1 + p2x2 respectively,Subtracting one from the other gives the change in money
income required to enable the consumer to just afiord bundle x = (x1;x2):
￠m = mb?ma = x1(pb1?pa1) = x1￠p1
For convenience,deflne ￠xm1 =?￠xn1,the negative of the income efiect,￠xm1 =￠m is then the change in demand
when income changes,So ￠xn1=￠p1 =?￠xm1 =￠p1 =?x1￠xm1 =￠m.
The endowment income efiect is given by the change in demand when income changes multiplied by the change in
income when price changes,The flrst term is ￠xm1 =￠m.
The second term is ￠m=￠p1,Now,ma = pa1!1 + p2!2 and mb = pb1!1 + p2!2,So ￠m = mb?ma = !1￠p1.
Therefore ￠m=￠p1 = !1,Finally:
￠x1
￠p1 =
￠xs1
￠p1 +
￠xn1
￠p1 +
￠x!1
￠p1 =
￠xs1
￠p1 +(!1?x1)
￠xm1
￠m
This is the standard way to write the Slutsky equation | usually with difierentials,It can be applied easily.
Consumption | Applications 4
Tax Rebates
The government decides to tax beer,They return all the revenue to the consumer | making them no worse ofi?
The amount of beer bought before the tax was b,The amount of other consumption was c,Make ‘other
consumption’ the numeraire,The price of beer is p,Adding a tax to the price of beer efiectively increases the price
the consumer faces to p + t.
The consumer will change consumption due to the price change and due to their increased income from the tax
rebate,Suppose the new bundle is (b0;c0),The government raises tb0 in tax.
Income before the tax was m,After the tax rebate they will get m + tb0,The budget line before the tax was
m = pb + c,After the tax it is m + tb0 = (p + t)b0 + c0,This equation simplifles to m = pb0 + c0.
So bundle (b0;c0) was afiordable when the consumer chose bundle (b;c),They cannot prefer it and must be worse ofi.
Consumption | Applications 5
Labour Supply | Budgets
Suppose a consumer has savings m,Income is spent on c lots of consumption goods | the numeraire good,The
consumer works for L hours,earning wage w per hour,There are only L hours available.
The budget line is given by c = m + wL,This can be rewritten as c + w(L?L) = m + wL,This has a better
interpretation,w,the wage,is the price of leisure and m + wL is the endowment income.
Since both are goods plot leisure against consumption to give the budget set.
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Leisure = L?L
L
BS
Budget line slope =?w
Endowment
m
m + wL
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Notice the consumer cannot have more than L hours of leisure or work.
Consumption | Applications 6
Labour Supply | Choice
Since the consumer values consumption and leisure,both are goods,Indifierence curves are easily drawn.
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Leisure = L?L
L?L?L?
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Reading from the graph,L?L? is the optimal amount of leisure and L? is the optimal amount of labour supplied.
c? is the optimal amount of consumption,Notice if Leisure = L,no labour is supplied | a corner solution.
It is usually assumed that leisure is a normal good,Leisure time should increase as income increases.
What happens when wages increase?
Consumption | Applications 7
The Labour Supply Curve
As wages increase the budget line becomes steeper as in the diagram below,Indifierence curves are again suppressed.
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LLeisure
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As wages increase from w1 to w2,choice of L rises initially (as might be expected) from L?1 to L?2,An increase in
the wage rate causes the consumer to supply more labour.
However as wages rise further from w2 to w3,labour supply falls from L?2 back to L?1,A wage increase causes the
consumer to supply less labour to the market.
The price efiect can be decomposed to show how this happens,The endowment income efiect of the wage rise
outweighs the substitution efiect,The consumer chooses to work less,\taking advantage" of the higher wage.
A backward bending labour supply curve is generated in the second graph.
Consumption | Applications 8
Intertemporal Budgets
The model developed so far can be used to analyse choices over time | intertemporal choice.
There are two goods,consumption today and consumption tomorrow,The consumer buys c1 units today and c2
tomorrow at a constant price level of 1,(This will be allowed to vary later).
The consumer receives an income of m1 today and m2 tomorrow but can save (and borrow) at an interest rate r.
