Constrained Maximisation I

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EC115 - Methods of Economic AnalysisSpring Term, Lecture 7Constrained Maximisation I

(Basic Concepts)

Renshaw - Chapter 16

University of Essex - Department of Economics

Week 22

Domenico Tabasso (Uni versity of Essex - Depart ment of Economics)

Lecture 7 - Spring Term Week 22 1 / 33
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Topics for this week

IntroductionConstrained Utility Maximization

The ConstraintThe Objective FunctionRelation Between Constrained and Unconstrained OptimizationProblems


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Introduction

Through the study of functions with several independent variableswe have been able to understand how economists use them:

to represent production technologies and consumer preferences(production and utility functions); and

to analyse their properties using partial differentiation and totaldifferentiation

When studying maximum points we have used productionfunctions to analyse rms optimal decisions (when they seek to

maximise prots) in situations where they do not face anyconstraints.


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Firms in a competitive market choose output to maximise protsat given prices.Monopolists choose output to maximise prots with a given

market demand function.We now go one step further and analyse maximisation andminimisation problems when agents face constraints.

Domenico Tabasso (Uni versity of Essex - Depart ment of Economics)Lecture 7 - Spring Term Week 22 4 / 33
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Constrained Utility Maximization

A typical problem we have in mind is of the following nature.A consumer wants to choose a combination of two goods X andY to maximise her utility,

u = U (x , y ),

where x and y represent the quantities of the goods.Typically this consumer will have a x monthly income to spend,M .

If prices are positive, i.e. p x > 0 and p y > 0, then her decision onhow much to consume of each good must be constrained by theamount of money she can spend.

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Everytime the consumer has to decide how much to consume,she faces the following budget constraint :

p x x + p y y M .

Note that by writing her budget constraint in this form, we areimplicitly assuming that she cannot borrow or lend money (so nonancial markets).Clearly the assumption of no nancial markets is restrictive as itdoes not capture many real world situations. However,analysing a consumers behaviour when there are no nancialmarkets is helpful for understanding how a budget constraintaffects her consumption choices.

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The consumers maximisation problem is described by:

maxx ,y

U (x , y ) s .t . p x x + p y y m.

The consumers objective is to maximise her utility and thus theobjective function of her maximisation problem problem is her

utility function U (x , y ).The consumer is constrained by her available income and thusthe constraint of her maximisation problem is given by herbudget constraint p x x + p y y m.

In almost all of the remainder of this course we will assume thatthe constraint binds so that it becomes p x x + p y y = m.

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The Constraint

The constraint is an implicit relation, (x , y ) = 0, between x and y that must be satised by the values of x and y that maximisethe objective function.In the case of the consumers problem we can derive acorresponding explicit relation, y = g (x ), given by:

y =mp y

p x p y

x .

The term (p x / p y ) describes the slope of the g () function:

dy dx

= p x p y

.

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The slope tells us how many units of Y the consumer to give upin order to buy one extra unit of X while keeping expenditureconstant at m (so it measures opportunity cost).The term (m/ p y ) describes the intercept of the g () function, i.e.the value of y when x = 0.The intercept tells how many units of Y the consumer can buywhen she has income m and spends all of it on Y .Similarly (m/ p x ) is the number of units of X the consumer canbuy when she has income m and spends all of it on X .

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10/35Domenico Tabasso (Uni versity of Essex - Depart ment of Economics)Lecture 7 - Spring Term Week 22 10 / 33
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Analysing the Constraint

Changing p x , p y and/or m changes the consumers budgetconstraint.Reducing m shifts the budget line in towards the origin(increasing m shifts it out away from the origin).

Reducing p x rotates the budget line by increasing the slope butkeeping the intercept constant (increasing p x reduces the slopebut keeps the intercept constant).Reducing p y rotates the budget line by reducing the slope and

adjusting the intercept accordingly (increasing p y increases theslope and adjusts the intercept accordingly).

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We can also use total differentiation to examine the slope of theconstraint.

We can rewrite the budget line as G (x , y ) = 0 whereG (x , y ) = p x x + p y y m.The total differentiation gives:

G x dx +

G y dy = 0.

But G x = p x and G

y = p y so we have:

p x dx + p y dy = 0and thus dy dx = (p x / p y ).

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The Objective Function

The objective function describes the function that the agent wantto maximise or minimise.This function is normally of several independent variables.Hence our previous techniques for analysing this functions apply.Assume the consumer has a Cobb-Douglas utility function

u = U (x , y ) = x 1/ 2 y 1/ 2

This function describes the preferences the consumer has forgoods x and y .

