Lecture 5, Consumption theory

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```							                     Lecture 5, Consumption theory
Varian 7 and 8 (G & R ch 3)

Preference orders:
In order to examine the optimal choice of the consumer we need to be
able to describe the consumer’s preferences over different consumption
bundles in a systematic way.

We say that x   š'   x means that the bundle x is weakly preferred to x .
''                                            '                                  ''

1 Completeness
For any pair of bundles x and x , either x
'          ''              š'   x or x
''         š''   x (or both).
'

2 Reflexivity
x x
š'   '

3 Transitivity
For three bundles, x , x , x
'       ''       '''   , if xš'   x and x
''         š''   x
'''   then x   š'   x
'''      .

4 Continuity
For all budles x the better set {x: x
'                                      š    x } and the worse set {x: x
'                                ˜       x}
'
are closed.

5 Strong monotonicity
x x if x > x
™'   ''    '    ''

6 Strict convexity
Given any bundle x its better set (the set of bundles preferred or
'
indifferent to x ) is strictly convex.
'
The utility function:
A utility function is a function u assigning a real number to each
consumption bundle x so that

(a)   u(x ) = u(x ) iff x
'        ''      -'   x
''
(b)   u(x ) > u(x ) iff x
'        ''      ™'   x
''

A utility function is an ordinal function, and it is unique up to a positive
monotonic transformation.

What is needed in order to ensure that consumer preferences can be
represented by a utility function? 1-5 ensure existence of a continuous
utility function. (4 rules out lexicographic preferences). 5 and 6 imply
that u(x) is strictly quasi-concave.

The consumption decision
The consumer faces the following problem:

The budget set is bounded from below, since negative consumption is
not allowed, and from above by m. It is closed since all quantities on the
boundaries are available. It is convex. Any convex combination of
bundles belongs to the set. Finally, it is non-empty provided that m > 0.

The slope of the budget line is,

Along the indifference curves utility is constant, u(x) = u0. Thus,
Consequently the marginal rate of substitution between two goods, say
1 and 2, is given by,

Interior solutions and corner solutions:

Interior solutions - the Lagrange problem.

which implies that the optimal bundle x* must satisfy

Also by the envelope theorem it must be the case that
Example: Cobb-Douglas utility, e.g. x1 x21- . We simplify comptutations
"       "
by taking the logarithm (a positive monotone transformation) so that
u(x1, x2) = lnx1 + (1- )ln x2.
"          "

Thus the consumer spends the fraction    "   of her income on good 1 and
the fraction (1- ) on good 2.
"

The indirect utility function
By substituting the consumption bundles x in u(x) for the optimal
bundles x* we obtain the indirect utility function. The optimal
consumption choices, or the demand, depends on prices and income, i.e.
x*(p,m). Thus the indirect utility function gives the maximum utility
that can be achieved for certain prices and a certain income. The
indirect utility function is often denoted v(p,m).
Example: In the Cobb-Douglas case above the indirect utility function
is given by,

Note that the partial derivative with respect to m is equal to 1/m which
is presicely the value of the Lagrange multiplier in the example above.

Properties of the indirect utility function

!     v(p,m) is nonincreasing in p and nondecreasing in m.

!     v(p,m) is homogeneous of degree 0 in (p,m).

!     v(p,m) is quasiconvex in p.

!     v(p,m) is continuous in p >>0, m > 0.

The expenditure minimization problem
Instead of maximizing utility given a budget constraint we can consider
the dual problem of minimizing the expenditure necessary to obtain a
given utility level. Specifically, if we would like to reach the utility level
that results in the first problem it turns out that the bundle that
minimizes the cost of doing so coincides with the solution to the first
problem.

