MICROECONOMICS I.
Sponsored by a Grant TÁMOP-4.1.2-08/2/A/KMR-2009-0041 Course Material Developed by Department of Economics,
Faculty of Social Sciences, Eötvös Loránd University Budapest (ELTE) Department of Economics, Eötvös Loránd University Budapest
Institute of Economics, Hungarian Academy of Sciences Balassi Kiadó, Budapest
Authors: Gergely K®hegyi, Dániel Horn, Klára Major Supervised by Gergely K®hegyi
June 2010
ELTE Faculty of Social Sciences, Department of Economics
MICROECONOMICS I.
week 8
Consumption and demand, part 2
Gergely, K®hegyiDániel, HornKlára, Major
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).
Income and substitution of a price change
Approaches to income compensation
How should the government compensate for the eects of price change, which made some groups in the society worse o?
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. IfX is a superior good, the consumer will use some of the excess to buy moreX This is called the income eect of the fall inPx
• Furthermore, at the lowerPxthe substitution balance equation tells us that even if real income or utility had remained the same, moreXwould have been purchased. This is called the pure substitution eect of the price change.
Hicks decomposition
A fall in price Px with income andPy held constant shifts in the budget line fromKLto 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 lineM N parallel toKL0 and tangent to the original indierence curveU0. The income eect of the price change is thereforexS−xR and the pure substitution eect of the price change isxR−xQ.
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.)
At the initial high bread price the budget line is KLand the optimum is Q. A fall in the price of bread shifts the budget line toKL0. The consumer is suciently enriched to prefer buying less bread and more meat at point R. The movement fromQ to R consists of a small substitution eect (Qto S) and a large negative income eect (S toR). For this Gien result to occur, bread must be strongly inferior.
Utility maximization
• maximize: U(x, y)→maxx,y
• subject to: pxx+pyy=I Lagrange-function:
L(x, y, λ) =U(x, y)−λ(pxx+pyy−I)
• ∂L∂x = ∂U∂x −λpx= 0
• ∂L∂y = ∂U∂y −λpy= 0
• ∂L∂λ =Pxx+Pyy−I= 0 The Marshall demand function:
xM(px, py, I) yM(px, py, I)
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 =py−λ∂U∂y = 0
• ∂L∂λ =U(x, y)−U¯ = 0 The Hicks demand function:
xH(px, py,U)¯ yH(px, py,U¯) Expenditure function and indirect utility
Denition 1. Indirect utility function: The value of the utility maximization in the optimum, which depends on the exogenouspx, 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 2. 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}
Duality
Statement 1. SHEPHARD-LEMMA
∂e(px, py,U¯)
∂px
=xH(px, py,U¯)
∂e(px, py,U)¯
∂py
=yH(px, py,U¯)
Proof 1. Let(p0x, p0y) 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(px, py)≡pxx0+pyy0−e(px, py,U)¯
function for an arbitrary(px, py). Since e(px, py,U¯)is the minimal expenditure for (px, py,U¯) f(px, py)≥0
always holds. And if (px, py) = (p0x, p0y), thenf(p0x, p0y) = 0takes its minimum. Then
Duality Proof 2.
∂f(px, py)
∂px
=x0−∂e(px, py,U¯)
∂px
= 0
∂f(px, py)
∂py
=y0−∂e(px, py,U¯)
∂py
= 0
Since (p0x, p0y)were arbitrary prices, the above relation holds for any set of prices:
∂e(px, py,U¯)
∂px
=xH(px, py,U¯)
∂e(px, py,U)¯
∂py =yH(px, py,U¯) Duality
Duality
e(px, py, v(px, py, I)) =I v(px, py, e(px, py,U)) = ¯¯ U
e(px, py,U¯) =pxxH(px, py,U¯) +pyyH(px, py,U¯) v(px, py, I) =U xM(px, py, I), yM(px, py, I)
Slutsky-theorem
Statement 2. Slutsky-theorem
∂xM px, py, e(px, py,U¯)
∂px
= ∂xH(px, py,U¯)
∂px
−∂xM
∂e xM
∂xM px, py, e(px, py,U¯)
∂py
= ∂xH(px, py,U)¯
∂py
−∂xM
∂e yM
∂yM px, py, e(px, py,U¯)
∂px
= ∂yH(px, py,U¯)
∂px
−∂yM
∂e xM
∂yM px, py, e(px, py,U¯)
∂py
=∂yH(px, py,U¯)
∂py
−∂yM
∂e yM
Slutsky-theorem
Proof 3. Due to duality relations:
xM px, py, e(px, py,U)¯
=xH(px, py,U¯) yM px, py, e(px, py,U)¯
=yH(px, py,U¯) Slutsky-theorem
Proof 4. Partially dierentiating both equations with both prices:
∂xM
∂px +∂xM
∂e
∂e
∂px = ∂xH
∂px
∂xM
∂py
+∂xM
∂e
∂e
∂py
=∂xH
∂py
∂yM
∂px +∂yM
∂e
∂e
∂px =∂yH
∂px
∂yM
∂py
+∂yM
∂e
∂e
∂py
=∂yH
∂py
Slutsky-theorem
Proof 5. Using the Shephard-lemma and rearranging the equation:
∂xM
∂px
= ∂xH
∂px
−∂xM
∂e xH
∂xM
∂py
= ∂xH
∂py
−∂xM
∂e yH
∂yM
∂px
= ∂yH
∂px
−∂yM
∂e xH
∂yM
∂py
=∂yH
∂py
−∂yM
∂e yH
Slutsky-theorem
Proof 6. Using duality relations we get the Slutsky-theorem:
∂xM
∂px
= ∂xH
∂px
−∂xM
∂e xM
∂xM
∂py = ∂xH
∂py −∂xM
∂e yM
∂yM
∂px
= ∂yH
∂px
−∂yM
∂e xM
∂yM
∂py =∂yH
∂py −∂yM
∂e yM
A little more precise, mathematically
xM =
xM1
...
