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1
Chapter 4
UTILITY MAXIMIZATION
AND CHOICE
2
Our Consumption Choices
Suppose that each month we have a
stipend of $1250.
What can we buy with this money?
3
What can we buy with this money?
Pay the rent, 750 $.
Food at WholeFoods, 300 $
Clothing 50 $
Gasoline, 40 $
Movies and Popcorn, 60 $
CD’s, 50 $
Tennis racket ?
Cable TV?
4
What can we buy with this money?
Pay the rent, 700 $.
Food at Ralph’s, 200 $
Clothing 40 $
Gasoline, 80 $
Movies and Popcorn, 60 $
CD’s, 20 $
Tennis racket 150 $
Cable TV 50 $
5
Which is Better?
If we like healthy food and our own bedroom, we will choose the first option
If we like crowded apartments and we absolutely need the Wilson raquet, we will choose the second option
The actual choice depend on our TASTE, on our PREFERENCES
6
Our Consumption Choices
Our decisions can be divided in two parts
We have to determine all available choices, given our income: BUDGET CONSTRAINT
Of all the available choices we choose the one that we prefer: PREFERENCES and INDIFFERENCE CURVES
7
UTILITY MAXIMIZATION PROBLEM
8
Basic Model• There are N goods.
• Consumer has income m.
• Consumer faces linear prices {p1,p2,…,pN}.
• Preferences satisfy completeness, transitivity
and continuity.
• Preferences usually satisfy monotonicity and
convexity.
• Consumer’s problem: Choose {x1,x2,…,xN} to
maximize utility u(x1,x2,…,xN) subject to budget
constraint and xi ≥ 0.
9
Budget Constraints• If an individual has m dollars to allocate
between good x1 and good x2
p1x1 + p2x2 m
Quantity of x1
Quantity of x2The individual can afford
to choose only combinations
of x1 and x2 in the shaded
triangle
If all income is spent
on x2, this is the amount
of x2 that can be purchased2p
m
If all income is spent
on x1, this is the amount
of x1 that can be purchased
1p
m
Slope=-p1/p2
10
Budget Constraints
• Suppose income increases
– Then budget line shifts out
• Suppose p1 increases
– Then budget line pivots around upper-left
corner, shifting inward.
11
Nonlinear Budget Constraints
• Example: Quantity Discounts.
• m=30
• p2=30
• p1=2 for x1<10
• p1=1 for x1>10
12
Consumer’s Problem (2 Goods)• Consumer chooses {x1,x2} to solve:
Max u(x1,x2) s.t. p1x1+p2x2 ≤ m
• Solution x*i(p1,p2,m) is Marshallian Demand.
Results:
1. Demand is homogenous of degree zero:
x*i(p1,p2,m)= x*i(kp1,kp2,km) for k>0
Idea: Inflation does not affect demand.
2. If utility is monotone then the budget binds.
13
UMP: The Picture
• We can add the individual’s preferences
to show the utility-maximization process
Quantity of x1
Quantity of x2
U1
A
The individual can do better than point A
by reallocating his budget
U3
C The individual cannot have point C
because income is not large enough
U2
B
Point B is the point of utility
maximization
14
Soln 1: Graphical Method• Utility is maximized where the indifference curve
is tangent to the budget constraint
• Equate bang-per-buck, MU1/p1 = MU2/p2
Quantity of x1
Quantity of x2
U2
B
constraintbudget of slope2
1
p
p
21constant 1
2 curve ceindifferen of slope
x
U
x
U
dx
dx
U
212
1
x
U
x
U
p
p
At optimum:
15
Soln2: Substitution Method• Rearrange budget constraint: x2 = (m-p1x1)/p2
• Turn into single variable problem:
Max u(x1,(m-p1x1)/p2)
• FOC(x1): MU1 + MU2(-p1/p2) = 0.