The FOC for expenditure minimization imply the same relation between
the prices and the marginal utilities as the FOC for utility maximization.
The solution to this problem is the optimal consumpion bundles as
functions of p and u. Income is adjusted so the consumer can afford the
cheapest possible bundle that yields u. These demand functions (one for
each good) are called compensated or Hicksian demand functions and
are denoted h(p,u). The minimal expenditure necessary to reach u is,

Example: Minimize expenditure p1x1+p2x2 subject to x1 x21- = u.
"   "

The first two conditions gives us x1 as a function of x2 and prices.
Inserting this into the constraint yields x2 as a function of prices and u.
We can then solve for x1 in terms of of prices and u. Finally, insertion
of the optimal demands into p1x1+p2x2 gives us the expenditure function.

Local non-satiation
This assumption implies that v(p,m) is strictly increasing in m. Thus we
can derive the minimal expenditure necessary to reach u, e(p,u), simply
by inverting v(p,m). It follows that e(p,u) is strictly increasing in u.
Properties of the expenditure function

!           e(p,u) is nondecreasing in p.

!           e(p,u) is homogeneous of degree 1 in p.

!           e(p,u) is concave in p.

!           e(p,u) is continuous in p.

!           M   e(p,u)/ pi = hi(p,u).
M
These are the same properties that cost functions have. To reach a higher
utility requires higher expenditures, otherwise we would not have been
optimizing before. The expenditure function is homogenous of degree
1 in prices since a proportional increase of all prices leaves relative
prices, and thus the optimal bundle, unchanged. Hence, the expenditure
increases in the same proportion as the prices. Concavity means that,

If x and x solve the min problem for p and p respectively then
'              ''                              '           ''
p x* p x and p x* p x , where x* solves the problem for p*=
'          \$    ' '            ''   \$   '' ''
tp + (1-t)p . Multiplying the inequalities by t and 1-t and summing
'                  ''
yields: (tp +(1-t)p )x*=p*x* tp x +(1-t)p x .
'            ''          \$    ' '       '' ''

Shepherd’s lemma again...

where e/ pi 0 and strictly so if xi*> 0.
\$ M M
Roy’s identity
If we insert the expenditure function into the indirect utility function we
must get the utility level in the cost minimization constraint, i.e.,

Differentiating this allows us to obtain Roy’s identity in an easy way.

Money metric utility functions
As was noted above local non satiation implies that e(p,u) is strictly
increasing in u. Since utility functions are only unique up to positive
monotone transformations we can use the expenditure function to define
m(p,x) = e(p,u(x)). For given p this is a “money metric” utility function
and for given x it is an expenditure function. Similarly, we can define
: (p;q, m) = e(p,v(q, m)) which measures the income required at prices
p to be as well off as with the income m at prices q. This measure is
useful in welfare analysis.

Comparative statics of consumer behavior
The solution to the consumer’s optimization problem gives us the
optimal demand for goods as functions of prices and income, xi(p,m).
An income expansion path depicts how consumption changes with
income and slopes upwards for normal goods. (Necessities & Luxury
goods)
Price offer curves trace out how consumption changes as prices change.
Demand decreases in price for ordinary good and increases for a Giffen
good.

Income and substitution effects
The own substitution effect: The change in consumption caused by the
change in relative prices keeping utility constant (by adjusting income).
The income effect: The difference in consumption between the above
point and the new optimal consumption bundle.

!    A normal good cannot be a Giffen good.

!    The own substitution effect is always opposite to the price change.

Deriving the Slutsky equation
Note that if we insert the expenditure function into the Marhallian
demand function this will give us the same demand as the Hicksian
demand function.
or on matrix form in the two good case,

Properties of demand functions

!    Since the e(p,m) is concave the matrix of substitution terms is
negative semi definite. Thus the diagonal terms - the own price
effects - are negative.

!    The matrix of substitution terms is symmetric.

If a set of demand functions give rise to symmetric and negative
semidefinite matrix of substitution terms then we can solve for the
indirect utility function and the expenditure function. (c.f. the condition
determining whether we can go from conditional demand functions to
the technology).

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