1. Demand Relationships among Goods
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2. The Plan Cross-price effects Substitutes and complements
Composite commodities
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3. The Two-Good Case
How the quantity of x chosen might be affected by a decrease in the price of y
Shifting the budget constraint outward
The quantity of good y chosen has increased
In (a), small substitution effect
x and py move in opposite directions
In (b), large substitution effect
x and py move in the same direction
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4. Differing Directions of Cross-Price Effects
6.1
Differing Directions of Cross-Price Effects
Quantity of x
(a)
Quantity of y
Quantity of x
(b)
Quantity of y I1 I1 U1 U1 U0 U0 I0 I0 y1 y1 x1 x1 y0 x0 y0 x0
In both panels, the price of y has decreased. In (a), substitution effects are small; therefore, the quantity of x consumed increases along with y. Because ∂x/∂py < 0, x and y are gross complements. In (b), substitution effects are large; therefore, the quantity of x chosen decreases. Because ∂x/∂py > 0, x and y would be termed gross substitutes.
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5. The Two-Good Case In elasticity terms Slutsky-type equation
Substitution effect (+)
Income effect (-) if x is normal
Combined effect (ambiguous)
Just expand sufficiently with py/x on both sides, etc. etc.
& uncompensated elasticity and compensated elasticity are similar if income effect is small (=last RHS term)
In elasticity terms
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6. 6.1 Another Slutsky Decomposition for Cross-Price Effects
The cross-price effect of a change in y prices on x purchases: Cobb-Douglas utility
Uncompensated demand function:
x(px,py,I)=0.5 I/px
Compensated demand function:
xc(px,py,V)=Vpy0.5px-0.5
Marshallian demand function in this case yields:
∂x/∂py=0 !
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7. 6.1 Another Slutsky Decomposition for Cross-Price Effects
The cross-price effect of a change in y prices on x purchases
Changes in the price of y do not affect x purchases
Because the substitution and income effects of a price change are precisely counterbalancing
Substitution effect:
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8. 6.1 Another Slutsky Decomposition for Cross-Price Effects
Income effect:
Total effect of a change in the price of y:
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9. Substitutes and Complements
Many goods
Generalize the Slutsky equation for any two goods xi, xj
An elasticity relation
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10. Substitutes and Complements
If one good may, as a result of changed conditions, replace the other in use
Substitute for one another in the utility function
Complements
Goods that ‘‘go together’’
Complement each other in the utility function
Query: Provide a few examples for both.
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11. Substitutes and Complements
Gross substitutes
Two goods, xi and xj, are said to be gross substitutes if: ∂xi/∂pj>0
An increase in the price of one good causes more of the other good to be bought
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12. Substitutes and Complements
Gross complements
Two goods, xi and xj, are said to be gross complements if: ∂xi/∂pj<0
An increase in the price of one good causes less of the other good to be purchased
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13. Substitutes and Complements
Asymmetry of the gross definitions
Gross definitions of substitutes and complements - are not symmetric
It is possible, by the definitions
For x1 to be a substitute for x2
And at the same time
For x2 to be a complement of x1
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14. 6.2 Asymmetry in Cross-Price Effects
Utility function, U(x,y) = ln x + y
The Lagrangian, ℒ = ln x + y + λ(I – pxx – pyy)
First-order conditions
∂ℒ /∂x = 1/x - λpx = 0
∂ℒ /∂y = 1 - λpy = 0
∂ℒ /∂λ= I - pxx - pyy = 0
So, pxx = py
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15. 6.2 Asymmetry in Cross-Price Effects
Substitution into the budget constraint
Solve for the Marshallian demand function for y
I=pxx+pyy = py+pyy
So, y=(I-py)/py
An increase in py - must decrease spending on y
Because px and I are unchanged, spending on x must increase
∂x/∂py>0, gross substitutes
∂y/∂px=0, independent (no complement!)
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16. Net (Hicksian) Substitutes and Complements
Goods xi and xj are said to be net substitutes if:
Goods xi and xj are said to be net complements if:
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17. Net (Hicksian) Substitutes and Complements
These definitions
Look only at the substitution terms to determine whether two goods are substitutes or complements
Look only at the shape of the indifference curve
This definition is unambiguous because the definitions are perfectly symmetric (why?)
Youngs theorem: E_ij = E_ji
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18. Composite Commodities
In the most general case
An individual who consumes n goods
Will have demand functions that reflect n(n+1)/2 different substitution effects
Query. Verify this formula for n=2
Convenience: group goods into larger aggregates
Examples: food, clothing, “all other goods”
S11, s12, s21, s22 – but s12 = s21 due to Young’s theorem -> 3 different substitution effects = 2(2+1)/2=3.
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19. Composite Commodities
Consumers choose among n goods
The demand for x1 will depend on the prices of the other n-1 commodities (why?)
If all of these prices move together: lump them into a single composite commodity, y
Let p20…pn0 represent the initial prices of these other commodities
Assume they all vary together (so that the relative prices of x2…xn do not change)
Due to budget constraint
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20. Composite Commodities
Define the composite commodity y
To be total expenditures on x2…xn at the initial prices,
y = p20x2 + p30x3 +…+ pn0xn
The individual’s budget constraint is
I = p1x1 + p20x2 +…+ pn0xn = p1x1 + y
Assumption; all of the prices p20…pn0 change by same factor t (t > 0)
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21. Composite Commodities
Budget constraint (after changes in prices: 0 to 1)
I = p1x1 + tp20x2 +…+ tpn0xn = p1x1 + ty
Changes in p1 or t induce substitution effects
As long as p20…pn0 move together
We can confine our examination of demand to choices between buying x1 and “everything else”
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22. Composite Commodities
A composite commodity
A group of goods for which all prices move together
These goods can be treated as a single commodity
The individual behaves as if he is choosing between x1 and the entire composite group
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23. 6.3 Housing Costs as a Composite Commodity
Suppose that an individual receives utility from three goods:
Food (x)
Housing services (y), measured in hundreds of square feet
Household operations (z), measured by electricity use
CES utility function
Query. Determine the elasticity of substitution.
Delta = -1, sigma = 1/(1-delta) = 1/2
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24. 6.3 Housing Costs as a Composite Commodity
Lagrangian technique
Can be used to calculate Marshallian demand functions for these goods as
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25. 6.3 Housing Costs as a Composite Commodity
If initially I = 100, px = 1, py = 4, pz = 1
The demand functions predict: x* = 25, y* = 12.5, z* = 25
$25 is spent on food and $75 is spent on housing-related needs
Assume that the py and pz move together
Define the “composite commodity” housing (h)
h = 4y + 1z (expenses, where ph=1, initially)
Initial quantity of housing = total spent on housing (75)
Observe that ph = 1 (defined!) arbitrarily, each different normalization would do as well!
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26. 6.3 Housing Costs as a Composite Commodity
py and pz always move together
ph will always be related to these prices,
ph=pz=0.25py
Recalculate the demand function for x as a function of I, px, and ph:
As t ph = t pz (move together), ph = pz
As t py = 4 ph -> ph = ¼ py
If I = 100, px = 1, py = 4, and ph=1, x*=25, h* =75
Query. What happens if ph=2 (rises)?
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27. Theory of two-stage budgeting
Partition of goods
Into m nonoverlapping groups (r =1, m)
A separate budget (lr)
Devoted to each category
Demand functions for the goods within any one category
Depend on the prices of goods within the category
And on the category’s budget allocation
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28. Theory of two-stage budgeting
Demand is given by:
xi(p1,…pn,I)=xiЄr(piЄr,Ir), r=1,m
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