The amount that can be consumed next period is c2? m2 +(1 + r)(m1?c1),So the budget line is given by:
c1 + c21 + r = m1 + m21+ r
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c1
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c1
m1
m2
m1
m2
BS BS
Endowment
Slope =?(1+ r) Slope =?(1+ r)
m1 + m2=(1 + r)
(1 + r)m1 + m2
(Future value)
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The flrst graph shows the budget set with no borrowing,The second shows the budget set when there is.
Consumption | Applications 9
Intertemporal Preferences
The case of perfect substitutes represents someone who doesn’t care whether they consume today or tomorrow.
The case of perfect complements represents someone who wants to consume exactly the same amount on each day.
Concavity is quite appropriate | the consumer wishes to split consumption over the two days,Averages (more
equal amounts of consumption on each day) are preferred to extremes (a more unequal split over the two days).
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Consumption | Applications 10
Intertemporal Choice
Optimal choice takes place as before,The consumer is either a lender or a borrower.
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The flrst graph shows the case of a lender,the second a borrower,The flrst agent consumes less than they start
with today and more tomorrow,the second agent consumes more than they start with today and less tomorrow.
Notice how this corresponds with net buyers and sellers when discussing endowments in earlier examples.
If the consumer is a lender and r rises they are better ofi and they continue to be a lender,If the consumer is a
borrower and r rises,they need not continue to be a borrower | but if they do they are worse ofi.
Consumption | Applications 11
In ation
So far,consumption prices were assumed to be 1,Allowing for in ation in the price level is straightforward.
The price today is 1 and tomorrow is p2,Tomorrow,the consumer can spend p2c2? p2m2 +(1 + r)(m1?c1).
The in ation rate,…,is the increase in the price level,p2 = 1 + …,Therefore the budget line equation is:
c2 = m2 + 1 + r1+ …(m1?c1) () c1 + c21 + ‰ = m1 + m21 + ‰
Where 1 + ‰ = (1+ r)=(1 + …) | ‰ is the real interest rate,r is the nominal interest rate.
Rearranging to flnd ‰ gives ‰ = (r?…)=(1+ …) … r?…,Often the second approximation is used.
Consumption | Applications 12
Present Value
Ignoring in ation,how much is a pound tomorrow worth in terms of pounds today? What is its present value?
A pound today is worth 1 + r tomorrow (from saving it | its future value),so the present value of a pound
tomorrow is 1=(1 + r),Budget sets are when \present value of consumption is at most present value of income".
Present value can be generalised to many periods,If there were 3 days,the present value of an income m3 on day 3
would be m3=(1 + r)2,(If m3 were saved on day 1 it would grow to m3(1 + r) on day 2,and m3(1 + r)2 on day 3).
If there were t periods,the present value (PV ) of an income stream m1;:::;mt is:
PV = m1 + m21 + r + m3(1+ r)2 +￠￠￠+ mt(1 + r)t?1 () PV =
tX
i=1
mi
(1+ r)i?1
This is the correct way to value income streams,It is easy to generalise to allow for interest rates that change over
time,t can be flnite or inflnite,This formula will appear again and again | remember it.
Hicks Decomposition
There is another way to decompose the price efiect,For simplicity consider a consumer with flxed income m.
The Hicks decomposition involves pivoting the budget line around the initial indifierence curve rather than the
initial bundle,The diagram below illustrates for a price decrease in good 1.
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The consumer now chooses (as before) xb1 on the outermost budget line,The Hicks substitution efiect is the change
in demand due to the relative price change given a constant level of utility rather than income.
Notice the consumer is indifierent between bundles xa and xc,On this new parallel budget line,the consumer can
no longer afiord xa | unlike before,The Hicks substitution efiect is then given by ￠xs1 = xc1?xa1.
The Hicks income efiect is ￠xn1 = xb1?xc1,These can be difierent in size from the Slutsky efiects,The price efiect is
the same of course,This means a Slutsky inferior good is not always a Hicks inferior good and vice-versa.
The substitution efiect is still always in the opposite direction to the price change,Why?
