Econ 604 Advanced Microeconomics
Davis
Spring 2004
10 February 2004
Reading. Chapter 4 (pp. 91 to 113) today
Chapter 5 (pp. 116-144) for next time
Problems: To collect: Ch. 3: 3.2, 3.4, 3.5, 3.7
Next time: Ch.4 4.2, 4.4, 4.6, 4.9
Lecture #4.
REVIEW
III. Choice and Demand.
A. Chapter 3. Preference and Utility.
1. Axioms of Rational Choice
Completeness
Transitivity
Continunity
2. Utility
a. Nonuniqueness of Utility Measures
b. The Ceteras Paribus Assumption
c. Utility form Consumption of Goods
d. Arguments of Utility Functions
e. Economic Goods
3. Trades and Substitution
a. Indifference curves and the Marginal Rate of Substitution
b. Indifference Curve Map
c. Indifference Curves and Transitivity
d. Convexity of Indifference Curves.
e. Convexity and Balance in Consumption
4. An Alternative Derivation
5. Examples of Utility Functions
a. Cobb-Douglas Utility
b. Perfect Substitutes
c. Perfect Complements
d. CES Utility
6. Generalizations to More than Two Goods
Some observations:
1) The relationship between quasi-concavity and convexity. Notice that convexity in x1, x2 space has a symmetrical notion of concavity. However, in U(x1, x2 space, strict concavity implies something a bit weaker than concavity in the utility function. This is the quasi-concavity condition we developed last time in class.
2) The notion of utility that we’ve developed is based on relatively weak assumptions, and we can include a number of unconventional arguments. Consider some of these.
a. Addictive Behavior: If utility for a good at a given time is a function of past consumption of the good, , then we write
Ut(Xt, Yt, St) where St = SXt-1 where St denotes all previous consumption. In fact consumption in previous periods is often unobservable, so economists often write,
Ut(Xt*, Yt,) where Xt* = Xt - Xt -1.
b. Second Party Preferences. It is not necessary that individuals confine attention only to individual well being. For example, we might write
Ui(X, Y,, Uj) where i and j denote different individuals. Ui>0 implies altruistic tendencies, Uj<0 implies malevolence.
Xt* = Xt - Xt -1.
Sometimes, related functions, such as those illustrating tastes for fairness are also of interest.
PREVIEW
IV. Utility Maximization and Choice
A. An Introductory Illustration. The two good case.
1. The Budget Constraint
2. First Order Conditions for a Maximum
3. Second Order Conditions for a Maximum
4. Corner Solutions
B. The n-Good Case
1. First Order Conditions
2. Implications of First Order Conditions
3. Interpreting the LaGrangain Multiplier
4. Corner Solutions
C. Indirect Utility Function
D. Expenditure Minimization
LECTURE______
IV. Utility Maximization and Choice. This chapter continues our development of rational choice from a utility function. In the last chapter, we introduced the utility function. Now we consider rational decisions, given a budget constraint. As we will see, although the theory has several variants, they all appeal to a single basic theme.
The basic idea is simple: Suppose that we can divide each of n possible goods into “boxes” that cost $1 each. Now suppose that the individual goes to a store with a fixed budget. The individual will maximize utility by purchasing those “$1 boxes” that yield the highest marginal utility. Now, of course, goods don’t all cost $1 per box. In this case, we can convert them all to dollar boxes by dividing the marginal utility by the price. Thus, given a vector of choices, 1, …., n, the consumer will maximize utility by picking so that
U1/p1 = U2/p2 = … = Un/pn.
If this condition does not hold, the consumer will increase utility the most by purchasing the product that yields the highest MU per dollar spent. This will, incidentally, reduce the marginal utility, and bring the equation back toward equality. On the other hand, if a price falls suddenly the equation is out of equality, an the newly inexpensive good yields higher marginal utility per dollar spent. The rational consumer will purchase more units of the good.
Observe that humans need not have super calculating capabilities to make decisions that follow with this rule. All they need to be able to decide is which of the n available options yields the most “bang for the buck”
In what follows, we develop this rationale in more detail, starting with the 2 good case, and then proceeding to the more general case of n goods.
