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Topic 2: Constrained optimisation

ECON30010 Microeconomics

9 - 16 March

ECON30010 Topic 2: Constrained optimisation 9 - 16 March 1 / 48

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Topic 1 refresherIn most economic problems, the following assumptions seem reasonable:

Completeness : any two bundles x = ( x 1, x 2) and y = ( y 1, y 2) can becompared.

Transitivity : if bundle x is at least as good as y and y is at least asgood as bundle z, then x is at least as good as z.

Continuity (not an exact denition): if bundle x is better than bundley and bundle z is sufficiently close to y, then x is better than z.

TheoremIf preferences over bundles of goods x satisfy completeness, transitivity,and continuity, then there exists a continuous utility function u (x) that represents these preferences.

Why is this theorem important? Because mathematicians are very good atworking with functions, and we can use a lot of their tools.


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The plan and the why Having a continuous utility function is very useful, but not enough.

Budget set Utility maximisation problem◦ Strict monotonicity We cover these two axioms here because they◦ Convexity are related to the maximisation problem.

The Lagrange method

Marshallian demandThe name of this topic sounds like math. And our main focus will bemath: it is going to be applied to an economics problem, but my maingoal is to make sure you are comfortable with constrained optimisation.

Why?We ourselves will need it later, but, more importantly, other subjects willneed it, and much more desperately than we would.So, any effort to become fully comfortable with constrained optimisationwill pay off handsomely in the future (when your macro lecturer will zip pastthe calculations and straight to the optimal solution).


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Constraint

Constraint A typical agent wants “stuff”. If fth TV set in a room does not

make an agent any happier, something else (more time? extra hike inmountains?) would make the agent more satised. If an agent to face a meaningful problem – where the solution is notgiving an agent an innite amount of something – we need tointroduce some constraint.

Earlier, we have dealt with a simple constraint: an agent had $5 thatshe needed to allocate.

This is not the most typical constraint the agent face, although wewill see plenty of similar simple constraints in this subject.

We now turn to a more typical one: budget constraint.Budget constraint is often thought of as income and prices, but youwill see on a tutorial that we can interpret the same constraint assomething completely different.

Even though I “focus” on budget constraint, everything we study

applies to other “types” of constraints, including $5 one.ECON30010 Topic 2: Constrained optimisation 9 - 16 March 4 / 48

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Constraint

Budget ConstraintBudget constraint: good 1 and good 2 cost money and an agent (evenWarren Buffet) has a limited amount of it.

A consumer is restricted to choose a consumption bundle q = ( q 1, q 2)such that p 1q 1 + p 2q 2 ≤ Y . Right now, we are not interested where pricesp 1, p 2 and income Y come from.

q 1

q 2

Y / p 1

Y p 2

Budget set p 1q 1 + p 2q 2 ≤ Y

Budget line p 1q 1 + p 2q 2 = Y

If q 1 = 0, p 2q 2 = Y ⇒ q 2 = Y / p 2

If q 2 = 0 , p 1q 1 = Y ⇒ q 1 = Y / p 1


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Consumer maximisation problem


Formally, we can write this problem as:

maxq 1 ,q 2

u (q 1, q 2)

subject to p 1q 1 + p 2q 2 ≤ Y

Note that I can write my $5 problem as: maxx , y u (x , y )

subject to x + y = 5


bl

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Indifference curves: ECON20002 recapIn ECON20002 you have learned1 that the indifference curve is the

solution to u (x , y ) = u 0, for different levels of u 0.

q1

q 2

Example for u (q 1, q 2) = q 1q 2

1In Tutorial 1 you have looked at a similar problem.ECON30010 Topic 2: Constrained optimisation 9 - 16 March 7 / 48

C i i i bl

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Solving consumer maximisation problem graphically:ECON20002 recap

q 1

q 2

Y / p 1

Y p 2

We are looking for the North-East-most indifference curve2, such that ittouches budget constraint (to guarantee the feasibility of the solution).