Consumption | Applications 2
Some Examples
Returning to Slutsky decompositions,consider the cases of perfect complements and perfect substitutes.
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0 0
x2
x1
x2
x1
xa1 xb1 xa1 xb1
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......
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.....
In both cases there is a decrease in p1,The consumer changes from xa1 to xb1.
The flrst graph shows the perfect complement case | it is all income efiect,The substitution efiect is zero.
The second graph shows the perfect substitute case | it is all substitution efiect,The income efiect is zero.
Note the Slutsky and Hicks decompositions are identical for both cases,Why?
Consumption | Applications 3
Rates of Change
Mathematically,the Slutsky decomposition can be written in the following way.
￠x1 = ￠xs1 + ￠xn1 +￠x!1
What is the rate of change of demand with respect to price,￠x1=￠p1?
Suppose price changes from pa1 to pb1,The old and new amounts of money income required to purchase x are
ma = pa1x1 + p2x2 and mb = pb1x1 + p2x2 respectively,Subtracting one from the other gives the change in money
income required to enable the consumer to just afiord bundle x = (x1;x2):
￠m = mb?ma = x1(pb1?pa1) = x1￠p1
For convenience,deflne ￠xm1 =?￠xn1,the negative of the income efiect,￠xm1 =￠m is then the change in demand
when income changes,So ￠xn1=￠p1 =?￠xm1 =￠p1 =?x1￠xm1 =￠m.
The endowment income efiect is given by the change in demand when income changes multiplied by the change in
income when price changes,The flrst term is ￠xm1 =￠m.
The second term is ￠m=￠p1,Now,ma = pa1!1 + p2!2 and mb = pb1!1 + p2!2,So ￠m = mb?ma = !1￠p1.
Therefore ￠m=￠p1 = !1,Finally:
￠x1
￠p1 =
￠xs1
￠p1 +
￠xn1
￠p1 +
￠x!1
￠p1 =
￠xs1
￠p1 +(!1?x1)
￠xm1
￠m
This is the standard way to write the Slutsky equation | usually with difierentials,It can be applied easily.
Consumption | Applications 4
Tax Rebates
The government decides to tax beer,They return all the revenue to the consumer | making them no worse ofi?
The amount of beer bought before the tax was b,The amount of other consumption was c,Make ‘other
consumption’ the numeraire,The price of beer is p,Adding a tax to the price of beer efiectively increases the price
the consumer faces to p + t.
The consumer will change consumption due to the price change and due to their increased income from the tax
rebate,Suppose the new bundle is (b0;c0),The government raises tb0 in tax.
Income before the tax was m,After the tax rebate they will get m + tb0,The budget line before the tax was
m = pb + c,After the tax it is m + tb0 = (p + t)b0 + c0,This equation simplifles to m = pb0 + c0.
So bundle (b0;c0) was afiordable when the consumer chose bundle (b;c),They cannot prefer it and must be worse ofi.
Consumption | Applications 5
Labour Supply | Budgets
Suppose a consumer has savings m,Income is spent on c lots of consumption goods | the numeraire good,The
consumer works for L hours,earning wage w per hour,There are only L hours available.
The budget line is given by c = m + wL,This can be rewritten as c + w(L?L) = m + wL,This has a better
interpretation,w,the wage,is the price of leisure and m + wL is the endowment income.
Since both are goods plot leisure against consumption to give the budget set.
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Leisure = L?L
L
BS
Budget line slope =?w
Endowment
m
m + wL
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Notice the consumer cannot have more than L hours of leisure or work.
Consumption | Applications 6
Labour Supply | Choice
Since the consumer values consumption and leisure,both are goods,Indifierence curves are easily drawn.
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Leisure = L?L
L?L?L?
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Reading from the graph,L?L? is the optimal amount of leisure and L? is the optimal amount of labour supplied.
c? is the optimal amount of consumption,Notice if Leisure = L,no labour is supplied | a corner solution.
It is usually assumed that leisure is a normal good,Leisure time should increase as income increases.
What happens when wages increase?
Consumption | Applications 7
The Labour Supply Curve
As wages increase the budget line becomes steeper as in the diagram below,Indifierence curves are again suppressed.