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Applying partial differentiation we obtain that

MU x (x , y ) = U (x , y )

x =

12

y x

1/ 2> 0

MU y (x , y ) = U (x , y )

y =

1

2

x

y

1/ 2

> 0

The marginal utilities are positive for positive values of x and y .Recall that the marginal utilities are a measure of how much doesthe utility changes when the consumer increases the consumption

of x or y .

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Applying partial differentiation one more time we get MU x (x , y )

x =

2U (x , y ) x 2

= y 1/ 2

4x 3/ 2 < 0

MU y (x , y ) y =

2U (x , y ) y 2 =

x 1/ 2

4 y 3/ 2 < 0

That is, the marginal utilities are downward sloping for positivevalues of x and y .

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Finally, using cross partial differentiation we obtain that MU y (x , y )

x =

2U (x , y ) y x

= 2U (x , y )

x y

= MU x (x , y )

y = 14

1xy

1/ 2

> 0

Thus increasing consumption of X while keeping consumption of Y xed will increase the marginal utility of Y (and vice versa).

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Using total differentiation, we have that for any utility level theindifference curve is given by the implicit function:

U (x , y ) u 0 = x 1/ 2 y 1/ 2 u 0 = 0du 0 = MU x (x , y )dx + MU y (x , y )dy = 0

The marginal rate of substitution describes how much of good y the consumer is willing to exchange for more of good x holdingutility constant:

MRS x ,y =

dy

dx u = u 0 =

MU x (x , y )MU y (x , y ) =

u 0x

2

=

y

x .

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That the change in the MRS increases with x , implies that as weconsume more of x the consumer is less willing to substitutemore of good y :

d 2 y dx 2

u =

u 0

= 2u 20x 3

> 0.

This is a reection that the consumer prefers averageconsumption of the two goods to extreme bundles in which thereis a lot of one good and only few units of the other.

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Relation Between Constrained and Unconstrained Optimization Problems

In the unconstrained optimisation problem, we want to obtain themaximum or minimum values of a function.In the constrained optimisation problem, we want to obtain the

maximum or minimum values of the the function subject to the constraint .In general, the solutions will be different (in general, the solutionto the unconstrained optimisation problem does not satisfy theconstraint).


B t h i ll thi t ff f l?
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But why is all this stuff useful?

Because we can prove that the optimal solution for themaximization of a consumers utility is given by thepoint where the slope of the indifference curve and theslope of the budget constraint are identical.


B t h is all this st ff sef l?
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But the slope of the indifference curve in every point ithe MRS!And the slope of the budget constraint is given by p x p y .


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But the slope of the indifference curve in every point ithe MRS!And the slope of the budget constraint is given by p x p y .

So for our solution to dene a maximum it is necessary(but not sufficient!) that MRS = p x p y

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The other condition that need to be satised is that the

optimal point MUST lay on the budget constraint.So the system:

MRS = p x p y M = p x X + p y Y

gives us the necessary and sufficient conditions for ndingthe constrained maximum.


So can points A and B constitute optimal solutions?
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So can points A and B constitute optimal solutions?

Y

X

NO!!! In both A and B , MRS = p x / p y .Domenico Tabasso (Uni versity of Essex - Depart ment of Economics)Lecture 7 - Spring Term Week 22 26 / 33

And how about points C and D?
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And how about points C and D?

Y

C

X

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In points C and D the slope of the indifference curves (i.e.the slope of the lines tangent to the indifference curves inthose points) is equal to the slope of the budget constraint.So:

MRS A = MRS B = p x p y

The problem is that points C and D are NOT on thebudget constraint. So C and D do not satisfy the second

equation of the system.


And the winner is...
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Y

E

X


An example
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An example

A consumer problem is:

maxX ,Y

U (X , Y ) = X 1/ 2Y 1/ 2

s .t . 100 = 2X + Y

First step: Find the MRS associated with the aboveutility function:

MRS = dY dX =

U / X U / Y

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In our case:

MRS = U / X U / Y

= 12X

1/ 2Y 1/ 212X

1/ 2Y 1/ 2 =

Y X

Second step: Set the system: MRS = p x p y

Y X =

21

M = p x X + p y Y 100 = 2X + Y

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Third Step: Solve the system

From the rst equation we get: X = 0.5Y .Plugging this in the second equation (the budget

constraint) we get:100 = Y + Y .

So Y = 50 and since X = 0.5Y X = 25.


Graphically:
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120

100 U=U*

80

60

40

20

0

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

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Constrained Maximisation I

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