Consumption | Budgets 1
The Budget Set
The last lecture investigated which bundle of goods the consumer prefers,However,goods cost money and the
consumer cannot afiord to buy indeflnite amounts of each good.
Suppose the consumer has an income of m,Then the amount of money the consumer spends cannot exceed m.
Suppose that the two goods have (non-negative) prices p1 and p2 respectively,The total amount of money spent on
x1 units of good 1 is then p1x1 and on x2 units of good 2 is p2x2,This yields the equation:
p1x1 + p2x2? m
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BS
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Usually it is impossible to consume negative amounts of goods and so to be exact the budget set includes two other
constraints,x1?0 and x2?0,The budget set is therefore the triangle BS in the diagram.
Consumption | Budgets 2
Properties of Budget Sets
When prices change the budget set will clearly change,The budget line is the diagonal boundary line of the budget
set,given by p1x1 + p2x2 = m.
The intercepts with the x1 and x2 axis can be calculated,For example,when x2 =0 the equation reduces to
x1 = m=p1,This is the intercept with the x1 axis.
The slope of the budget line can be calculated by dividing the vertical axis intercept by the horizontal axis intercept
yielding?p1=p2 (recall this is a downward sloping straight line).
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x2 = m=p2 Budget Line,slope =?p1=p2
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Now it is possible to investigate what will happen to the budget set when prices change.
Consumption | Budgets 3
Changing Prices and Income
Suppose income m is increased,In this case,the budget line shifts outward as below.
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m Increases p1 Increases
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Suppose one of the prices p1 increases,The intercept of the budget line with the horizontal axis decreases and the
budget pivots inwards,The same operation can be done for a price decrease,where the budget line pivots outward.
Changes in p2 are treated in a similar way.
Budget sets increase in size if income increases and decrease in size if either of the prices increase | as expected.
Notice the slope changes when prices change,The slope represents the market rate of exchange between the two
goods or the opportunity cost of consuming good 1,It is the rate at which the market substitutes good 1 for good 2.
Consumption | Budgets 4
Taxes and the Budget Set
What happens to the budget set in the case of taxes,subsidies and rationing?
1,Quantity Tax,For each unit (of good 1,say) the consumer has to pay a tax of size t,This is just like a price
rise | the consumer now pays p1 + t.
2,Ad Valorem Tax,This is a tax on the value of the purchase,like VAT,Say the tax rate is set at t then the
consumer now pays (1+ t)p1 for each unit,p1 to the producer and tp1 to the state.
3,Lump Sum Tax,The consumer now would have to pay a total of t to the government regardless of quantity
bought,Hence income is reduced to m?t but prices are unafiected.
4,Subsidies,Subsidies are simply the opposite of taxes and can take any of the three forms above.
5,Rationing,Consumers are not allowed more than a certain level of a particular good | say r1 for good 1.
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Rationing Variable Taxes
r1
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Slope =?p1=p2
Slope =?(1+ t)p1=p2
The flrst picture shows simple rationing,The second shows an ad valorem tax that only operates if the consumer
purchases more than r1 of the good,All forms of tax afiect the budget line | the flrst two operate like a price
change,the third like an income change,Subsidies work in reverse.
Consumption | Budgets 5
The Numeraire
A good is often referred to as a numeraire,What does this mean?
Prices are relative things,They are exchange rates between goods,revealing how much of good 1 is required to buy
some of good 2 and so on,If there were only 1 good,the price would be meaningless.
In the model so far there are three variables,p1,p2 and m,One of these is redundant,For example,setting p2 =1
does not alter anything,Mathematically:
p1x1 + p2x2 = m =) p1p
2
x1 + x2 = mp
2
This flnal equation can be written as px1 + x2 = y where p = p1=p2 and y = m=p2,This formulation contains the
same information as before but is simpler,It is equivalent to setting p2 =1,Good 2 is the numeraire.
All prices are in terms of the price of the numeraire,Notice?p is the slope of the budget line,It is the price of
good 1 in terms of good 2,y is the value of income in terms of the numeraire.
Consumption | Budgets 6
Endowments
Suppose the consumer didn’t start with an income m,but rather had an endowment of good 1 and of good 2,How
would this alter the budget set?
