Chapter 5, Income and Substitution Effects

• Marshallian demands(consumption varies with price and income) $x_1^* =g_1(p_1, p_2,..., p_n, I)$

• Hicksian Demands, (consumption varies with price and utility) $x_1^* =h_1(p_1, p_2,..., p_n, U_-)$

Changes in Income

• Increase income → budget constrain swifts out in parallel. higher levels of utility.

• $\frac{p_x}{p_y}$ does not change, the MRS at a optimal will not change.

• The change in consumption caused by a small change in income can be computed using the Marshallian Demands(Uncompensated demands) $g_x(p_x, p_y, l') - g_x(p_x, p_y, l)$

• inferior & normal:

• x and y are normal goods $\frac{\partial g_x}{\partial I} ≥ 0$ when increasing income, choose to consume more x and y.

• x is an inferior goods $\frac{\partial g_x}{\partial I} ≤ 0$ when increasing income, choose to consume less x and more y.
  
  

Changes in a Good’s price

Alters the slope of the budget constraint.

• $p_x$ change, cause $x$ change, use Marshallian demand function for $x=g_x(p_x^, ,p_y, I) - g_x(p_x, p_y, I)$

• $p_x$ change, cause $y$ change, use Marshallian demand function for $y=g_y(p_x^, ,p_y, I) - g_y(p_x, p_y, I)$

• Small changes in $p_x$, cause change in $x$, use Marchallian demand function for $x=\frac{\partial g_x(p_x, p_y, I)}{\partial p_x} dp_x$

• Small changes in $p_x$, cause change in $y$, use Marchallian demand function for $y=\frac{\partial g_y(p_x, p_y, I)}{\partial p_x} dp_x$

Substitution Effect:

• A is always cheaper than B, people choose to buy cheaper things more. insert substitution effect pic form ppt

• substitution effect caused by a change in $p_x$ can be computed using the hicksian demand function: $x=h_x(p_x^, ,p_y, U_-) - h_x(p_x, p_y, U_-)$

• caused by a small changes in $p_x$ , $x=\frac{\partial h_x(p_x, p_y, U_-)}{\partial p_x}$

Income Effect

• When income does not change, and the price of A is dropped, purchasing power increased, achieve hight utility. insert pic form ppt

• income effect caused by $p_x$ is the total change-substitution effect: $g_x(p_x^, ,p_y, I) - g_x(p_x, p_y, I) - h_x(p_x^, ,p_y, U_-) - h_x(p_x, p_y, U_-)$

• caused by a small change in the price $p_x$ ia rhw difference between the total change and the substitution effect: $\frac{\partial g_x}{\partial p_x}dp_x - \frac{\partial h_x}{\partial p_x}dp_x$

Slutsky Equation for $x, y, p_x, p_y$:

1. we have $x^*=g_x(p_x, p_y, I)$, $x^*=h_x(p_x, p_y, U_-)$, and $E(p_x, p_y, U_-)=I$

2. We replace $I$ by Expenditure function: $g_x(p_x, p_y, I)=g_x(p_x, p_y, (E(p_x, p_y, U_-)) = h_x(p_x, p_y, U_-)$

3. Take partial derivatives of the function, we get: $\frac{\partial h_x}{\partial p_x}= \frac{\partial g_x}{\partial p_x}+\frac{\partial g_x}{\partial E} \frac{\partial E}{\partial p_x}$, and because of $E=I$, we replace the E: $\frac{\partial h_x}{\partial p_x}= \frac{\partial g_x}{\partial p_x}+\frac{\partial g_x}{\partial I} \frac{\partial E}{\partial p_x}$.

4. Since $E(p_x, p_y, U_-)=p_x x^x*, p_y y^*$, we get $\frac{\partial E}{\partial p_x}= x^*$ → (Envelop Theorem)

5. plug back in to the function $\frac{\partial h_x}{\partial p_x}= \frac{\partial g_x}{\partial p_x}+\frac{\partial g_x}{\partial I} x^*$

6. $\frac{\partial g_x}{\partial p_x}=\frac{\partial h_x}{\partial p_x}-\frac{\partial g_x}{\partial I} x^*$, → Slutsky Equation, where total effect = substitution effect - income effect.
   

