Two types of eects of price change

(1)

MICROECONOMICS I.

(2)

(3)

(4)

ELTE Faculty of Social Sciences, Department of Economics

Microeconomics I.

week 8

CONSUMPTION AND DEMAND, PART 2 Authors:

Gergely K®hegyi, Dániel Horn, Klára Major Supervised by

Gergely K®hegyi

June 2010

(5)

week 8

K®hegyi-Horn-Major

Income and substitution eects of a price change

The course was prepaerd by Gergely K®hegyi, using Jack Hirshleifer, Amihai Glazer and David Hirshleifer (2009) Mikroökonómia. Budapest: Osiris Kiadó, ELTECON-books (henceforth HGH), and Gábor Kertesi (ed.) (2004) Mikroökonómia el®adásvázlatok.

http://econ.core.hu/ kertesi/kertesimikro/ (henceforth KG).

(6)

week 8

K®hegyi-Horn-Major

Approaches to income compensation

How should the government compensate for the eects of price change, which made some groups in the society worse o?

(7)

week 8

K®hegyi-Horn-Major

Two types of eects of price change

The eect of price change upon consumer demand may be separated into two components.

Fall in Px increases the consumer's real income. He or she could buy the same bundle of goods as before, and have something left over. If X is a superior good, the consumer will use some of the excess to buy more X This is called the income eect of the fall in Px

Furthermore, at the lower P_x the substitution balance equation tells us that even if real income or utility had remained the same, more X would have been purchased. This is called the pure substitution eect of the price change.

(8)

week 8

K®hegyi-Horn-Major

Two types of eects of price change (cont.)

(9)

week 8

K®hegyi-Horn-Major

Hicks decomposition

(10)

week 8

K®hegyi-Horn-Major

Hicks decomposition (cont.)

A fall in price Px with income and Py held constant shifts in the budget line from KL to KL⁰ so that the consumption optimum changes from Q to S. Because S lies on a higher indierence curve, there has been an increase in real income. We construct an articial budget line MN parallel to KL⁰ and tangent to the original indierence curve U₀. The income eect of the price change is therefore x_S−x_R and the pure substitution eect of the price change is x_R−x_Q.

(11)

week 8

K®hegyi-Horn-Major

How can the Gien case come about?

A Gien good must have the following properties.

It must be inferior, so that the income eect of a price change is negative.

It must account for a large fraction of the budget. This makes the "perverse" income eect large in magnitude. (It has to be large if it is to overcome the pure substitution eect.)

(12)

week 8

K®hegyi-Horn-Major

How can the Gien case come about? (cont.)

At the initial high bread price the budget line is KL and the optimum is Q. A fall in the price of bread shifts the budget line to KL⁰. The consumer is suciently enriched to prefer buying less bread and more meat at point R. The movement from Q to R consists of a small substitution eect (Q to S) and a large negative income eect (S to R). For this Gien result to occur, bread must be strongly inferior.

(13)

week 8

K®hegyi-Horn-Major

Utility maximization

maximize: U(x,y)→max_x,y

subject to: pxx+pyy=I Lagrange-function:

L(x,y, λ) =U(x,y)−λ(pxx+pyy−I)

∂L

∂x = ^∂_∂^U_x −λp_x =0

∂L

∂y = ^∂_∂^U_y −λpy =0

∂L

∂λ =P_xx+P_yy−I =0

The Marshall demand function:

x^M(p_x,p_y,I) y^M(p_x,p_y,I)

(14)

week 8

K®hegyi-Horn-Major

Expenditure minimization

minimize: pxx+pyy →minx,y

subject to: U(x,y) = ¯U =U(x₀,y₀) Lagrange-function:

L(x,y, λ) =pxx+pyy−λ(U(x,y)−U¯)

∂L

∂x =px−λ^∂_∂^U_x =0

∂L

∂y =p_y−λ^∂_∂^U_y =0

∂L

∂λ =U(x,y)−U¯ =0

The Hicks demand function:

x^H(p_x,p_y,U¯) y^H(p_x,p_y,U¯)

(15)

week 8

K®hegyi-Horn-Major

Expenditure function and indirect utility

Denition

Indirect utility function: The value of the utility maximization in the optimum, which depends on the exogenous px,py,I variables and which shows the maximal level of utility of the consumer at the given prices and income:

v(px,py,I) ={max(U(x,y))|pxx+pyy=I}

Denition

Expenditure function: The value of the expenditure minimization in the optimum, which depends on the exogenous px,py,U¯ variables and which shows the minimal amount of expenditure by which the given utility level can be achieved (at given prices):

e(px,py,U¯) ={min(pxx+pyy)|U(x,y) = ¯U}

(16)

week 8

K®hegyi-Horn-Major

Duality

Statement

SHEPHARD-LEMMA

∂e(px,py,U¯)

∂p_x =x^H(p_x,p_y,U¯)

∂e(p_x,p_y,U¯)

∂py =y^H(px,py,U¯)

(17)

week 8

K®hegyi-Horn-Major

Duality (cont.)

