ECO3009F
Topic 2- Theoretical foundations
Topic 2a – Residential water demand
Water as a consumer good
Definitions & assumptions
The following notation is common when setting up a consumer Lagrange problem
Consists of a utility fn and budget constraint
• U ( x 1 x 2 )=utility function
• M¿ income , alternatively income is Y
• x i=good i , alternatively y i∨q i
• pi=the price of good i
• λ=Lagrange multiplier
• The subscript i runs from 1 ¿ n goods (usually only 2 goods)
The utility fn is a Cobb Douglas utility fn:
We assume that preferences are
• Complete (if you confront consumer with a list of goods, they will have an idea of
what every good means and they won’t be consuming any good that isn’t on the list)
• Transitive (if a person prefers apples to pears and pears to cucumbers, they will also
prefer apples to cucumbers)
• Continuous (always say if they prefer one over another. Indifference is a possibility)
• strictly monotonic (more is better)
• strictly convex (people tend to prefer mixtures of goods rather than extremes)
Utility maximisation is subject to a budget constraint:
Consumer Lagrange written in general terms
L=U ( x 1 , x 2 ) + λ ( M −p 1 x 1− p2 x 2 )
First order condition:
∂L
1. =U 1 ( x 1 , x 2 ) −λ p 1=0
∂ x1
The slope of the Lagrange expression, in the x1 direction, is 0
There are 2 terms
1st term: partial derivative of the utility fn with respect to x1 (subscript 1 on U)
2nd term: partial derivative of the 2nd term above
The slope must be equal to 0
1
, ∂L
2. =U 2 ( x1 , x2 ) −λ p2=0
∂ x2
The slope of the Lagrange expression, in the x1 direction, is 0
There are 2 terms
1st term: partial derivative of the utility fn with respect to x2 (subscript 2 on U)
2nd term: partial derivative of the 2nd term above
The slope must be equal to 0
∂L
3. =M − p1 x1 −p 2 x 2=0
∂λ
The partial derivative of the Lagrange fn, with respect to lambda
Simply gives us the bracket next to lambda
If we solve these 3 equations for the 2 unknowns (x1 & x2) we end up with a demand fn
Trying with real numbers: pg 1 & 2 written notes
- functionally separable=
- Marshallian demand function=
- Ordinary demand function=
- Expansion path=
- Marginal utility of money=
- Shadow price=
Solving the consumer Lagrange:
1. Set up the Lagrange. The utility function + λ (income bracket)
2. Set up the first order conditions for a maximum (= to zero). There is one for each
variable (good 1, good 2 and λ)
3. Divide FOC 1 by FOC 2. Lamba drops out and you get MRS = relative prices.
4. Rearrange to have x 2on the LHS and x 1on the RHS. This is called the expansion path.
5. Substitute expansion path into FOC 3
¿
6. Solve for x 1
¿ ¿
7. Insert x 1 into the expansion path and solve for x 2
Graphically:
Constrained budget in two dimensions
∂L
=MU 1 ( x 1 , x 2 )−λ p1 =0
∂ x1
∂L
=MU 2 ( x 1 , x 2 )− λ p 2=0
∂ x2
∂L
=M −p 1 x 1− p2 x 2=0
∂λ
2
, - Red= indifference curve
- Black= budget line
- These are derived from the FOC
- If all 3 of these FOC hold at the same time we are referring to the tangency
• slope of the budget line (p1/p2) = MRS (MU1/MU2)
From solid red to dashed red- the price of x1 is higher than it was before
Price corresponding to the dashed line is higher
Ordinary/ Marshallian demand fn
Instead of maximising utility subject to a fixed
budget, expenditure is minimised subject to a
fixed level of utility (indifference curve)
The solution is a Hicksian or compensated
demand function
This 2nd formulation you can think of holding the
green IC constant and asking how small the
expenditure triangle could be, provided it reaches
the green level of utility
Expenditure Minimisation
Page 3 of written notes
Hicksian demand function=
Maximising utility vs Minimising expenditure:
3
, Variables that belong in a model of residential water demand:
• Water price
• Other prices
• Household disposable income
• Habits e.g. gardening, car washing
• Size of the plot
• Washing machines, dishwashers
• Seasonal effects (water gardens more in summer, or take more showers in summer)
• Onsite water harvesting (Jo-jo tanks)
• Population- Number of people that live in residence (demand curve shifted out by
no.)