Normal goods: $\frac{\partial g_x}{\partial p_x}<= \frac{\partial h_x}{\partial p_x}$,

Inferior goods: $\frac{\partial g_x}{\partial p_x} > \frac{\partial h_x}{\partial p_x}$.

Example of a Cobb-Douglas Function: $U(x,y) = x^{0.5}y^{0,5}$

$x^*=g_x(p_x, p_y, I)= \frac{0.5}{p_x}I$,

$x^*=h_x(p_x, p_y, U_-) = (\frac{p_y}{p_x})^{0.5}U_-$,

Total Effect: $\frac{\partial g_x}{\partial p_x}=0.5 I(- \frac{1}{(p_x)^2})=-\frac{0.5}{p_x^2}I<0$

Substitution effect: $\frac{\partial h_x}{\partial p_x} = -0.5 \frac{p_y^{0.5}}{p_x^{1.5}} U_- < 0$

$U_- = V(p_x, p_y, I)=\frac{I}{2p_x^{0.5} p_y^{0.5}}$ plug in to above equation, we get $\frac{\partial h_x}{\partial p_x} = -0.25 \frac{I}{p_x^2}$

Income effect: $- \frac{\partial g_x}{\partial I} x^* = -\frac{0.5}{p_x} \frac{0.5}{p_x} I = -0.25 \frac{0.5}{p_x^2}$

Therefore, Sluskey Equation: $-\frac{0.5}{p_x}I = -0.25 \frac{I}{p_x^2}- 0.25 \frac{I}{p_x^2}$

Prices changes for Normal goods:

• Normal Goods:

  • price falls, both effects lead to a rise in quantity demanded;
  • price rises, both effects lead to a drop in quantity demanded;

• Inferior Goods: substitution and income effects move in opposite directions:

  • price falls, substitution rises quantity demanded, income move to opposite;
  • price rises, substitution drops quantity demanded, income move to opposite.

Giffen’s Paradox

increase in price generates increase in real income

inferior: drop income = rise demanded

income effect greater than the substitution, consumption increase.

Uncompensated & Compensated Curve

Uncompensated Demand Curve

• Uncompensated demand curve for good x is the Marshallian demand function with fixed price of the other good and income: $x^* = g_x(p_x,p_{y-},I_-)$;

• Increase $p_x$:

  • lower indifference curve,
  • Nominal income is constant, but real income falls as price of x rises.

Compensated Demand Curve

• Compensated Demand curve for good x is the Hicksian demand function with fixed price of the other good and utility level: $x^* = h_x(p_x,p_{y-},U_-)$;

NOTE: For normal good:

• compensated demand curve is less responsive of price changes than the uncompensated demand curve.
• the uncompensated demand curve reflects both income and substitution effects:
• compensated demand curve only reflect only substitution effects.
  
  

Demand Elasticities

Since the size of partial derivatives depends on the unit we use to consume goods, it is hard to compare:
(changing currency, lB-kg, apple in 1980s and apple today. )

Marshallian (Uncompensated) Demand Elasticities

• Income elasticity of demand $e_{x, I}=\frac{\partial g_x}{\partial I} \frac{I}{g_x}= \frac{\partial ln g_x}{\partial ln I}$,
  measures the percentage change in consumption of x in response to one percent change in income.

• Size of the elasticity:

  • $e_{x, I} > 0$, demand is normal;
  • $e_{x, I} < 0$, demand is inferior;
  • $e_{x, I} < 1$, the good is necessity;
  • $e_{x, I} > 1$, the goos is luxury.
    

Own Price Elasticity of Demand:

• Own price elasticity of demand is negative, the only exceptions are the Giffen goods.

• $e_{x, p_x}=\frac{\partial g_x}{\partial p_x} \frac{P}{g_x}= \frac{\partial ln g_x}{\partial ln p_x}$

• the size of the elasticity is important,

  • $e_{x, I} < -1$, demand is elastic.
  • $e_{x, I} > -1$, demand is inelastic.
  • $e_{x, I} = -1$, demand is unit elastic.