A. An Introductory Illustration. The two good case.
1. The Budget Constraint. Assume consumers have available 2 goods, X and Y that may be purchased at price PX and PY, respectively. They may spend up to their total income I on these products. This budget constraint may be written
PXX + PYY I
To graph this in a two dimensional space, write in slope/intercept form
Y = I/PY - (PX/PY)X
Notice that the slope of this constraint is -PX/PY
All consumption bundles up to those along the budget constraint are available. However, only the assumption that more is preferred to less is necessary in order to conclude that a rational consumer will confine his or her attention only to points on the budget frontier.
2. First Order Conditions for a Maximum Now overlay on this budget constraint some indifference curves, as shown in the pane on the next page. Observe that with the usual ordering (U3>U2>U1)
the consumer maximizes utility at point C. At this point, the slope of the budget constraint and the MRS are equal. Or
Ux/Uy = - Px/Py.
Or, alternatively,
Ux/Px = Uy/Py.
3. Second Order Conditions for a Maximum As long as the indifference curves are convex, this point of tangency will be a unique solution. (Thus, the second derivative of Y with respect to X must be strictly positive.) “Wiggles” in indifference curves undermine this unique tangency result. For example, in the figure shown in to the left, a non-convexity in the indifference curves implies that the point C is a point of tangency, but not a point of maximum utility. Rather D maximizes Utility.
4. Corner Solutions Also, it may be the case that due either to relative prices or to weak preferences for a good, consumers may consume only one good. This is a “corner solution.” In the illustration, only units of good X are optimally consumed.
Our first order conditions must be modified to allow for this solution.
B. The n-Good Case. The graphical results derived above carry over directly to the case of n goods, a case which is better evaluated analytically.
1. First Order Conditions
Consider the utility function
Utility = U(x1, x2, …., xn)
We maximize subject to the constraint (expressed in implicit form)
I- p1x1 - p2x2 - …. - pnxn = 0
Following the Lagrangian technique, we write
L = U(x1, x2, …., xn) + l(I-p1x1-p2x2 -…. - pnxn)
and taking first order conditions, yields
¶L/¶x1 = ¶U/¶x1 - lp1 = 0
¶L/¶x2 = ¶U/¶x2 - lp2 = 0
.
.
.
¶L/¶xn = ¶U/¶xn - lpn = 0
¶L/¶l = I-p1x1-p2x2 -…. - pnxn = 0
This is n+1 equations in the n+1 variables x1, x2,… xn, l, which can usually be solved for a maximum.
2. Implications of First Order Conditions. Notice that for any two goods i and j above, we solve the above as
Uxi/Uxj = pxi/pxj
This is the tangency condition derived for the case of two goods.
3. Interpreting the LaGrangain Multiplier. Notice also, solving for l yields
l = Ux1/px1 = Ux2/px2 =…. = Uxn/pxn.
Or, equivalently
l = MUx1/px1 = MUx2/px2 =…. = MUxn/pxn.
Thus, l reflects the marginal utility of an extra dollar of income. Notice that in an equilibrium, this marginal utility is equal for all goods.
Finally, taking l and any particular good i yields
px1 = MUx1/l
That is, for any good purchased, the price of the good reflects the individual’s utilty for the last unit consumed.
4. Corner Solutions The first order conditions hold only for interior maxima. When “corner” solutions arise, we have to rewrite the first order conditions
¶L/¶xi = ¶U/¶xi - lpi 0 (i= 1,… n)
and if
¶L/¶xi = ¶U/¶xi - lpi 0
then xi = 0.
These conditions, called Kuhn-Tucker conditions are solutions for linear programming. To interpret the latter condition (e.g,. the strict inequality) rewrite it as
pi MUxi/l
That is, the price exceeds the marginal utility of a good, so none of it is purchased.
5. Example: Cobb Douglas Demand Functions.
a. General case.To illustrate these points, consider constrained optimization for the Cobb-Douglas utility function, with 2 argurments. Specifically, let
U(x,y) = xayb where a+b=1
Define the income constraint as
I - pxx - pyy
To optimize, write the Lagrangian expression
L= = xayb + l( I - pxx - pyy)
Taking First order conditions
¶L/¶x = axa-1yb - lpx = 0
¶L/¶ y = bxayb-1 - lpy = 0
.
¶L/¶l = I-pxx-pyy = 0
The ratio of the first 2 equations yields
ax/by = px/py
or pyy = (1-a)pxx/a
Substituting this expression along with pxx into the budget constraint yields
I = pxx + (1-a)pxx/a = pxx/a
Solving for optimal x* and y* yields
x* = I a/px y* = I b/py
Notice what this solution implies. An individual will always spend a percent of his or her income (pxx/I) on good x. Similarly, this individual will spend b percent of his or her income on y pyy/I.
