Micro I. Lesson 5 : Consumer Equilibrium

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5.1 Optimal Choice

If preferences are well behaved (smooth, convex, continuous and negatively sloped), then at the optimal choice of the consumer, Slope of i.c. = Slope of b.c. MRS = Price ratio MUp xx= MUpyy

Optimal Choice

x Why? Consider a situation in which the above equality does not hold. Also, remember that moving along the bc is equivalent to using the market. Moving along the ic shows the minimum amount of y I need to compensate for loss of x. If the market gives me an amount of y that is greater than the minimum I require, I will follow the market. Microeconomics I. Antonio Zabalza. University of Valencia 2

U(C) U(B) B B’ U(A) A x B’’

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At A, for instance, if I give up 1 unit of x (distance AB’’), the market gives me 1 unit of y (distance B’’B). But I would be satisfied with less; say, 0.25 units of y (distance B’B’’). Then, it is optimal for me to trade in the market, and go to point B where my utility U(B) is higher than that at point a, U(A).

If I keep applying this reasoning I end up at point C. (Check that you understand this). Point C represents the best I can do, given my opportunities. Point C therefore represents the optimal choice, the equilibrium, of the consumer. Going beyond point C, would lower again my utility.

Mathematics: Maximization of utility subject to a given budget constraint. Method of Lagrange.

Max U=Uxy(,) xy, s.t. px+=pym xy

éù L=Ux(,)y+l ëûm--pxyxpy

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First order (necessary) conditions

ddLU =-=l p 0 (1) ddxx x ddLU =-=l p 0 (2) ddyy y d L =m-px-=py 0 (3) d y xy

This is a system of 3 equations in 3 unknowns: x, y and l. From (1) and (2) we have that

ddUxMU l ==x pp xx ddUyMU l ==y ppyy

MUMU MUp x=Þ=y xx pxpyMUpyx

Then, to solve for x and y we consider equation (3) to get this more simplified form of the above system.

MUp xx= (4) MUpyx

m=+pxyxpy (5) Microeconomics I. Antonio Zabalza. University of Valencia 5

This is a system of 2 equations with two unknowns (x,y). (Notice that in general the marginal utilities will depend on x and y). Equation (4) is the equality of slopes of bc and ic discussed above. What the result above says is that this condition is not enough; we need also that the budget constraint is fulfilled (equation 5).

Solving this system will, in general, give us the two demand functions for x and y that we are after.

x= x(ppmxy,,)

y= y(,ppmxy,)

The purpose of this lesson is to find out how the three variables (,ppmxy,) influence the demand for x and y. Before, we give an example of this derivation for a particular utility function: Cobb- Douglas.

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Example: Cobb-Douglas (CD) utility function: Ux(,)y= xyab Max U= xyab xy, s.t. px+=pym xy

ab éù L=+xyl ëûm--pxyxpy

Necessary conditions: d L =axab-1yp-=l 0 (1) d x x d L =bxabyp-1 -=l 0 (2) d y y d L =m--=pxpy 0 (3) d y xy

Eliminate l from (1) and (2), and together with (3) you obtain axa--11ybbxyab = (4) ppxy m=+pxyxpy (5)

System of two equations with two unknowns. Microeconomics I. Antonio Zabalza. University of Valencia 7

Equation (4) can be expressed in the form MUp xx= , MUpyy which for this particular case is ab-1 axy px ab-1 = bxypy or, ay p = x . bxpy

So the system, in this simplified form is ay p = x (4) bxpy

m=+pxyxpy (5)

Solving for x and for y, we find the two demand equations: am x = a+ bp x bm y = a+ bpy

With a CD utility function, the demand for each good depends on income (positively) and its own price (negatively). It does not depend on the price of the other good.

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Another characteristic of this utility function is that the parameters of the function give information about the expenditure shares on each good.

px a Share of expenditure on x : x = mab+

py b Share of expenditure on y: y = mab+

Sufficient condition

Equations (1), (2) and (3) are the necessary conditions. They are not sufficient. For instance, consider the following situation:

Point of

tangency

Point of maximum A utility

At A, the first order conditions are met and yet utility is not maximized. You need also another set of conditions which are sufficient. These conditions boil down to the requirement that preferences have to be convex. See that in the figure they are concave.