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c w
LLeisure
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w1
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As wages increase from w1 to w2,choice of L rises initially (as might be expected) from L?1 to L?2,An increase in
the wage rate causes the consumer to supply more labour.
However as wages rise further from w2 to w3,labour supply falls from L?2 back to L?1,A wage increase causes the
consumer to supply less labour to the market.
The price efiect can be decomposed to show how this happens,The endowment income efiect of the wage rise
outweighs the substitution efiect,The consumer chooses to work less,\taking advantage" of the higher wage.
A backward bending labour supply curve is generated in the second graph.
Consumption | Applications 8
Intertemporal Budgets
The model developed so far can be used to analyse choices over time | intertemporal choice.
There are two goods,consumption today and consumption tomorrow,The consumer buys c1 units today and c2
tomorrow at a constant price level of 1,(This will be allowed to vary later).
The consumer receives an income of m1 today and m2 tomorrow but can save (and borrow) at an interest rate r.
The amount that can be consumed next period is c2? m2 +(1 + r)(m1?c1),So the budget line is given by:
c1 + c21 + r = m1 + m21+ r
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m2
m1
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BS BS
Endowment
Slope =?(1+ r) Slope =?(1+ r)
m1 + m2=(1 + r)
(1 + r)m1 + m2
(Future value)
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The flrst graph shows the budget set with no borrowing,The second shows the budget set when there is.
Consumption | Applications 9
Intertemporal Preferences
The case of perfect substitutes represents someone who doesn’t care whether they consume today or tomorrow.
The case of perfect complements represents someone who wants to consume exactly the same amount on each day.
Concavity is quite appropriate | the consumer wishes to split consumption over the two days,Averages (more
equal amounts of consumption on each day) are preferred to extremes (a more unequal split over the two days).
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Consumption | Applications 10
Intertemporal Choice
Optimal choice takes place as before,The consumer is either a lender or a borrower.
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The flrst graph shows the case of a lender,the second a borrower,The flrst agent consumes less than they start
with today and more tomorrow,the second agent consumes more than they start with today and less tomorrow.
Notice how this corresponds with net buyers and sellers when discussing endowments in earlier examples.
If the consumer is a lender and r rises they are better ofi and they continue to be a lender,If the consumer is a
borrower and r rises,they need not continue to be a borrower | but if they do they are worse ofi.
Consumption | Applications 11
In ation
So far,consumption prices were assumed to be 1,Allowing for in ation in the price level is straightforward.
The price today is 1 and tomorrow is p2,Tomorrow,the consumer can spend p2c2? p2m2 +(1 + r)(m1?c1).
The in ation rate,…,is the increase in the price level,p2 = 1 + …,Therefore the budget line equation is:
c2 = m2 + 1 + r1+ …(m1?c1) () c1 + c21 + ‰ = m1 + m21 + ‰
Where 1 + ‰ = (1+ r)=(1 + …) | ‰ is the real interest rate,r is the nominal interest rate.
Rearranging to flnd ‰ gives ‰ = (r?…)=(1+ …) … r?…,Often the second approximation is used.
Consumption | Applications 12
Present Value
Ignoring in ation,how much is a pound tomorrow worth in terms of pounds today? What is its present value?
A pound today is worth 1 + r tomorrow (from saving it | its future value),so the present value of a pound
tomorrow is 1=(1 + r),Budget sets are when \present value of consumption is at most present value of income".
Present value can be generalised to many periods,If there were 3 days,the present value of an income m3 on day 3
would be m3=(1 + r)2,(If m3 were saved on day 1 it would grow to m3(1 + r) on day 2,and m3(1 + r)2 on day 3).
If there were t periods,the present value (PV ) of an income stream m1;:::;mt is:
PV = m1 + m21 + r + m3(1+ r)2 +￠￠￠+ mt(1 + r)t?1 () PV =
tX
i=1
mi
(1+ r)i?1
This is the correct way to value income streams,It is easy to generalise to allow for interest rates that change over
time,t can be flnite or inflnite,This formula will appear again and again | remember it.