The consumer starts with !1 of good 1 and !2 of good 2,But a unit of good 1 is worth p1 and of good 2 is worth
p2,The total worth of the consumers endowment is p1!1 + p2!2,The endowment is written as ! =(!1; !2).
The total amount spent cannot exceed the value of the endowment,Hence:
p1x1 + p2x2? p1!1 + p2!2
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x1 =(p1!1 + p2!2)=p1
x2 =(p1!1 + p2!2)=p2 Budget Line,slope =?p1=p2
!1
!2
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The slope clearly remains the same,The endowment lies on the budget line,This is simply because the endowment
is always just afiordable | by deflnition,Price changes now alter the budget set in a more complicated way.
Consumption | Budgets 7
Price Changes and Endowments
An increase in either of the endowments is just like an increase in income,In the graph below the endowment of
good 1 increases from !1 to !01,The endowment of good 2 and both prices remain constant.
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Slope =?p1=p2 Old slope =?p1=p2
New slope =?p01=p2
!1
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!01 !1
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An increase in the price of good 1 is slightly more complicated,Suppose the price increases from p1 to p01,Recall
the slope of the budget line is given by?p1=p2,This will increase (in absolute value) and hence the budget line
becomes steeper.
However,it does not pivot around the intercept with the vertical axis,as before,The endowment must still be just
afiordable and so the budget line will still intersect with that point,as illustrated in the second graph above.
Notice that if !1 increased and !2 decreased (or vice-versa) it is possible for the budget line to shift outward,
inward or even remain where it started.
Consumption | Budgets 8
Optimal Choice
An economic model of consumption | \Consumers choose the most preferred bundle from their budget sets."
This can be illustrated graphically by the following important picture.
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Higher indifierence curves correspond to higher utility levels,Therefore,the consumer chooses the bundle from the
budget set which lies on the highest indifierence curve,This is called the optimal choice | x? =(x?1; x?2).
Consumption | Budgets 9
The MRS Condition
In the case on the last slide the solution to the consumer’s problem is interior | optimal choice is given by a
tangency condition,The slope of the budget line at the solution is equal to the slope of the indifierence curve.
Recall both slopes had an interpretation,one was the price ratio and one the marginal rate of substitution.
MRS=?p1p
2
If preferences are convex and monotonic and the above MRS condition is satisfled at a particular point then that
point represents the optimal choice for the consumer.
With convex preferences the tangency condition is su–cient for optimality,The condition itself says that the
internal (private) rate of exchange | the MRS | equals the external (market) rate of exchange | the price ratio.
Everyone consuming the goods has the same MRS regardless of their preferences.
Consumption | Budgets 10
Corner Solutions
The MRS condition is not necessary for a solution with convex and monotonic preferences,In other words,a point
which represents an optimal choice does not imply the tangency condition at that point.
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In the flrst graph there is a corner solution | sometimes called a boundary solution,In the second graph,an
optimal solution with a \kinky" indifierence curve need not correspond to a tangency.
These examples have convex preferences,If preferences are non-convex the tangency condition is no longer su–cient.
Consumption | Budgets 11
Some Examples
Difierent preferences result in difierent optimal choices,Here are some examples.
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Non-convex preferences Perfect complements Perfect substitutes
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The flrst diagram illustrates an example of non-convex preferences,The tangency point is not the optimum.
The second is the case of perfect complements,The consumer purchases an equal amount of both goods.
The third is the case of perfect substitutes,The consumer spends all income on the cheaper of the two goods.
Consumption | Budgets 12
The Dual
The problem the consumer faces can be written mathematically as:
maxu(x1; x2) s.t,p1x1 + p2x2? m
The solution is a demand function telling the consumer how much of each good to purchase given prices and
income,for example x1 = x1(p1; p2; m) and x2 = x2(p1; p2; m).
The problem could be solved from an alternative perspective,Suppose the consumer wanted to know what the
minimum expenditure was to achieve a particular utility level u,then:
minp1x1 + p2x2 s.t,u(x1; x2)? u
This problem is called the Dual,Both are solved by the Lagrangian technique,The mathematics lectures will cover
this method,The diagram below illustrates the fact that both problems generate the same solution x?.