2Utility increases in that direction – see the picture on the previous slideECON30010 Topic 2: Constrained optimisation 9 - 16 March 8 / 48

Cons mer ma imisation problem

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What can go wrong? SatiationConsider an agent, Ann, with preferences:

u A(q 1, q 2) = − (q 1 − 1)2

− (q 2 − 1)2

A similar function appeared in Tutorial 1. We know that this function ismaximised at q = (1 , 1).

Suppose now that Ann has plenty of money($4) and prices are p 1 = p 2 = 1. Her budgetconstraint is q 1 + q 2 ≤ 4.From Tutorial 1, 3 we know that Ann’sindifference curves are circles. We can alsond that at point q = (2 , 2) indifference

curve touches the budget line.4

q 1

q 2

Y / p 1

Y p 2

But point (2 , 2) is not the maximum! What is going on here?. 3Note: tutorials are very useful!

4How: by symmetry, we can guess that the point is on the intersection of budget lineand a 45 ◦ line, then nd this po int.



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Strict monotonicity This would not happen if utility function u were increasing: then for

any point in the interior of the budget set you could increase utility bygiving more of some good.

However, we should not impose this assumption on utility function:utility function is a fake, and we need to know what it means in termsof something that is not a fake (something that we can observe, suchas choice/preferences).

This leads us to a new condition on preferences, strict monotonicity:5

Denition

If bundle x contains at least as much as bundle y of every good and strictly more of at least one good

then a consumer with strictly monotone preferences strictly prefers x to y.5This condition was introduced in ECON20002 under the name “more is better”.



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Implication of strict monotonicity.

Theorem: With strictly monotonic preferences, we can replace inequality≤ with equality = in the consumer maximisation problem, so thatp 1q 1 + p 2q 2 = Y .

Proof: Suppose not.6 That is, suppose that we cannot replace ≤ by =because the optimal consumption of a consumer, ( q 1, q 2), is such thatp 1q 1 + p 2q 2 = Y < Y . Then consumer has Y − Y of income left over,which can be spent on good 1 and good 2 (say, equally). Yet, if preferences are strictly monotonic, then the consumer is better off. So, if (q 1, q 2) is an optimal consumption bundle, preferences could not be

strictly monotonic.

6The proof technique where you assume that the statement of the theorem is wrongand then prove that the condition of the theorem is not satised is called “proof bycontradiction”. Here the statement is that “we can replace ≤ with =” and the condition

is “preferences are strictly monotonic”.ECON30010 Topic 2: Constrained optimisation 9 - 16 March 11 / 48


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Strict monotonicity, example 1, washing dishes

DenitionIf bundle x contains

at least as much as bundle y of every good and

strictly more of at least one good

then a consumer with strictly monotone preferences strictly prefers x to y.

Suppose that the (only) good is “washingdishes”. It does not satisfy monotonicity.

However, this is not a problem: we only need todene the good as “not washing dishes”.



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Strict monotonicity, example 2, exercise & chocolateDenition

If bundle x

contains at least as much as bundle y of every good and

strictly more of at least one good

then a consumer with strictly monotone preferences strictly prefers x to y.

Exercise is good, but too much of exercise is bad.Eating one chocolate bar is fantastic, but fth kilo maycause death.

In most cases, these examples are not a cause of concern

because: (1) goods would be more generally dened (e.g.consumed over longer periods of time) and (2) the range of consumption in the problem is where the satiation does nothappen (you won’t eat chocolate only).

You still need to examine your problem to see whether the assumption ts.ECON30010 Topic 2: Constrained optimisation 9 - 16 March 13 / 48


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p

What if no strict monotonicity?Strict monotonicity is a “technical” assumption: it is much easier to solvea problem if this assumption is imposed, but we are not doomed if thisassumption does not make sense in our problem. There are moresophisticated methods to nd a solution.With equality, we will use the Lagrange method. With inequalities, wewould need to use the Kuhn-Tucker method.

You can check when they lived and guess which method is easier.ECON30010 Topic 2: Constrained optimisation 9 - 16 March 14 / 48


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What can go wrong? Non-convexitySuppose that indifference curves look like this:

q 1

q 2

Y / p 1

Y p 2

Two optimal consumption bundles

We then have two optimal consumption bundles. This is not a big problemif nding optimal bundle is our ultimate goal. However, if we want toknow how consumption changes in response to change in prices, havingtwo bundles is not convenient: we do not know which one the agent wouldbe consuming.