Proof

Let(p⁰_x,p_y⁰)be arbitrary prices andU arbitrary level of utility, and¯ (x⁰,y⁰)be the solution of the expenditure minimization function for the three given parameters. Let's dene

f(p_x,p_y)≡p_xx⁰+p_yy⁰−e(p_x,p_y,U¯)

function for an arbitrary(p_x,p_y). Since e(p_x,p_y,U¯)is the minimal expenditure for(p_x,p_y,U¯)

f(p_x,p_y)≥0

always holds. And if (px,py) = (p_x⁰,p⁰_y), then f(p⁰_x,p_y⁰) =0 takes its minimum. Then

(18)

week 8

K®hegyi-Horn-Major

Duality

Proof

∂f(px,py)

∂p_x =x⁰−∂e(px,py,U¯)

∂p_x =0

∂f(px,py)

∂py =y⁰−∂e(px,py,U¯)

∂py =0

Since(p⁰_x,p_y⁰)were arbitrary prices, the above relation holds for any set of prices:

∂e(p_x,p_y,U¯)

∂p_x =x^H(p_x,p_y,U¯)

∂e(px,py,U¯)

∂py =y^H(px,py,U¯)

(19)

week 8

K®hegyi-Horn-Major

Duality

(20)

week 8

K®hegyi-Horn-Major

Duality

e(px,py,v(px,py,I)) =I v(p_x,p_y,e(p_x,p_y,U¯)) = ¯U

e(px,py,U¯) =pxx^H(px,py,U¯) +pyy^H(px,py,U¯) v(px,py,I) =U x^M(px,py,I),y^M(px,py,I)

(21)

week 8

K®hegyi-Horn-Major

Slutsky-theorem

Statement

Slutsky-theorem

∂x^M p_x,p_y,e(p_x,p_y,U¯)

∂p_x = ∂x^H(p_x,p_y,U¯)

∂p_x −∂x^M

∂e x^M

∂x^M px,py,e(px,py,U¯)

∂py =∂x^H(px,py,U¯)

∂py −∂x^M

∂e y^M

∂y^M px,py,e(px,py,U¯)

∂px =∂y^H(p_x,p_y,U¯)

∂px −∂y^M

∂e x^M

∂y^M p_x,p_y,e(p_x,p_y,U¯)

∂p_y =∂y^H(px,py,U¯)

∂p_y −∂y^M

∂e y^M

(22)

week 8

K®hegyi-Horn-Major

Slutsky-theorem

Proof

Due to duality relations:

x^M p_x,p_y,e(p_x,p_y,U¯)

=x^H(p_x,p_y,U¯) y^M p_x,p_y,e(p_x,p_y,U¯)

=y^H(p_x,p_y,U¯)

(23)

week 8

K®hegyi-Horn-Major

Slutsky-theorem

Proof

Partially dierentiating both equations with both prices:

∂x^M

∂px +∂x^M

∂e

∂px =∂x^H

∂px

∂x^M

∂py +∂x^M

∂e

∂py =∂x^H

∂py

∂y^M

∂px +∂y^M

∂e

∂px =∂y^H

∂px

∂y^M

∂p_y +∂y^M

∂e

∂p_y =∂y^H

∂p_y

(24)

week 8

K®hegyi-Horn-Major

Slutsky-theorem

Proof

Using the Shephard-lemma and rearranging the equation:

∂x^M

∂p_x = ∂x^H

∂p_x −∂x^M

∂e x^H

∂x^M

∂py = ∂x^H

∂py −∂x^M

∂e y^H

∂y^M

∂p_x = ∂y^H

∂p_x −∂y^M

∂e x^H

∂y^M

∂py = ∂y^H

∂py −∂y^M

∂e y^H

(25)

week 8

K®hegyi-Horn-Major

Slutsky-theorem

Proof

Using duality relations we get the Slutsky-theorem:

∂x^M

∂p_x =∂x^H

∂p_x −∂x^M

∂e x^M

∂x^M

∂py = ∂x^H

∂py −∂x^M

∂e y^M

∂y^M

∂p_x = ∂y^H

∂p_x −∂y^M

∂e x^M

∂y^M

∂py = ∂y^H

∂py −∂y^M

∂e y^M

(26)

week 8

K®hegyi-Horn-Major

A little more precise, mathematically

x^M =





 x₁^M x_i...^M x_n...^M





 ,x^H =





 x₁^H x..._i^H x..._n^H





 ,p=





 p1

p..._i p...n







Expenditure function:

e(p,U¯) ={min(px|U(x) = ¯U} Shephard-lemma:

∇e(p,U¯) =x^H

(27)

week 8

K®hegyi-Horn-Major

A little more precise, mathematically (cont.)

Marshall substitution matrix:

M=







∂x1^M

∂p1

∂x1^M

∂p2 · · · ^∂_∂^x_p¹^M

i . . . ^∂_∂^x_p¹^M

n

∂x₂^M

∂p1

∂x₂^M

∂p2 · · · ^∂_∂^x_p²^M

i . . . ^∂_∂^x_p²^M ... ... n

∂x₁^M

∂pi

∂x_i^M

∂pi ...

... ... ...

∂x₁^M

∂pn · · · ^∂_∂^x_pⁿ^M

n







(28)

week 8

K®hegyi-Horn-Major

A little more precise, mathematically (cont.)