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Extreme examples

Corner solutions

Maximum utility, but at this point Slope of ic > slope bc MRS > price ratio

A particular example of a corner solution is when the two goods are perfect substitutes: Suppose

U=x+y and pxy=2; pm==4; and 12.

ic slope: 1

bc slope: 1/2

Maximum U

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Here pMU 1 xx<<; 1 pyyMU 2 Maximum is obtained at point A, where only x is consumed. So the demand function in this case is m x = px Kinky solutions

Suppose the two goods are perfect complements with the following utility function: U= min,{xy} and pxy=2; pm==4; and 24.

x 4 12

The optimal choice must lie on the diagonal and on the budget constraint. Therefore, the optimal choice is found by solving the system

Microeconomics I. Antonio Zabalza. University of Valencia 11 yx=

m=+pxyxpy The solution is m xy== ppxy+ For the particular example used here 24 xy===4 6

5.2 Changes in the equilibrium position

Now we want to investigate how the equilibrium just studied is altered (displaced) by changes in the exogenous variables of this problem. In particular, we want to know how the equilibrium changes when, m, ppxy and change. Or to put it in other words. The result of the previous analysis was the derivation of two demand curves

x= x(ppmxy,,)

y= y(,ppmxy,) We want to sign the partial effects of the three exogenous variables m, ppxy and on the demand of x and y.

We will consider three types of changes: a) Simultaneous change in prices and income by the same proportion. b) Change in income only. Microeconomics I. Antonio Zabalza. University of Valencia 12

c) Change in one price only.

Equiproportional change in prices and income

If m, ppxy and all move by the same proportion, the bc does not change and therefore the point of equilibrium does not change either.

Suppose initial bc is 000 m=+pxyxpy Multiply all prices and income by k (if, for instance, k=1.1, then all variables increase by 10%). The new bc is 000 km=+kpxyxkpy But k can be cancelled out by dividing both sides of the equation by k. So, the original bc remains unchanged. 000 m=+pxyxpy

Change in m only

We know that an increase in m moves the bc out. The position of the final equilibrium depends on whether the goods are normal or inferior. Suppose first that both goods are normal. Then, if there is an increase in income from mm0 to ¢, more of both goods will be bought. This is represented in the following figure.

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0 B m

Dy m¢ A

x Dx The demand curve that treats prices as given parameters and income as a variable, is known as the

Engle curve. It takes the form: x= xm(,,)ppxy.

(ppxy,)

m¢

0 m A

x 0 x x¢

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We say that a good is inferior if when income is raised, holding everything else constant, less of this good is bought. Say x is inferior. Then,

0 m Dy

m¢ A

x Ñx

Exercise: Derive the shape of the Engle curve corresponding to good x, when x is an inferior good.

Property: The sum of the income elasticities of each good, weighted by its corresponding expenditure share, must be equal to one.

We start with the budget constraint:

m=+pxyxpy

Then, differentiating both sides of the equality by m, we find: Microeconomics I. Antonio Zabalza. University of Valencia 15

dmxy¶¶ =+pp dmmmxy¶¶ ¶¶xy 1 =+ppxy ¶¶mm

pxx æö¶¶xmpyy æöym 1 =+ç÷ç÷ mèø¶¶mxmèømy

1 =+ssxeexmyym

As we were looking for, the weighted average of income elasticities must add up to 1.

Implications of this result: a) Not all goods can be inferior. Not all Engle curves can have negative slope. b) Goods that take a large share of expenditures are unlikely to have either very large or very low income elasticities, since the average must equal one.

Problem for home: Say we divide goods in two types: food and non-food. We know food takes 60% of expenditures, and the income elasticity of non- food is 2. What is the income elasticity of food?

Exercise: Find the slope and graphical shape of the Engle curve for good x when the utility function is Cobb – Douglas. What about when the goods are perfect substitutes? And perfect complements? You Microeconomics I. Antonio Zabalza. University of Valencia 16 will see in this exercise that all Engle curves for these specific cases are straight lines. This is like this because the points of equilibrium in the goods space when income rises is also a straight line from the origin. In these cases we say that preferences are homothetic.

Change in one price (holding constant m and the other price)

Suppose px decreases with p y and m constant.

B 0 p x ' p A x

x Dx

Normally if the price of one good goes down, the quantity consumed of that good goes up.

Ñpxx ÞD and viceversa.

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The demand curve The following graph shows how the demand curve is derived out of the consumer equilibrium. In fact, each point in the demand curve is a point of equilibrium for the consumer at different price levels.

B 0 px ' p A x

x Dx

0 A px

1 B px Demand curve x(ppm,,) x y

x Dx Microeconomics I. Antonio Zabalza. University of Valencia 18

Exercise: Draw the demand curves (and identify the slope of the curve) for CD preferences, for perfect substitutes and for perfect complements.