Hence, we want to rule out this situation too.ECON30010 Topic 2: Constrained optimisation 9 - 16 March 15 / 48


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Convexity in chocolate and candy (recap of ECON20002)The usual assumption is that our marginal utility is decreasing: you areecstatic about the rst bar of chocolate, but not so much about thesecond bar.

This naturally leads to convexity assumption: if you are indifferentbetween

x : 3 bars of chocolate & 1 candy andy : 3 candies & 1 bar of chocolate,

then you must prefer (z ) 2 bars of chocolate and 2 candies to x and y .

Does this assumption always hold? Not necessarily: both aspirin andparacetamol are good against a headache (so, you are, conceivably,indifferent), but having 1/2 of aspirin and 1/2 of paracetamol is a recipefor a disaster.Similarly to strict monotonicity: this is unlikely to be a problem with moregenerally dened goods. At the same time, as always , when you think about

your specic problem, you do need to think whether this assumption is satised.ECON30010 Topic 2: Constrained optimisation 9 - 16 March 16 / 48


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ConvexityI have told you what convexity means when we talk about 3 chocolates

and 1 candy. We need to be a bit more general than that.Formal denition of convexity:7

DenitionSuppose that a consumer is indifferent between bundle x and y. Considera bundle z := αx + (1 − α)y for any α such that 0 < α < 1.8

The preferences are convex if z x , z y .9

The preferences are strictly convex if z x , z y .

7

Do not confuse convex preferences and convex functions. These are two differentnotions. (Only if you are very curious: The set of better allocations – the inside of the indifference curve – is a convex set;the set above the convex function is a convex set. Convex set is the set that contains a straight line that connects any two

points of the set.)8This is called a convex combination of x and y.9I do not need the second relation, z y , because I have already assumed

transitivity. Check that you understand why. The same applies to strict convexity.ECON30010 Topic 2: Constrained optimisation 9 - 16 March 17 / 48


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Strict convexity on a graphDenitionSuppose that a consumer is indifferent between bundle x and y. Consider abundle z := αx + (1 − α)y for any α such that 0 < α < 1.


q 1

q 2

x

yz

Straight line connects x and y.

Any point on this line is “z” fromthe denition.

If utility increases in the northeast direction (as it usually does), then thewhole straight line should be above the indifference curve (recall that

points above indifference curve are more desirable bundles).ECON30010 Topic 2: Constrained optimisation 9 - 16 March 18 / 48


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Solution is unique with strict convexity

q 1

q 2

Y / p 1

Y p 2

x

y

z

Straight line connects x and y.Any point on this line is “z” fromthe denition.

With strict convexity, we will have a unique solution to our maximisationproblem.



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Convexity on a graphDenitionSuppose that a consumer is indifferent between bundle x and y. Consider abundle z := αx + (1 − α)y for any α such that 0 < α < 1.


The preferences are convex if z x , z y .

q 1

q 2

x

yz

These preferences are convex, but not strictly convex.ECON30010 Topic 2: Constrained optimisation 9 - 16 March 20 / 48


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No unique solution with convexity

q 1

q 2

Y / p 1

Y p 2

s

S

Convexity holds, but solution is not unique: there is an interval of solutions from s to S .



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Non-convexityDenitionSuppose that a consumer is indifferent between bundle x and y. Consider abundle z := αx + (1 − α)y for any α such that 0 < α < 1.

The preferences are convex if z x , z y .

q 1

q 2

Y / p 1

Y p 2

xy

z

These preferences are not convex.ECON30010 Topic 2: Constrained optimisation 9 - 16 March 22 / 48


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More assumptions

I will also assume that utility function is differentiable, or twice

differentiable. These assumptions are also traceable to assumptions onpreferences. However, it becomes too technical and I will skip that.