• Rearrange: MU1/MU2 = p1/p2
• Example: u(x1,x2) =x1x2
16
Soln3: Lagrange Method
• Maxmize the Lagrangian:
L = u(x1,x2) + [m-p1x1-p2x2]
• First order conditions:
MU1 - p1 = 0
MU2 - p2 = 0
• Rearranging: MU1/MU2 = p1/p2
• Example: u(x1,x2) =x1x2
17
LIMITATIONS OF FIRST ORDER
APPROACH
18
1. Second-Order Conditions
• The tangency rule is necessary but not
sufficient
• It is sufficient if preferences are convex.
– That is, MRS is decreasing in x1
• If preferences not convex, then we must
check second-order conditions to ensure
that we are at a maximum
19
1. Second Order Conditions
• Example in which the tangency condition is
satisfied but we are not at the optimal bundle.
Quantity of x1
Quantity of x2
B
A
A = local min
B = global max
C = local maxB
A
C
20
1. Second-Order Conditions
• Example: u(x1,x2) = x12 + x2
2
• FOCs from Lagrangian imply that
x1/x2 = p1/p2
• But SOC is not satisfied: L11 = L22 = 2 > 0
• Actual solution:
– x1 = 0 if p1>p2
– x2 = 0 if p1<p2
21
2. Corner Solutions• In some situations, individuals’ preferences
may be such that they can maximize utility
by choosing to consume only one of the
goods
Quantity of x
Quantity of yAt point A, the indifference curve
is not tangent to the budget constraintU2U1 U3
A
Utility is maximized at point A
22
2. Corner Solutions
• Boundary solution will occur if, for all (x1,x2)
MRS>p1/p2 or MRS<p1/p2
• That is bang-per-buck from one good is always
bigger than from the other good.
• Formally, we can introduce Lagrange multipliers
for boundaries.
• If preferences convex, then solve for optimal
(x1,x2). If find x1<0, then set x1*=0.
23
2. Corner Solutions
• Example: u(x1,x2) = x1 + x2
• MRS = - /
• Price slope = -p1/p2
• Which is bigger?
24
3. Non-differentiable ICs
• Suppose preferences convex.
• At kink MRS jumps down.
• Solution occurs at kink if
MRS- ≥ p1/p2 ≥ MRS+
25
3. Non-differentiable ICs
• Perfect complements: U(x1,x2) = min (x1, x2)
• Utility not differentiable at kink.
• Solution occurs at:
x1 = x2
26
GENERAL THEORY
27
The n-Good Case
• The individual’s objective is to maximize
utility = U(x1,x2,…,xn)
subject to the budget constraint
m = p1x1 + p2x2 +…+ pnxn
• Set up the Lagrangian:
L = U(x1,x2,…,xn) + (m - p1x1 - p2x2 -…- pnxn)
28
The n-Good Case• First-order conditions for an interior
maximum:
L/x1 = U/x1 - p1 = 0
L/x2 = U/x2 - p2 = 0•••
L/xn = U/xn - pn = 0
L/ = m - p1x1 - p2x2 - … - pnxn = 0
29
Implications of First-Order Conditions
• For any two goods,
j
i
j
i
p
p
xU
xU
/
/
• This implies that at the optimal
allocation of income
j
iji
p
pxxMRS ) for (
30
Indirect Utility Function• Substituting optimal consumption in the utility
function we find the indirect utility function
V(p1,p2,…,pn,m) = U(x*1(p1,…,pn,m),…,x*n(p1,…,pn,m))
• This measures how the utility of an individual
changes when prices/income varies
• Enables us to determine the effect of
government policies on utility of an individual.
31
Indirect Utility: Properties1. v(p1,..,pn,m) is homogenous of deg 0:
v(p1,..,pn,m) = v(kp1,..,kpn,km)
Idea: Inflation does not affect utility.
2. v(p1,..,pn,m) is increasing in income and
decreasing in prices.
3. Roy’s identity:
If p1 rises by $1, then income falls by x1×$1.