Hicks substitution matrix:

H=







∂x₁^H

∂p1

∂x₁^H

∂p2 · · · ^∂_∂^x_p¹^H

i . . . ^∂_∂^x_p¹^H

n

∂x2^H

∂p1

∂x2^H

∂p2 · · · ^∂_∂^x_p²^H

i . . . ^∂_∂^x_p²^H ... ... n

∂x1^H

∂pi

∂xi^H

∂pi ...

... ... ...

∂x1^H

∂pn · · · ^∂_∂^x_pⁿ^H

n







(29)

week 8

K®hegyi-Horn-Major

A little more precise, mathematically (cont.)

Slutsky-matrix

S=







∂x₁^M

∂p1 +^∂_∂^x¹_e^Mx₁^M · · · ^∂_∂^x_p¹^M

n +^∂_∂^x_e¹^Mx_n^M

... ... ...

... ^∂_∂^x_pⁱ^M_i +^∂_∂^x_eⁱ^Mx_i^M ...

... ... ...

∂x_n^M

∂p1 +^∂_∂^xⁿ_e^Mx₁^M · · · ^∂_∂^x_pⁿ^M

n +^∂_∂^x_eⁿ^Mx_n^M







Slutsky-theorem:

S=H

Statement

The Hicks substitution matrix is symmetrical ^∂_∂^x_p^Hⁱ

j = ^∂_∂^x_p^j^H

i and has non-positive elements in the main diagonal ^∂_∂^x_pⁱ^H_i ≤0.

(30)

week 8

K®hegyi-Horn-Major

A little more precise, mathematically (cont.)

(31)

week 8

K®hegyi-Horn-Major

A little more precise, mathematically (cont.)

Proof (draft)

Symmetry comes from the Young-theorem.

py≤px qx≤qy (q−p)(y−x)≤0

(q₁−p₁)(y₁−x₁) + (q₂−p₂)(y₂−x₂)≤0 Let's assume that q₂=p₂, then

(q₁−p₁)(y₁−x₁)≤0

∆p_i∆x_i|_U=U0≤0

∆limpi→0

∆xi

∆pi|_U=U0 ≤0

∂x_i^H

∂p_i ≤0

(32)

week 8

K®hegyi-Horn-Major

A little more precise, mathematically

Consequence

The main diagonal of the Slutsky matrix has non-positive elements.

Consequence

LAW of DEMAND: If good i is a normal good, then ^∂_∂^x_pⁱ^M_i ≤0, which means that its demand curve has a negative slope.

Proof

Since in the Hicks matrix ^∂_∂^x_pⁱ^H

i ≤0, therefore in the Slutsky matrix

∂x_i^M

∂pi +^∂_∂^xⁱ_e^Mx_i^M ≤0. For normal goods ^∂_∂^x_eⁱ^M >0, thus necessarily

∂x_i^M

∂pi ≤0.

(33)

week 8

K®hegyi-Horn-Major

Slutsky decomposition

maximize: U(x,y)→max_x,y

subject to: pxx+pyy=pxx0+pyy0

Lagrange-functions:

L(x,y, λ) =U(x,y)−λ(pxx+pyy−pxx0−pyy0)

∂L

∂x = ^∂_∂^U_x −λp_x =0

∂L

∂y = ^∂_∂^U_y −λpy =0

∂L

∂λ =P_xx+P_yy−p_xx₀−p_yy₀=0

Slutsky demand functions:

x^S(p_x,p_y,x₀,y₀) y^S(p_x,p_y,x₀,y₀)

(34)

week 8

K®hegyi-Horn-Major

Slutsky decomposition (cont.)

Statement

Slutsky theorem[with Slutsky decomposition]

∂x^M(px,py,pxx0+pyy0))

∂p_x = ∂x^S(px,py,x0,y0)

∂p_x −∂x^M

∂I x0

∂x^M(p_x,p_y,p_xx₀+p_yy₀))

∂py = ∂x^S(p_x,p_y,x₀,y₀)

∂py −∂x^M

∂I y₀

∂y^M(px,py,pxx0+pyy0))

∂p_x = ∂y^S(px,py,x0,y0)

∂p_x −∂y^M

∂I x₀

∂y^M(p_x,p_y,p_xx₀+p_yy₀))

∂py = ∂y^S(p_x,p_y,x₀,y₀)

∂py −∂y^M

∂I y₀

(35)

week 8

K®hegyi-Horn-Major

Market demand

Summing individual demands

X ≡

N

X

i=1

xi

Here d1and d2are demand curves for two individuals. If these are the only two potential purchasers of the good, the overall market demand curve D is the horizontal sum of d₁ and d₂.

(36)

week 8

K®hegyi-Horn-Major

Subsidy versus voucher

(37)

week 8

K®hegyi-Horn-Major

Subsidy versus voucher (cont.)

Voucher

The initial optimum is a corner solution at K; no education is purchased. A voucher gift of income in the amount KK⁰ leads to a new optimum at K⁰⁰ The voucher leads to an increased consumption of education, provided only that education is a good rather than a bad for this individual.