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Solving consumer maximisation problemmaxq 1 ,q 2

u (q 1, q 2)

subject to p 1q 1 + p 2q 2 = Y

This is called constrained maximisation problem because you need to (i)nd max u (q 1, q 2) and (ii) satisfy your constraint (here p 1q 1 + p 2q 2 = Y ).Problem (i) alone is called unconstrained optimisation problem. You can

guess that constrained maximisation problem is usually more difficult thanan unconstrained one.

How do you solve constrained maximisation problem? Sometimes you can “see” what the solution is (for example, when

goods are perfect complements or perfect substitutes); You can substitute in budget constraint ( q 1 = 1p 1 (Y − p 2q 2)) (this iswhat you have done in ECON20002)

Or you can use the Lagrange methodYou can also use a “shortcut” which comes from the Lagrange method;

this is what you’ve also done in ECON20002.ECON30010 Topic 2: Constrained optimisation 9 - 16 March 24 / 48

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Constrained Optimisation: Lagrange’s Method, Example 1

maxq 1 ,q 2 ,λ L (q 1, q 2, λ ) = u (q 1, q 2) + λ[Y − p 1q 1 − p 2q 2]Solve for u (q 1, q 2) = q 1q 2:

∂ L (q 1,q 2,λ )∂ q 1 = 0∂ L (q 1,q 2,λ )

∂ q 2 = 0∂ L (q 1,q 2,λ )

∂λ = 0 .



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Lagrangian: A more careful look

∂ L

(q 1 ,q 2 ,λ )∂ q 1 = ∂ u (q 1 ,q 2)∂ q 1 − λ p 1 = 0∂ L (q 1 ,q 2 ,λ )

∂ q 2= ∂ u (q 1 ,q 2)∂ q 2 − λ p 2 = 0

∂ L (q 1 ,q 2 ,λ )∂λ = Y − p 1q 1 − p 2q 2 = 0 .

⇓

MU 1(q 1, q 2) = λp 1MU 2(q 1, q 2) = λp 2p 1q 1 + p 2q 2 = Y

⇒MRS = MU 1(q 1 ,q 2)MU 2(q 1 ,q 2) =

p 1p 2

p 1q 1 + p 2q 2 = Y .

λ = MU 1

p 1=

MU 2p 2

λ is equal to marginal utility of good i divided by the price of thatgood; that is, an increase in utility if income Y is increased by 1 cent.



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Shadow value of a constraint: an exampleRevisit our utility function u (q 1, q 2) = q 1q 2.

Recall that MU 1 = q 2, MU 2 = q 1, q 1 = Y 2p 1 , and q 2 = Y 2p 2 ; from theprevious slide λ = q 2/ p 1 = q 1/ p 2

Let Y = 1 , p 1 = 1 / 2, p 2 = 2. Then q 1 = 1 , q 2 = 1 / 4, u (q 1, q 2) = 0 .25 andthe shadow value of a constraint is λ = 1 / 2.

Let Y = 1 .01 now; p 1 = 1 / 2, p 2 = 2 as before. Then q 1 = 1 .01,q 2 = 1 .01/ 4, u (q 1, q 2) = 1 .012/ 4 = 0 .255025.

Note that

∆u

= 0 .255025−

0.25 = 0 .005025≈

0.005 = 1 / 2·

(1.01−

1) = λ·

∆Y

Change in u is almost equal to λ times the change in “how binding” theconstraint is. It would have been exactly 0.005 if the increase in income Y has been even smaller (you can try that at home).



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Constrained Optimisation: Lagrange’s Method, Example 2Consider another utility function: u (q 1, q 2) = q 1q 2 + 3 q 1. Let p 1 = p 2 = 1

and Y = 2The Lagrangian is

L (q 1, q 2, λ ) = q 1q 2 + 3q 1 + λ[2 − q 1 − q 2],

∂ L (q 1 ,q 2 ,λ )∂ q 1 = q 2 + 3 − λ = 0

∂ L (q 1 ,q 2 ,λ )∂ q 2 = q 1 − λ = 0

∂ L (q 1 ,q 2 ,λ )∂λ = 2 − q 1 − q 2 = 0 .