*look out if it is a population demand curve (seasonality, population size will play into it) or
household/ per capita curve (house hold size doesn’t matter)
Interpreting possible regression results:
4
Topic 2- Theoretical foundations
Topic 2a – Residential water demand
Water as a consumer good
Definitions & assumptions
The following notation is common when setting up a consumer Lagrange problem
Consists of a utility fn and budget constraint
• U ( x 1 x 2 )=utility function
• M¿ income , alternatively income is Y
• x i=good i , alternatively y i∨q i
• pi=the price of good i
• λ=Lagrange multiplier
• The subscript i runs from 1 ¿ n goods (usually only 2 goods)
The utility fn is a Cobb Douglas utility fn:
We assume that preferences are
• Complete (if you confront consumer with a list of goods, they will have an idea of
what every good means and they won’t be consuming any good that isn’t on the list)
• Transitive (if a person prefers apples to pears and pears to cucumbers, they will also
prefer apples to cucumbers)
• Continuous (always say if they prefer one over another. Indifference is a possibility)
• strictly monotonic (more is better)
• strictly convex (people tend to prefer mixtures of goods rather than extremes)
Utility maximisation is subject to a budget constraint:
Consumer Lagrange written in general terms
L=U ( x 1 , x 2 ) + λ ( M −p 1 x 1− p2 x 2 )
First order condition:
∂L
1. =U 1 ( x 1 , x 2 ) −λ p 1=0
∂ x1
The slope of the Lagrange expression, in the x1 direction, is 0
There are 2 terms
1st term: partial derivative of the utility fn with respect to x1 (subscript 1 on U)
2nd term: partial derivative of the 2nd term above
The slope must be equal to 0
1
, ∂L
2. =U 2 ( x1 , x2 ) −λ p2=0
∂ x2
The slope of the Lagrange expression, in the x1 direction, is 0
There are 2 terms
1st term: partial derivative of the utility fn with respect to x2 (subscript 2 on U)
2nd term: partial derivative of the 2nd term above
The slope must be equal to 0
∂L
3. =M − p1 x1 −p 2 x 2=0
∂λ
The partial derivative of the Lagrange fn, with respect to lambda
Simply gives us the bracket next to lambda
If we solve these 3 equations for the 2 unknowns (x1 & x2) we end up with a demand fn
Trying with real numbers: pg 1 & 2 written notes
- functionally separable=
- Marshallian demand function=
- Ordinary demand function=
- Expansion path=
- Marginal utility of money=
- Shadow price=
Solving the consumer Lagrange:
1. Set up the Lagrange. The utility function + λ (income bracket)
2. Set up the first order conditions for a maximum (= to zero). There is one for each
variable (good 1, good 2 and λ)
3. Divide FOC 1 by FOC 2. Lamba drops out and you get MRS = relative prices.
4. Rearrange to have x 2on the LHS and x 1on the RHS. This is called the expansion path.
5. Substitute expansion path into FOC 3
¿
6. Solve for x 1
¿ ¿
7. Insert x 1 into the expansion path and solve for x 2
Graphically:
Constrained budget in two dimensions
∂L
=MU 1 ( x 1 , x 2 )−λ p1 =0
∂ x1
∂L
=MU 2 ( x 1 , x 2 )− λ p 2=0
∂ x2
∂L
=M −p 1 x 1− p2 x 2=0
∂λ
2
, - Red= indifference curve
- Black= budget line
- These are derived from the FOC
- If all 3 of these FOC hold at the same time we are referring to the tangency
• slope of the budget line (p1/p2) = MRS (MU1/MU2)
From solid red to dashed red- the price of x1 is higher than it was before
Price corresponding to the dashed line is higher
Ordinary/ Marshallian demand fn
Instead of maximising utility subject to a fixed
budget, expenditure is minimised subject to a
fixed level of utility (indifference curve)
The solution is a Hicksian or compensated
demand function
This 2nd formulation you can think of holding the
green IC constant and asking how small the
expenditure triangle could be, provided it reaches
the green level of utility
Expenditure Minimisation
Page 3 of written notes
Hicksian demand function=
Maximising utility vs Minimising expenditure:
3
, Variables that belong in a model of residential water demand:
• Water price
• Other prices
• Household disposable income
• Habits e.g. gardening, car washing
• Size of the plot
• Washing machines, dishwashers
• Seasonal effects (water gardens more in summer, or take more showers in summer)
• Onsite water harvesting (Jo-jo tanks)
• Population- Number of people that live in residence (demand curve shifted out by
no.)
*look out if it is a population demand curve (seasonality, population size will play into it) or
household/ per capita curve (house hold size doesn’t matter)
Interpreting possible regression results:
4
