Constrained Maximisation II

8/10/2019 Constrained Maximisation II

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EC115 - Methods of Economic AnalysisSpring Term, Lecture 8

Constrained Maximisation II

Renshaw - Chapter 16

University of Essex - Department of Economics

Week 23

Domenico Tabasso (University of Essex - Department of Economics)

Lecture 8 - Spring Term Week 23 1 / 41

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Recap - 1

In the last weeks we have tried to answer the following

question:

How can we maximize a function under a constraint?

The answer we gave so far is:1 In case of a utility function U (x , y ) the maximum is given by the

solution of the system: Slope of the IC (≡ MRS ) = Slope of the BC

Equation of the Budget Constraint

2 In case of a production function Y (K , L) the maximum is given by the

solution of the system: Slope of the Isoquant(≡ MRTS ) = Slope of the Isocost

Equation of the Isocost



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Recap - 2

The only difficulty with this method consists in calculating

the MRS (or the MRTS).

Reminder:

MRS = −dy dx

MRTS = −dK

dLSo how can we obtain an expression for dy

dx or dK

dL that we

can then insert in our systems?



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Recap - 3

The way to obtain dy dx

or dK dL

is to use total

differentiation.

Domenico Tabasso (University of Essex - Department of Economics)Lecture 8 - Spring Term Week 23 4 / 41

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Recap - 3


or dK dL

is to use total

differentiation.Really???


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Recap - 3


or dK dL

is to use total

differentiation.Really???

Thanks to the implicit function theorem we know that:

MRS = −∂ U /∂ x ∂ U /∂ y

MRTS = −∂ Y /∂ L

∂ Y /∂ K So we don’t need to totally differentiate the functions.We can simply calculate the ratio of the two partial

derivatives and we are done.Domenico Tabasso (University of Essex - Department of Economics)Lecture 8 - Spring Term Week 23 4 / 41

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Solving Through Direct Substitution

Recall that the consumer’s problem is described by:

maxx ,y

U (x , y ) = x 1/2 y 1/2 s .t . p x x + p y y = m.

It is important to realise that the budget constraint

requires:

y = g (x ) = −

p x

p y

x +

m

p y

must hold at all times!!!The problem is thus to obtain the highest level of utility, u ∗ = U (x , y ), such that x and y satisfy thebudget constraint.


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Now substitute y = g (x ) into the utility function togive:

U (x ) = U (x , g (x )) = x 1/2

−

p x

p y x +

m

p y

1/2

where p x , p y and m are positive constants.Then the consumer’s problem can be re-written as anunconstrained maximisation problem of one variable:

maxx

U (x ), i.e. maxx

x 1/2 −p x

p y x +

m

p y

1/2


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The first order condition implies that the quantity of good X that maximises

U (x ), namely x ∗, is given by

solving:

d U (x )

dx

x =x ∗

= 0.

Note that U is not a function of y, but only of x, so ourderivative is denoted by the symbol d and not ∂ .

The second order condition implies that x ∗ indeed

describes a maximum if:

d U 2(x )

dx 2

x =x ∗

< 0.


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Example

Consider the case in which:

p x = 1, p y = 1 and m = 10. Then: U (x ) = x 1/2 [−x + 10]1/2 so the problem is now:

maxx

x 1/2 [−x + 10]1/2

Using the product rule, the first order condition impliesthat x ∗ is determined by:

d U (x )dx = 0⇒

12 (x ∗)−1/2 [−x ∗ + 10]1/2 − 1

2 (x ∗)1/2 [−x ∗ + 10]−1/2 = 0


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It is important to realise that we have obtained the

individual demand functions for goods X and Y ,namely:

x ∗ = 1

2

m

p x

y ∗ = 12

m

p y

Of course, when we were given the values of

p x = 1, p y = 1 and m = 10, we obtained a point ineach demand function.


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Solving through Total Differentiation

To solve this problem we can use total differentiation in

the following way:To obtain the highest level of utility such that thebudget constraint is satisfied,

first we have to look for all those indifference curves that satisfy thebudget constraint then we can look for the highest indifference curve or highest level of

utility.