⇓

q 1 = q 2 + 3q 2 + 3 + q 2 = 2

⇒ 2q 2 = − 1

Not the best news. . .ECON30010 Topic 2: Constrained optimisation 9 - 16 March 31 / 48


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q 1

q 2

Y / p 1

Y p 2

Optimal consumption bundle

(corner solution)Optimal consumption bundle

if can consume negative quantities

Lagrange method is not aware that you cannot consume negative

quantities.ECON30010 Topic 2: Constrained optimisation 9 - 16 March 32 / 48


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A diversion: a complete constrained problem

maxq 1 ,q 2

u (q 1, q 2)

subject top 1q 1 + p 2q 2 = Y q 1 ≥ 0

q 2 ≥ 0Yet, the solution to this problem, using Kuhn-Tucker conditions, would betoo messy. Hence, we ignore some of the constraints, with a hope thatthey would not matter, but need to revisit our problem once we obtained

the solution to a simplied problem.

Surprisingly, a lot of problems in economics are solved this way: Originalproblem is too hard; let us try to solve a simplied problem, hoping that itssolution will satisfy other constraints; if not, try a different simplication.



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Marshallian demand function

Revisit our solution in Example 1:

q 1 = Y 2p 1

q 2 = Y

2p 2

Note that we obtained the solution for any prices p 1, p 2 and any incomeY . That is, we could write the solution as functions q 1(p 1, p 2, Y ) andq 2(p 1, p 2, Y ). These functions are called Marhsallian, or uncompensated, 10

demand.

10You can look up your notes from ECON20002 if you want to know right now why

the demand is called uncompensated, or you can wait a little.ECON30010 Topic 2: Constrained optimisation 9 - 16 March 34 / 48

Summary

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Summary and what’s next:

Main points:

Assumptions on individual preferences allow us to represent thesepreferences by a continuous utility function;

Maximisation of a utility function subject to a budget constraint(using Lagrange method) leads to Marshallian demand functions.

Lagrange method: turn constrained optimisation problem intounconstrained maximisation problem.Solve unconstrained maximisation problem using standard methods.

What’s next:

A more difficult constrained maximisation problem. An odd maximisation problem. Normal, inferior and Giffen goods.

Testing consumer demand theory: a search for Giffen goods.


Utility maximisation problems

l bl ( )

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Utility maximisation problem 1 (Ii)

u (q 1, q

2) = ln( q

1− b

1) − 2ln(b

2− q

2)

For numerical calculations I use b 1

= 1 , b 2

= 3 , p 1

= 1 , p 2

= 1 , Y = 4



U ili i i i bl 1 (II)

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Utility maximisation problem 1 (II)

maxq 1 ,q 2

u (q 1, q 2) = ln( q 1 − b 1) − 2ln(b 2 − q 2)

subject to p 1q 1 + p 2q 2 = Y

Corresponding Lagrangian

L = ln( q 1 − b 1) − 2ln(b 2 − q 2) + λ [Y − p 1q 1 − p 2q 2]



S l i bl 1 FOC

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Solving problem 1: FOC∂ L∂ q 1

= 1

q 1 − b 1− λp 1 = 0

∂ L∂ q 2

= 2

b 2 − q 2− λp 2 = 0

∂ L∂λ

= Y − p 1q 1 − p 2q 2 = 0

1

q 1 − b 1= λp 1

2b 2 − q 2

= λp 2

p 1q 1 + p 2q 2 = Y

b 2 − q 2q 1 − b 1

= 2p 1p 2

p 1q 1 + p 2q 2 = Y

q 2 = b 2 − 2 p 1p 2(q 1 − b 1)

Y = p 1q 1 + p 2q 2 = p 1q 1 + p 2(b 2 − 2p 1p 2

(q 1 − b 1))

= p 1q 1 + p 2b 2 − 2p 1(q 1 − b 1) = p 2b 2 − p 1q 1 + 2p 1b 1

q 1 = 2b 1 − Y − p 2b 2

p 1; q 2 = b 2 − 2

p 1p 2

b 1 − Y − p 2b 2

p 1

= b 2 − 2 1p 2

(b 1p 1 − Y + p 2b 2) = 2Y − b 1p 1

p 2− b 2

Reminder: ln(x ) is a dream for differentiation: ln( x ) = 1 / x ECON30010 Topic 2: Constrained optimisation 9 - 16 March 38 / 48