Agent also changes demand, but effect small
m
Vmppx
p
VNi
i
),,...,( 1
32
EXPENDITURE MINIMISATION
PROBLEM
33
Expenditure Minimization
• We can find the optimal decisions of our
consumer using a different approach.
• We can minimize her/his expenditure
subject to a minimum level of utility that
the consumer must obtain.
• This is important to separate income
and substitution effects.
34
Basic Model• There are N goods.
• Consumer has utility target u.
• Consumer faces linear prices {p1,p2,…,pN}.
• Preferences satisfy completeness, transitivity
and continuity.
• Preferences usually satisfy monotonicity and
convexity.
• Consumer’s problem: Choose {x1,x2,…,xN} to
minimise expenditure p1x1+…+pNxN subject to
u(x1,x2,…,xN) ≥ u and xi ≥ 0.
35
Consumer’s Problem (2 Goods)• Consumer chooses {x1,x2} to solve:
Min p1x1+p2x2 s.t. u(x1,x2) ≥ u
• Solution h*i(p1,p2,u) is Hicksian Demand.
• Expenditure function
e(p1,p2,u) = min [p1x1+p2x2] s.t. u(x1,x2) ≥ u
= p1h*1(p1,p2,u) +p2h*2(p1,p2,u)
• Problem is dual of UMP
36
Expenditure level E2 provides just enough to reach U
EMP: The Picture
Quantity of x1
Quantity of x2
U
Expenditure level E1 is too small to achieve U
Expenditure level E3 will allow the
individual to reach U but is not the
minimal expenditure required to do so
A
• Point A is the solution to the dual problem
37
Graphical Solution
• Slope of Indifference Curve = -MRS
• Slope of Iso-expenditure curves = -p1/p2
• At optimum
• Idea: equate bang-per-buck
• If p2=2p1 then MU2=2MU1 at optimum.
212
1
x
U
x
UMRS
p
p
38
Lagrangian Solution
• Minimize the Lagrangian:
L = p1x1+p2x2 + [u-u(x1,x2)]
• First order conditions:
p1 - MU1 = 0
p2 - MU2 = 0
• Rearranging: MU1/MU2 = p1/p2
• Example: u(x1,x2) =x1x2
39
EMP & UMP• UMP: Maximize utility given income m
– Demand is (x1,x2)
– Utility v(p1,p2,m)
• EMP: Minimize spending given target u
– Suppose choose target u = v(p1,p2,m).
– Then (h1,h2) = (x1,x2)
– Then e(p1,p2,u) = m
• Practically useful: Inverting Expenditure Fn
– Fixing prices, e(v(m)) = m, so v(m)=e-1(m)
– If u=x1x2 then e=2(up1p2)1/2. Inverting, v(m)=m2/(4p1p2)
40
Expenditure Function: Properties1. e(p1,p2,u) is homogenous of degree 1 in (p1,p2)
– If prices double constraint unchanged, so need
double expenditure.
2. e(p1,p2,u) is increasing in (p1,p2,u)
3. Shepard’s Lemma:
– If p1 rises by ∆p, then e(.) rises by ∆p×h1(.)
– Demand also changes, but effect second order.
4. e(p1,p2,u) is concave in (p1,p2)
),,(),,( 2121 upphuppep
i
i
41
e(p1,…)
But the consumer can do
better by reallocating
consumption to goods
that are less expensive.
Actual expenditures will
be less than esub
esub
If he continues to buy the
same set of goods as p*1
changes, his expenditure
function would be esub
Expenditure Fn: Concavity and Shepard’s Lemma
p1
e(p1,…)
At p*1, the person spends
e(p*1,…)=p*1x*1+ … + p*nx*n
e(p*1,…)
p*1
42
Hicksian Demand: Properties1. h(p1,p2,u) is homogenous of degree 0 in (p1,p2)
– If prices double constraint unchanged, so demand
unchanged.
2. Symmetry of cross derivatives
– Uses Shepard’s Lemma
3. Law of demand
– Uses Shepard’s Lemma and concavity of e(.)
2
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