Cross price elasticity of demand

$e_{x, p_y}=\frac{\partial g_x}{\partial p_y} \frac{p_y}{g_x}= \frac{\partial ln g_x}{\partial ln p_y}$,

Hicksian (Compensated) price elasticities

Compensated own price elasticity

$e^c_{x, p_x} = \frac{\partial h_x}{\partial p_x} \frac{p_x}{h_x}= \frac{\partial ln h_x}{\partial p_x}$

cross-price elasticity

$e^c_{x, p_y} = \frac{\partial h_x}{\partial p_y} \frac{p_y}{h_x}= \frac{\partial ln h_x}{\partial p_y}$

Slusky Equation

• Difference between hicksian and marshallian price elasticities:

  • income elasticity;
  • the share of income spent on the good.

• hicksian and marshallian price elasticities will be similar if:

  • the income elasticity of x is small;
  • the share of income devoted to x is small.
    
    

• Slusky Equaiton: $\frac{\partial g_x}{\partial p_x}=\frac{\partial h_x}{\partial p_x}-\frac{\partial g_x}{\partial I} x^{* }$, multiply both sides by $\frac{p_x}{x^* }$,

• $\frac {p_x}{x^* } \frac{\partial g_x}{\partial p_x}=\frac {p_x}{x^* } \frac{\partial h_x}{\partial p_x} - \frac{p_x x^* }{x^* } \frac{\partial g_x}{\partial I} \frac{I}{x^* }$

• if we set $s_x = \frac{p_x x^* }{I}$, then $e_{x, p_x} = e_{x, p_x}^c - s_x e_{x, I}$

Example:

Known: $U(x,y)= x^\alpha y^\beta$, and $\alpha + \beta =1$

1. solve for marshallian $x^* = g_x(p_x, p_y, I)=\frac{\alpha}{p_x}I$,
   → $e_{x, p_x} = \frac{\partial g_x}{\partial p_x} \frac{p_x}{g_x}= -\frac{\alpha I}{p_x(\frac{\alpha I}{p_x})} = -1$

2. Aanother way to solve: $ln x^* =ln g_x(p_x, p_y, I) = ln \alpha + lnI - ln p_x$

3. $e_{x, p_x} = \frac{\partial ln g_x}{\partial ln p_x} = -1$
   $e_{x, p_y} = 0$
   $e_{x, I} = \frac{\partial g_x}{\partial I} \frac{I}{g_x}=\frac{\alpha}{p_x}\frac{I}{g_x} = 1$

Homogeneity, between demand elasticities, Euler’s theorem (relations)

• Marshallian demand function are homogeneous of degree zero in all price and income;

The Euler’s Theorem

For homogeneous function implies that: $0 = p_x \frac{\partial g_x}{\partial p_x} + p_y \frac{\partial g_x}{\partial p_y} + I \frac{\partial g_x}{\partial I}$

• Devide by $g_x$, we get $0= e_{x, p_x} + e_{x, p_y} +e_{x, I}$

• A 1 percentage change in all prices and income will leave the quantity demanded of x unchanged.

Engel Aggregation

• Income elasticity of demand for food items is less than one,
• Income elasticity of demand for non food items must be greater than one;

Proof: $I$ = Expenditure $= p_x g_x(p_x, p_y, I) + p_y g_y(p_x, p_y, I)$ $1=p_x \frac{\partial g_x}{\partial I} + p_y \frac {\partial g_y}{\partial I}$ mutiply $\frac{g_x I}{g_x I}$ for each term:

$1= \frac{p_x g_x}{I} \frac{\partial g_x}{\partial I} + \frac{p_y g_y}{I} \frac {\partial g_y}{\partial I}$

$1= s_x e_{e, I} + s_y e_{y,I}$ $s_x=$ share on food, $s_y=1-s_x$

$ (1- s_x) e_{y,I} = 1- s_x e_{x,I} > 1-s_x $, since $e_{x,I} < 1$; cancel out $(1- s_x)$, therefore, $e_{y,I} > 1$.