Recall that the indifference curves can describedimplicitly by:

U (x , y ) − u 0 = x 1/2 y 1/2 − u 0 = 0.


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From the budget line we get:

dy dx

= −p x p y

.

Re-arranging we obtain the condition for a maximum in

the consumer’s problem:

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

MU y (x , y ) =

p x

p y .

The point on the budget line at which MRS x ,y = p x /p y then gives the highest possible utility we can achieve.


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When the consumer chooses x ∗ and y ∗ such that:

MRS x ,y = MU x (x , y )p x

= MU y (x , y )p y

then she has no incentive to change his consumption.

At this level of consumption, the rate of substitution of y for x determined by her preferences is the same asthe opportunity cost x in terms of y dictated by themarket through the relative prices.

In this cases we say that x ∗ and y ∗ describe an interior

solution.


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Tangency condition : Optimal consumption of x and y

Y

Preference

direction

EOptimal Choice

Internal solutionY*

XX*


P bl i h h T l Diff i i A h

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Problems with the Total Differentiation Approach

To better understand the intuition behind total

differentiation approach, suppose thatMRS x ,y = (p x /p y ). What happens here?

First consider the case in which:

MU x (x , y )MU y (x , y ) > p x

p y =⇒ MU x (x , y )

p x > MU y (x , y )

p y

This expression describes implies that the consumer

obtains a higher marginal utility from spending onemore unit of his income in good x than y .

What do you think this consumer will do? Recall thatshe is spending all of m.


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Similarly, consider the case in which:

MU x (x , y )

MU y (x , y ) <

p x

p y =⇒

MU x (x , y )

p x <

MU y (x , y )

p y

This expression describes implies that the consumerobtains a higher marginal utility from spend one moreunit of her income in good y than x .

What do you think this consumer will do? Recall that

she is spending all of m.


Wh if h di i f il ?

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What if the tangency condition fails?

To determine which good is bought we use our previousinequalities:

if MU x

p x

> MU y

p y

=⇒ consume all in x

if MU x

p x <

MU y

p y =⇒ consume all in y .

If the consumer only buys one of the two goods, thenthe optimal solution will lay on one of the axes. In thiscase we deal with a corner solution.


P f t b tit t d l ti

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Perfect substitutes and corner solutions

Y

20

n erence

Curve

12

u ge

Constraint

20 X


P f t b tit t d l ti

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What if the goods are perfect substitutes but MRS = p x p y

?

Y

Budget Constraint

X0



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How about the goods are perfect complements(U = min{x α, y β })?

Y

Optimum

X0


Example 1: MRS or Substitution?

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A consumer problem is given by:

maxx ,y

U (x , y ) = x α y β

s . t . p x x + p y y = m

where x , y , m, p x , p y > 0Find the optimal consumption bundle (x ∗, y ∗) using both

the substitution method and the MRS approach.


Example 1: The substitution method

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1 Obtain y = f (x ) from the Budget Constraint

y = m

p y −

p x

p y x

2

Plug y = f (x ) into the utility function U (x , y ) andobtain a new expression for the utility function:

U (x ) = x αm

p y −

p x

p y x

β

3 Maximize U (x )



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First Order Condition:

d U (x )

dx = 0

=⇒

αx α−1 mp y

− p x

p y x β

− β p x

py x αm

p y − p x

p y x β −1

= 0

Rearranging:

αx α−1 mp y

− p x p y

x β

= β p x py x αm

p y − p x

p y x β −1



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The last equation can be rewritten as:

p y αx α−1

p x β x α =

mp y − p x

p y x β −1

mp y −

p x p y x β

Which can be rewritten as:

p y αm

p y − p x

p y x p x β x

= 1



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Or:

αm − αp x x = p x β x which implies

(α + β )p x x = αm

So finally:x ∗ =

αm

(α + β )p x

Which substituted back into the budget constraint gives us:

y ∗ = β m

(α + β )p y


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Little homework:

Check the sign of the second order conditions and verifyunder which conditions x ∗, y ∗ define a maximum.


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Little homework:

Check the sign of the second order conditions and verifyunder which conditions x ∗, y ∗ define a maximum.