xMi ...
xMn
,xH=
xH1
...
xHi ...
xHn
,p=
p1
...
pi ...
pn
Expenditure function:
e(p,U¯) ={min(px|U(x) = ¯U} Shephard-lemma:
∇e(p,U¯) =xH Marshall substitution matrix:
M=
∂xM1
∂p1
∂xM1
∂p2 · · · ∂x∂pM1
i . . . ∂x∂pM1
n
∂xM2
∂p1
∂xM2
∂p2 · · · ∂x∂pM2
i . . . ∂x∂pM2 ... ... n
∂xM1
∂pi
∂xMi
∂pi ...
... ... ...
∂xM1
∂pn · · · ∂x∂pMn
n
Hicks substitution matrix:
H=
∂xH1
∂p1
∂xH1
∂p2 · · · ∂x∂pH1
i . . . ∂x∂pH1
n
∂xH2
∂p1
∂xH2
∂p2 · · · ∂x∂pH2
i . . . ∂x∂pH2 ... ... n
∂xH1
∂pi
∂xHi
∂pi ...
... ... ...
∂xH1
∂pn · · · ∂x∂pHn
n
Slutsky-matrix
S=
∂xM1
∂p1 +∂x∂eM1 xM1 · · · ∂x∂pM1
n +∂x∂eM1 xMn
... ... ...
... ∂x∂pMii +∂x∂eMi xMi ...
... ... ...
∂xMn
∂p1 +∂x∂eMn xM1 · · · ∂x∂pMn
n +∂x∂eMn xMn
Slutsky-theorem:
S=H
Statement 3. The Hicks substitution matrix is symmetrical ∂x∂pHij = ∂x
H j
∂pi and has non-positive elements in the main diagonal ∂x∂pHii ≤0.
Proof 7 (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 thatq2=p2, then
(q1−p1)(y1−x1)≤0
∆pi∆xi|U=U0 ≤0 lim
∆pi→0
∆xi
∆pi
|U=U0≤0
∂xHi
∂pi
≤0
A little more precise, mathematically
Consequence 1. The main diagonal of the Slutsky matrix has non-positive elements.
Consequence 2. LAW of DEMAND: If good i is a normal good, then ∂x∂pMii ≤ 0, which means that its demand curve has a negative slope.
Proof 8. Since in the Hicks matrix ∂x∂pHii ≤ 0, therefore in the Slutsky matrix ∂x∂pMii + ∂x∂eMi xMi ≤ 0. For normal goods ∂x∂eMi >0, thus necessarily ∂x∂pMii ≤0.
Slutsky decomposition
• maximize: U(x, y)→maxx,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 −λpx= 0
• ∂L∂y = ∂U∂y −λpy= 0
• ∂L∂λ =Pxx+Pyy−pxx0−pyy0= 0 Slutsky demand functions:
xS(px, py, x0, y0) yS(px, py, x0, y0) Statement 4. Slutsky theorem[with Slutsky decomposition]
∂xM(px, py, pxx0+pyy0))
∂px = ∂xS(px, py, x0, y0)
∂px −∂xM
∂I x0
∂xM(px, py, pxx0+pyy0))
∂py = ∂xS(px, py, x0, y0)
∂py −∂xM
∂I y0
∂yM(px, py, pxx0+pyy0))
∂px =∂yS(px, py, x0, y0)
∂px −∂yM
∂I x0
∂yM(px, py, pxx0+pyy0))
∂py = ∂yS(px, py, x0, y0)
∂py −∂yM
∂I y0
Market demand
Summing individual demands
X ≡
N
X
i=1
xi
Hered1 andd2 are demand curves for two individuals. If these are the only two potential purchasers of the good, the overall market demand curveD is the horizontal sum ofd1 andd2.
Subsidy versus voucher
Voucher
The initial optimum is a corner solution atK; no education is purchased. A voucher gift of income in the amount KK0 leads to a new optimum atK00 The voucher leads to an increased consumption of education, provided only that education is a good rather than a bad for this individual.