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The Budget Set
The last lecture investigated which bundle of goods the consumer prefers,However,goods cost money and the
consumer cannot afiord to buy indeflnite amounts of each good.
Suppose the consumer has an income of m,Then the amount of money the consumer spends cannot exceed m.
Suppose that the two goods have (non-negative) prices p1 and p2 respectively,The total amount of money spent on
x1 units of good 1 is then p1x1 and on x2 units of good 2 is p2x2,This yields the equation:
p1x1 + p2x2? m
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BS
x2
x1
Usually it is impossible to consume negative amounts of goods and so to be exact the budget set includes two other
constraints,x1?0 and x2?0,The budget set is therefore the triangle BS in the diagram.
Consumption | Budgets 2
Properties of Budget Sets
When prices change the budget set will clearly change,The budget line is the diagonal boundary line of the budget
set,given by p1x1 + p2x2 = m.
The intercepts with the x1 and x2 axis can be calculated,For example,when x2 =0 the equation reduces to
x1 = m=p1,This is the intercept with the x1 axis.
The slope of the budget line can be calculated by dividing the vertical axis intercept by the horizontal axis intercept
yielding?p1=p2 (recall this is a downward sloping straight line).
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x2
x10
x1 = m=p1
x2 = m=p2 Budget Line,slope =?p1=p2
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Now it is possible to investigate what will happen to the budget set when prices change.
Consumption | Budgets 3
Changing Prices and Income
Suppose income m is increased,In this case,the budget line shifts outward as below.
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x2
x1
x2
x10 0
m Increases p1 Increases
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Suppose one of the prices p1 increases,The intercept of the budget line with the horizontal axis decreases and the
budget pivots inwards,The same operation can be done for a price decrease,where the budget line pivots outward.
Changes in p2 are treated in a similar way.
Budget sets increase in size if income increases and decrease in size if either of the prices increase | as expected.
Notice the slope changes when prices change,The slope represents the market rate of exchange between the two
goods or the opportunity cost of consuming good 1,It is the rate at which the market substitutes good 1 for good 2.
Consumption | Budgets 4
Taxes and the Budget Set
What happens to the budget set in the case of taxes,subsidies and rationing?
1,Quantity Tax,For each unit (of good 1,say) the consumer has to pay a tax of size t,This is just like a price
rise | the consumer now pays p1 + t.
2,Ad Valorem Tax,This is a tax on the value of the purchase,like VAT,Say the tax rate is set at t then the
consumer now pays (1+ t)p1 for each unit,p1 to the producer and tp1 to the state.
3,Lump Sum Tax,The consumer now would have to pay a total of t to the government regardless of quantity
bought,Hence income is reduced to m?t but prices are unafiected.
4,Subsidies,Subsidies are simply the opposite of taxes and can take any of the three forms above.
5,Rationing,Consumers are not allowed more than a certain level of a particular good | say r1 for good 1.
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Rationing Variable Taxes
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Slope =?p1=p2
Slope =?(1+ t)p1=p2
The flrst picture shows simple rationing,The second shows an ad valorem tax that only operates if the consumer
purchases more than r1 of the good,All forms of tax afiect the budget line | the flrst two operate like a price
change,the third like an income change,Subsidies work in reverse.
Consumption | Budgets 5
The Numeraire
A good is often referred to as a numeraire,What does this mean?
Prices are relative things,They are exchange rates between goods,revealing how much of good 1 is required to buy
some of good 2 and so on,If there were only 1 good,the price would be meaningless.
In the model so far there are three variables,p1,p2 and m,One of these is redundant,For example,setting p2 =1
does not alter anything,Mathematically:
p1x1 + p2x2 = m =) p1p
2
x1 + x2 = mp
2
This flnal equation can be written as px1 + x2 = y where p = p1=p2 and y = m=p2,This formulation contains the
same information as before but is simpler,It is equivalent to setting p2 =1,Good 2 is the numeraire.
All prices are in terms of the price of the numeraire,Notice?p is the slope of the budget line,It is the price of
good 1 in terms of good 2,y is the value of income in terms of the numeraire.
Consumption | Budgets 6
Endowments
Suppose the consumer didn’t start with an income m,but rather had an endowment of good 1 and of good 2,How
would this alter the budget set?