Example check the relationships:

• Homogeneity: $0= e_{x, p_x} + e_{x, p_y} + e_{x, I}$, since: $e_{x, p_x} = -1, e_{x, p_y} = 0$

• Engel aggregation: $1= s_x e_{e, I} + s_y e_{y,I}$, $s_x= \frac{p_x x^* }{I}= \frac{p_x \frac{\alpha I}{p_x}}{I}= \alpha$ $s_y= \frac{p_y y^* }{I}= \beta$ $e_{x, I} = e_{y, I}=1$, $\alpha * 1+ \beta * 1 = \alpha + \beta = 1$

Slusky equation in elasticity form:

$e_{x, p_x}= e^c_{x, p_x} -s_x e_{x,I}$ → $e^c_{x, p_x} = e_{x, p_x} + s_x e_{x,I}$ $e^c_{x, p_x} = -1 + \alpha * 1 = \alpha -1 = - \beta$

Consumer Welfare

• Price increase from $p^0 _ x$ to $p^ 1 _ x$, to compare the expenditure for a given level of utility $U_-$.

• if $p^1 _ x > p^0 _ x$,then $E(p^1 _x, p_y, U_-)>E(p^0 _x, p_y, U_-)$

• The difference is called compensating variation (CV).

• the increase in income that the individual would need to achieve $U_1$. point B and C have same slope:
  
  
  
  
  
  
  
  
  

• compensate the consumer to stay at the same indifference curve.

One last result:
derivative of the expenditure function with respect to $p_x$ is the compensated demand function

$\frac {\partial E}{p_x} = h_x$

Some Price Elasticities:

Special brand:

Narrow Categories:

Example of Inferior Goods:

Fast Food, Used car, Lottery tickets, Fake brand, toilet paper…

chapter 6

Substitutes and Complements

• Substitute: One can replace the other in use. coffee&tea, butter&cream
• Complement: Used together. coffee&cream

Gross Substitutes and Complements:

$p_y$falls, $y=-\frac{p_x}{p_y}x+\frac{I}{p_y}$

• goods x and y are gross substitutes if $\frac{\partial g_x}{\partial p_y} > 0$
  $p_y$ increases, then consume more x;

• goods x and y are gross Complements if $\frac{\partial g_x}{\partial p_y} < 0$
  $p_y$ increases, then consume less x;
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  

Mathematical Treatment

the change in $x$ caused by a change in $p_y$, use Slutsky-type equation:

$\frac{\partial g_x}{\partial p_y}=\frac{\partial h_x}{\partial p_y} - \frac{\partial g_x}{\partial I}y$

Asymmetry of the Gross definitions

• Gross substitutes and gross complements are not symmetric

• It is possible

  • for x to be a substitute for y
  • at same time for y to be a complement of x.

• This occurs when the income effect dominates the substitution effect for one good but not for the other.

Example:
$U(x,y) = lnx+y$
solve the utility maximization, we get $y^* = \frac{(I-p_x)}{p_y}$, $x^* = \frac{p_y}{p_x}$
therefore, $\frac{\partial x^* }{\partial p_y} > 0, \frac{\partial y^* }{\partial p_x} = 0$

Net Substitutes and Complement:

focus solely on substitution effect

• net substitutes if $\frac{\partial h_x}{\partial p_x}>0$

• net complement if $\frac{\partial h_x}{\partial p_x}<0$

• This definition is unambiguous because $\frac{\partial h_i}{\partial p_j}=\frac{\partial h_j}{\partial p_i}$, since $h_i (p_1,..., p_n,U_-)=\frac{\partial E(p_1, p.., p_2, U_-)}{\partial p_i}$
  
  

Example: Gross Complements but Net substitutes
$p_y$ falls,

Substitutability with many goods

• “most” goods are net substitutes.

proof:

1. Start with Hicksian demand function $x_i = h_i(p_1, ..., p_n, U_-)$, homogeneous of degree zero in prices.

2. The Euler’s theorem yields $0= p_1 \frac{\partial h_i}{p_1} + ... p_i \frac{\partial h_i}{p_i}+...+ p_n \frac{\partial h_i}{p_n}$

3.Divide by $h^i$, same compensated elasticities $e^c_{i1}+...e^c_{ii}+...e^c_{in}=0$

4. Since the negativity of Substitution effect that $e^c_{ii}<0$,
   $∑_{i≠j} e^c_{ij} = - e^c_{ii}≥0$