The demand curve not always is downward sloping. This is an anomaly, but in principle it can happen.

0 px ' px A

x Dx Demand curve p x x(ppmx ,y ,)

0 A px

B 1 p x

x Ñx Microeconomics I. Antonio Zabalza. University of Valencia 19

Goods which display this anomalous behaviour are called Giffen goods. [Illustration concerning consumption of horse meat].

5.3 Income and substitution effects

Now we want to decompose the effect of the change in one price in two effects: a) a first effect which is equivalent to a change in relative prices holding income constant. This is called the substitution effect. b) A second effect which is equivalent to a change in income holding relative prices constant. This is called the income effect.

Suppose initially we are at point A (xy00,), with 00 0 prices (,)ppxy and income m . Therefore, the budget constraint at this point is: 00000 pxyx+=pym (1)

1 Suppose now the price of x goes down to px , and we ask what is the income the consumer would now need to buy the old bundle of goods. This income, m1, is 10001 pxyx+=pym (2)

Clearly mm10< . More precisely, the change in income can be found substracting equation (1) from equation (2). Microeconomics I. Antonio Zabalza. University of Valencia 20

1010000000 m-m=+pxxpyy-+()pxyxpy 010 =-x()ppxx 0 D=Dmxpx So, the change in income needed to buy the old bundle is equal to the initial quantity of x times the change in the price of x.

Suppose after the price change in x we take away Dm from the consumer, so his new bc passes through A but is flatter than his old bc. Will he remain at A? Clearly not. He can do better than this by going to B (xyss,). As compared with point A, the consumer has adjusted his consumption by buying more of the good that has become relatively cheaper (x) and less of the good that has become relatively more expensive (y). The change from point A to point B is the substitution effect.

We can represent it formally as a change in the demand for x from the initial position 0000 x= x(,,)ppmxy at point A, to the position s 101 s 0 x= x(,,)ppmxy at point B. If we denote ()xx- by Dxs , then

s 101000 Dx=-x(,,ppxxm)x(,ppmxy,)

s 101 x= x(,,)ppmxy is called the compensated demand for x. “Compensated” because is the demand for x as Microeconomics I. Antonio Zabalza. University of Valencia 21 a result of a fall in the price of x when the consumer is compensated for the increase in income generated by the fall in the price of x.

Income effect

The movement from A to B is a hypothetical movement. At his final choice, the consumer is spending all his income; so he will be at a point such as C in the final budget line, where the demand for x n 100 is x= x(,,)ppmxy . Now, observe that the move from B to C is like a pure income effect (income increases from mm10 to , while prices remain 10 nsn constant at ppxy and ). If we call x-xx=D , we have that the income effect is,

n 100101 Dx=-x(,,ppxym)x(ppmxy,,)

Total effect

The total effect is the move from A to C. That is from xx0 to n. Or using the same terminology as above, Dx=-xxn 0 100000 Dx=-x(,,ppxym)xp(xy, pm,)

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000 (,ppmxy,)

101 (,,)ppm xy

n C y 0 y A

B s y 100 (,ppmxy,)

x 0 s n x x x

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The Slutsky equation

Notice that the total effect can be written as the sum of the substitution and income effects.

(xns-=-+-x00)(xx)()xxns

Dx=Dxxsn+D

This expression is called the Slutsky equation.

Signing the substitution effect

If px goes down, as in the figure above, then the change in demand for x that results from the substitution effect must be non-negative.

10101000 If px

Why is this so? Because the indifference curves are well behaved (continuous, smooth, negatively sloped and convex). Convince yourself that with this type of i.c. it cannot be otherwise. Points to the left of A will lie on lower i.c. and therefore will not be chosen. Conclusion: The substitution effect always moves opposite to the price movement. We say the substitution effect is negative: if the price goes down, the demand for the good due to the substitution effect increases, and vice versa. Microeconomics I. Antonio Zabalza. University of Valencia 24

Signing the total effect

Contrary to what happens with the substitution effect, the total effect can be signed most of the times, but not always. It depends on whether the good is normal or inferior. To see that, recall the Slutsky equation.

Dx=Dxxsn+D

Normal goods Dx=Dxxsn+D

(-=-+-)()() Both substitution and income effects work in the same direction. Consequently, as price goes down, quantity demanded goes up, and vice versa. The total effect is negative.

Check you understand why income effect in this case is negative. px ¯Þmx•Þ•. Price and good demanded due to income effect move in opposite directions.