Solution: α, β < 1



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Again the same problem

maxx ,y

U (x , y ) = x α y β

s . t . p x x + p y y = m

where x , y , m, p x , p y > 0but now we find the optimal consumption bundle (x ∗, y ∗)

using the MRS approach.



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p S Su u

First Step: Find the MRSWe know that:

MRS = −∂ U /∂ x

∂ U /∂ y So in our case:

MRS = −

αx α−1 y β

β x α y β −1 = −

α y

β x



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p

Second Step: Set Up the Following System:

MRS = −p x

p y M = p x X + p y Y

So in our case:

−α y

β x = −

p x

p y M = p x x + p y y



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p

From the first equation we obtain:

y = p x

p y

β

αx

which we can substitute in the second equation (the budgetconstraint) and get:

m =

p x

x +

p y

p x

p y

β

αx



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p

Simplifying we obtain:

x ∗ = αm

(α + β )p x

(of course!) and substituting back into y = p x

p y

β

α

x we get:

y ∗ = β m

(α + β )p y

(of course!)



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Simplifying we obtain:

x ∗ = αm

(α + β )p x

(of course!) and substituting back into y = p x

p y

β

α

x we get:

y ∗ = β m

(α + β )p y

(of course!)

Which method do you find easier?


Distinguishing maxima and minima

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Unfortunately the tangency condition for an interior

solution does not tell us if we are really maximising orminimising the objective function.

Consider the following problem:

maxx ,y

H (x , y ) = x 2 + y 2 st . x + y = 10

The tangency condition implies x ∗ and y ∗ must satisfy:

∂ H

∂ x /

∂ H

∂ y =

2x ∗

2 y ∗ =

x ∗

y ∗ = 1


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Thus the solution to the first order conditions is givenby x ∗ = y ∗ = 5 with H (x ∗, y ∗) = H (5, 5) = 50.

But (x = 10, y = 0) also satisfies the constraint andH (10, 0) = 100 > H (5, 5)!!!


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Thus the solution to the first order conditions is givenby x ∗ = y ∗ = 5 with H (x ∗, y ∗) = H (5, 5) = 50.

But (x = 10, y = 0) also satisfies the constraint andH (10, 0) = 100 > H (5, 5)!!!

One way to go around this problem is by analysing thelevel curves of the function.

To guarantee that in the consumer problem we areindeed finding a maximum, we assume that theconsumer prefers average consumption of x and y

instead extreme consumption of x and y .


Hence in our standard maximization problem the

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indifference curves are convex towards the origin.


Hence in our standard maximization problem the

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indifference curves are convex towards the origin.

In the case of

u = H (x , y ) = x 2 + y 2

if we fix a level of utility u 0, we have that the associate

indifference curve is given by the implicit function:

H (x , y ) − u 0 = x 2 + y 2 − u 0 = 0

Applying total differentiation we obtain thatdu 0 = 2xdx + 2 ydy = 0


Therefore we find that:

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dy

dx

= −x

y

< 0

d 2 y

dx 2 = −

1

y < 0 for values of y > 0

But recall from the first term material (Lecture 8, slides38-45) that if a function y = f (x ) shows a negative

second derivative (

d 2y

dx 2 ) then the function is concave!!!

H (x , y ) has downward sloping and concave indifferencecurves.


Level Curves

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Y

Level Curves of

Z=X^2+Y^2

P5

p mum

X0 5

X+Y=10


Level curvesS d di h h ili f i l k lik i i l f

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So understanding how the utility function looks like is crucial forcorrectly identifying the optimal consumption bundle. Take twodifferent utility functions: u (x , y ) = x 2 + y 2 andu (x , y ) = −(x 2 + y 2). In both cases the indifference curves can berepresented as:

Y

X0


Level curves

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BUT:

=‐ + U=X^2+Y^2


Level curves

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In the case of u (x , y ) = −(x 2 + y 2) the two goods x and y

actually represent bads . The more we consume of them,the lower our utility. So the only optimal solution is:

x ∗ = 0, y ∗ = 0

and this solution is totally independent from the constraint.


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Constrained Maximisation II

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