Term 2 Summary - Microeconomics (ECON0013)

1. Budget constraints and consumer demand (10 pages)
1.1. Budget constraint
 Budget set: all economically viable combinations of consumption.
 Total spending has to remain within the budget y: p1 q1 + p2 q2 +…+ pm q m=∑ pi q i= p' q, where p ’ q is
i
the vector product of price and quantity.
 Budget line:
−p 1
o Slope: rate at which you can increase consumption of q 2 as you reduce consumption of q1  .
p2
 Linear budget constraint:

o
∂ qi
Slope:
∂qj B( )
| =−p j / pi  constant independent of budget.

y
o Maximum affordable quantity of any commodity: .
pi
 When budget changes:
o Budget constraint shifts in or out; slope does not change.
o Intercepts change.
 When price changes:
o Budget constraint rotates around the intercept of the q 1 axis.
o Slope changes: if p1 increases, q 1 end of diagram changes.
 y / p1 shifts in  total amount of p1 we can afford
falls.
−p 1
 Slope becomes steeper  becomes more
p2
negative.
 If the ratio of prices remain the same after a price change, the slope does not change.
 Taxation/Non-linear pricing – budget constraints are frequently kinked or discontinuous.
o Taxation in excess of E:
 Suppose q_1 is subject to tax t if q 1> E
 Beyond the line E, the consumer is happy to pay tax, so the price of q_1 increases from p1 to
p1 +t .





 Equations:
 If q 1 ≤ E , p 1 q 1+ p2 q 2= y
 If q 1> E , p1 E+ ( p1 +t )( q1−E )− p2 q2 = y
 ( p1 +t ) q1 + p2 q2= y +tE
o Tax paid on the whole:

, o




o
o There would be discontinuity if the discount applied to any amount of the goods. If it only applied
above a threshold, the budget constraint would be continuous.
1.2. Marshallian demands – uncompensated demands
 Income expansion path: path traced out by demands as y increases but prices are constant.
 Engel curve: the graph of f i ( y , p) as a function of y.




o
 Offer curve: path traced out by demands as pi increases.




o
 Demand curve: the graph of f i ( y , p) as a function of pi.

, o
o The uncompensated demand for a good is increasing in total budget IFF the compensated demand is
increasing in utility.

 Uncompensated own price elasticity: ηii = ( )
pi ∂ qi
=
∂ ln qi
qi ∂ pi ∂ ln pi
.

o Price elastic vs price inelastic: budget share of a good as price rises/falls (1. ηii ←1, 2. ηii >−1)
o Giffen good: if uncompensated demand for a good rises with own price ηii >0.

 Uncompensated cross price elasticity: ηij = ( )
pj ∂ qi
=
∂ ln qi
qi ∂ p j ∂ ln p j
.

o Substitute: uncompensated demand for a good rises with the price of another  ηij >0 .
o Complement: uncompensated demand for a good falls with the price of another  ηij <0.

 Total budget elasticity (TBE): ϵ i= ( )
y ∂ q i ∂ ln qi
qi ∂ y
=
∂ lny
.

o Normal (TBE>0) vs Inferior: amount of good consumed as y increases (1. Rises [ ϵ i >0 ], 2. Falls [ϵ i <0
]).
∂f ∂g
 Normal good: >0 and >0
∂y ∂υ
o Luxury (TBE>1) vs Necessity: share of income as y increases (1. Rises [ϵ i >1], 2. Falls [ϵ i <1]).
 Luxury: budget share ( pi f i ( y , p) / y is increasing in y).





o Cobb-Douglas: TBE=-1 for all goods; all cross price elasticities=0
1.3. Adding up
 Adding up: if consumer spending (demand) exhausts the total budget set, then p' f ( y , p ) = y .
o Not all goods can be inferior.
 If a good is inferior then income and substitution effects are in opposite directions. The
uncompensated demand curve can slope either way.
o Not all goods can be luxuries.
o Not all goods can be necessities.
o Demand systems: not possible for all goods to have constant income elasticities unless they are all 1.
 Requires that a budget share weighted average of TBE=1.