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Inferior goods Dx=Dxxsn+D

(?)=-++()()

Here the sign of the final effect depends on the relative strength of substitution and income effects. We have two possibilities:

a) Substitution effect dominates in absolute terms. Then the total effect is negative. This is what will usually happen. b) Income effect dominates in absolute terms. Then the total effect is positive: as price goes down, quantity demanded goes up and vice versa. This means that the demand curve is upward sloping. We call this type of goods, Giffen goods. They are very rare.

A conclusion regarding inferior and Giffen goods: a Giffen good must be an inferior good, but no all inferior goods are Giffen goods.

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Expressing the Slutsky equation in terms of rates of change with respect the change in price.

Dx=Dxxsn+D DDDxxxsn =+ (1) DDDpppxxx

Recall previous result,

Dm=Dxpx

If the price goes down, this expression gives us a negative number (the amount of income that, as a result of the price fall, has to be taken away from the consumer so that he can just afford the old bundle A). To identify the income effect from B to C we want to work with the negative of this (negative) amount: specifically, with the amount of money that is given to the consumer so that he can go from the budget AB to the final budget at C. For this purpose we define a new change in income, ()Dn , which is just the negative of the previously defined income change, ()Dm , Dnm=-D

Dn=-Dxpx Dn Dp =- (2) x x

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Substituting (2) into (1)

DxDDxxsn =-x DpxxDDpn

Which is the Slutsky equation expressed in terms of rates of change. Notice that now the income effect is expressed directly as a change in x due to a change in income and, therefore, for a normal good, is positive.

From this equation we state the Law of Demand: If the demand for a good increases when income increases (that is, if the good is normal), then the demand for that good must decrease when its price increases.

DxDDxxsn =-x DpxxDDpn (-=--)()(++)()

Naturally, this can be said because we know the substitution effect is always negative.

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Another way of measuring the substitution effect: the “Hicks” substitution effect.

The previous way of measuring the substitution effect was proposed by an economist called Slutsky. Another economist (John Hicks) proposed another way of identifying the substitution effect. To compare them we define both:

Slutsky substitution effect: Change in demand when prices change but the consumer’s purchasing power is held constant so that the original bundle remains affordable.

Hicks substitution effect: Change in demand when prices change but the consumer’s income is changed so that he can reach his original utility level. That is, change in demand when prices change but consumer’s utility is held constant at its original level.

For small (infinitesimal) changes in prices both measures coincide.

For large (non infinitesimal) changes in prices they differ. This can be seen graphically.

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000 (,ppmxy,)

101 (,,)ppmxy

B 100 B’ (,ppmxy,)

x x0 xs xn sH p x x

0 A p x

C 1 px B’ Demand curve (dc)

Compensated dc (Slutsky) Compensated dc (Hicks)

x Microeconomics I. Antonio Zabalza. University of Valencia 30

In the graph we identify three demand curves:

Usual demand curve (dc):

x= x(ppmx ,y ,)

Compensated dc (Slutsky):

x= x(ppx ,y ,purchasing power)

Compensated dc (Hicks):

x= x(ppux ,y ,)

Check you understand the ceteris paribus clause of each of these three demand curves.

Exercise: How would the graph below look like if good x instead of being normal was inferior? (What will be the relative configuration of the three demand curves?)

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5.4 Implications of the MRS conditions

Observation of demand behaviour can give us information about the underlying preferences of consumers who display that behaviour.

In a competitive market, prices are the same for everybody. Thus, if consumers are at equilibrium positions,

px MRS123==MRSMRS=... ==MRSn py Everybody will adjust their consumption of goods until their own “internal” marginal valuation (MRS) equals the market’s external valuation ()ppxy. Marginal changes in consumption, therefore, will be valued the same for everybody.

Example (Varian’s): Suppose that in a competitive market one bottle of milk costs 1€ and one pack of butter costs 2€. This means that,

MUp MRS =bb==2 MUpmm

Any project (policy) that gives people goods for more than what they value them is profitable, and vice versa.

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Project A: 1 pack of butter is produced with 3 bottles of milk. This is not a good project. Using market prices, it is equivalent to saying that 2€ are produced with 3€. People value more the inputs than the output of this project.

Project B: 3 bottles of milk are produced with 1 pack of butter. This project is OK. Using market prices, it is equivalent to produce 3€ with 2€. Here people value inputs less than output.

Conclusion: Prices are not arbitrary things; rather, they reflect how people value things at the margin.