The consumer starts with !1 of good 1 and !2 of good 2,But a unit of good 1 is worth p1 and of good 2 is worth
p2,The total worth of the consumers endowment is p1!1 + p2!2,The endowment is written as ! =(!1; !2).
The total amount spent cannot exceed the value of the endowment,Hence:
p1x1 + p2x2? p1!1 + p2!2
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x1 =(p1!1 + p2!2)=p1
x2 =(p1!1 + p2!2)=p2 Budget Line,slope =?p1=p2
!1
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The slope clearly remains the same,The endowment lies on the budget line,This is simply because the endowment
is always just afiordable | by deflnition,Price changes now alter the budget set in a more complicated way.
Consumption | Budgets 7
Price Changes and Endowments
An increase in either of the endowments is just like an increase in income,In the graph below the endowment of
good 1 increases from !1 to !01,The endowment of good 2 and both prices remain constant.
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Slope =?p1=p2 Old slope =?p1=p2
New slope =?p01=p2
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An increase in the price of good 1 is slightly more complicated,Suppose the price increases from p1 to p01,Recall
the slope of the budget line is given by?p1=p2,This will increase (in absolute value) and hence the budget line
becomes steeper.
However,it does not pivot around the intercept with the vertical axis,as before,The endowment must still be just
afiordable and so the budget line will still intersect with that point,as illustrated in the second graph above.
Notice that if !1 increased and !2 decreased (or vice-versa) it is possible for the budget line to shift outward,
inward or even remain where it started.
Consumption | Budgets 8
Optimal Choice
An economic model of consumption | \Consumers choose the most preferred bundle from their budget sets."
This can be illustrated graphically by the following important picture.
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Higher indifierence curves correspond to higher utility levels,Therefore,the consumer chooses the bundle from the
budget set which lies on the highest indifierence curve,This is called the optimal choice | x? =(x?1; x?2).
Consumption | Budgets 9
The MRS Condition
In the case on the last slide the solution to the consumer’s problem is interior | optimal choice is given by a
tangency condition,The slope of the budget line at the solution is equal to the slope of the indifierence curve.
Recall both slopes had an interpretation,one was the price ratio and one the marginal rate of substitution.
MRS=?p1p
2
If preferences are convex and monotonic and the above MRS condition is satisfled at a particular point then that
point represents the optimal choice for the consumer.
With convex preferences the tangency condition is su–cient for optimality,The condition itself says that the
internal (private) rate of exchange | the MRS | equals the external (market) rate of exchange | the price ratio.
Everyone consuming the goods has the same MRS regardless of their preferences.
Consumption | Budgets 10
Corner Solutions
The MRS condition is not necessary for a solution with convex and monotonic preferences,In other words,a point
which represents an optimal choice does not imply the tangency condition at that point.
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In the flrst graph there is a corner solution | sometimes called a boundary solution,In the second graph,an
optimal solution with a \kinky" indifierence curve need not correspond to a tangency.
These examples have convex preferences,If preferences are non-convex the tangency condition is no longer su–cient.
Consumption | Budgets 11
Some Examples
Difierent preferences result in difierent optimal choices,Here are some examples.
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Non-convex preferences Perfect complements Perfect substitutes
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The flrst diagram illustrates an example of non-convex preferences,The tangency point is not the optimum.
The second is the case of perfect complements,The consumer purchases an equal amount of both goods.
The third is the case of perfect substitutes,The consumer spends all income on the cheaper of the two goods.
Consumption | Budgets 12
The Dual
The problem the consumer faces can be written mathematically as:
maxu(x1; x2) s.t,p1x1 + p2x2? m
The solution is a demand function telling the consumer how much of each good to purchase given prices and
income,for example x1 = x1(p1; p2; m) and x2 = x2(p1; p2; m).
The problem could be solved from an alternative perspective,Suppose the consumer wanted to know what the
minimum expenditure was to achieve a particular utility level u,then:
minp1x1 + p2x2 s.t,u(x1; x2)? u
This problem is called the Dual,Both are solved by the Lagrangian technique,The mathematics lectures will cover
this method,The diagram below illustrates the fact that both problems generate the same solution x?.
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