This constant percentage expenditure of the Cobb-Douglas function makes it very easy to work with. However, it can be limiting. More generally, we would expect the share of income devoted to particular goods to change with income. As our next example, we consider a more general utility function. Consider first a numerical illustration of the above example.
b. A numerical example. Using the above functional form, suppose that x px = $.25 , py = $1.00 and I = $2.00. Suppose further that a=b=.5. Then
x* = 2(.5)/.25 = 4 and y* = 2(.5)/1 = 1
Given these optimal choice, U = 4.51.5 = 2.
Notice also that l = .5(4) -.5(1).5/.25 = 1.
This implies that small changes income will lead to comparable changes in utiltity. To see this, suppose that income increase to I= 2.1. Then
x* = 2.1(.5)/.25 =4.2 and y* =2.1(.5)/1 = 1.05
and U = 4.2.51.05.5= 2.10
]
6. Example: CES Utility.
CES Demand Now consider an example where expenditures are responsive to economic conditions. Suppose
U(x,y) = x.5+y.5
The Lagrangian expression becomes
L= = x1+y.5 +l( I - pxx - pyy)
Taking First order conditions
¶L/¶x = .5x-.5 - lpx = 0
¶L/¶y = 5y-.5 - lpy = 0
.
¶L/¶l = I-pxx-pyy = 0
Taking the ratio of the first two equations yields
(y/x).5 = px/py, or, for example, y = (px/py)2x
Substituting into the budget constraint yields
I = pxx + py(px/py)2x
Or x* = I/(px[1+px/py]) and y* = I/(py[1+py/px])
Notice here that these demand functions are not constant. Rather they depend on the ratio px/py, or the relative price of the two goods.
CES with less substitutibility. Consider
U(x,y) = -x.-1+y.-1
It is easy to show that the first order conditions imply
y/x = (px/py).5
Substituting into the budget constraint
Or x* = I/(px[1+py/px].5) and y* = I/(py[1+px/py] .5)
These demand functions are less responsive to relative price changes.
.
C. Indirect Utility Function. As the above examples illustrate, it is often possible to manipulate first order conditions for the optimal values of x1, x2, … xn. In general, these optimal values will depend on the prices of all goods and an individual’s income. That is,
x1* = x1(p1, p2,…. ,pn, I)
x2* = x2(p1, p2,…. ,pn, I)
.
.
.
xn* = xn(p1, p2,…. ,pn, I)
These expressions are demand functions, which illustrate the relationship between a good, own price, the price of complements and substitutes, and Income. We will study these in later chapters. For the time being, insert these back into the orginial utility function, to yield
Maximum utility = U(x1*, x2*, …., xn*)
= U(x1*(p1, p2,…. ,pn, I),
x2*(p1, p2,…. ,pn, I)
…..
xn* (p1, p2,…. ,pn, I))
= V(p1, p2,…. ,pn, I)
That is, because an individual optimizes utility in light of a budget constraint, we can write the individual an individual’s optimal utility in terms of prices and the income constraint, because these parameters indirectly determine utility. This indirect utility function will be useful later in the text. Of course, if either prices or income change, so in general will the level of utility. This indirect expression will be useful later in the course, when, for example, we attempt to evaluate the effects of a change in costs conditions on outcomes.
2. Example Indirect utility and the Cobb-Douglas function. Recall in the numerical example of the the Cobb Douglas function that
Using the above functional form, suppose that x px = $.25 , py = $1.00 and I = $2.00. Suppose further that a=b=.5. Then
x* = I/2px and y* = I/2py
Substituting into the utility function yields
Maximum Utility - U(x*,y*) = x.5y.5
= (I/2px).5(I/2py).5
= I/2px.5 py.5
Notice that with I=2, px = .25 and py=1, Utility can be computed as
= 2/2[(.25).5(1).5]
= 2
By stating utility in terms of the externally observable parameters, it is possible to study explicitly the effects of a parameter on a system
D. Expenditure Minimization. As we pointed out in chapter 2, maximization problems have associated with them “dual” minimization problems. That logic applies to the current situation as well. Rather than maximizing utility subject to a, we might vary possible expenditure levels E1, E2, and E3 in order to minimize the expenditures necessary to achieve a level of utility U This is illustrated in the above panel.
Mathematically, the problem is to mminimze total expenditures, E, subject to a given utility level, U