S l ti t tilit i i ti bl 1

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Solution to utility maximisation problem 1Marshallian (uncompensated) demand is:

q 1(p 1, p 2, Y ) = 2 b 1 − Y − p 2b 2

p 1

q 2(p 1, p 2, Y ) = 2Y − b 1p 1

p 2− b 2

However, this is not the end of the solution:11 suppose thatY = 3 , b 1 = b 2 = p 1 = p 2. Then q 1 = 2 − 3− 11 = 0. If we go back to theutility function, u (q 1, q 2) = ln( q 1 − b 1) − 2ln(b 2 − q 2), it is not dened if q 1 = 0 and b 1 = 1 because it leads to ln(0 − 1), which is undened. Toensure that q 1 − b 1 > 0 and b 2 − q 2 > 0 we need to impose the followingconditions:

For q 1 > b 1 we need 2b 1 − Y − p 2b 2

p 1> b 1 hence Y − p 2b 2 < p 1b 1 and

for 0 < q 2 < b 2 we need Y − p 2b 2 < p 1b 1 < Y − p 2b 2/ 211 We always need to check that our solution makes sense, but in simple examples it is

“obvious”.ECON30010 Topic 2: Constrained optimisation 9 - 16 March 39 / 48

Types of goods

T f g d l g d

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Types of goods: normal goods

Consider good 2:

∂ q 2(p 1, p 2, Y )∂ Y =

∂ ∂ Y 2

Y − b 1p 1p 2 − b 2 =

2p 2 > 0 because p 2 > 0.

The consumption of good 2 increases as the income of this individualincreases.These goods are called normal .


Types of goods

Types of goods: luxury goods

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Types of goods: luxury goodsIn fact, we can do even more math and look at what share of income thisindividual spends on good 2, and how it changes with income:

∂ ∂ Y

p 2q 2(p 1, p 2, Y )Y

= p 2Y ∂ ∂ Y q 2(p 1, p 2, Y ) − q 2(p 1, p 2, Y )

Y 2

= 2Y − (2(Y − b 1p 1) − b 2p 2)

Y 2 = b 1p 1 + b 2p 2

Y 2 > 0

This is a luxury good. For this good, the individual spends larger share of her income on the good as income rises. That is, if an individual spends

10% of income on good 2 at income level Y, if income increases by , theshare spent from this would be above 10%.12 In fact, in our case it ispossible that b 1p 1+ b 2p 2Y 2 > 1, so an individual spend the whole and more on good 2.

12

This sentence implies that luxury goods must also be normal.ECON30010 Topic 2: Constrained optimisation 9 - 16 March 41 / 48

Types of goods

Types of goods: inferior goods

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Types of goods: inferior goodsConsider good 1:

∂ q 1(p 1, p 2, Y )∂ Y

= ∂ ∂ Y

2b 1 − Y − p 2b 2

p 1= −

1p 1

< 0 because p 1 > 0

The consumption of good 1 decreases as the income of this individualincreases.These goods are called inferior .

Note that our formula implies that if the price of good 1 is very low, theconsumption of this good decreases extremely fast as income increases. It maysound peculiar on a rst sight, but should not be surprising: if price of good 1 isvery low, then this individual consumes a lot of good 1 to begin with, so the fallin consumption is also very large.


Types of goods

Giffen goods

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Giffen goodsConsider good 1 again:

∂ q 1(p 1, p 2, Y )∂ p 1

= ∂ ∂ p 1

2b 1 − Y − p 2b 2

p 1=

Y − p 2b 2(p 1)2

Suppose Y , p 2 and b 2 are such that Y − p 2b 2 > 0. Then ∂ q 1(p 1 ,p 2 ,Y )

∂ p 1 > 0.It means that as price of good 1 increases, the consumption of good 1increases. These goods are called Giffen goods. This is an unusualproperty and we will explore it in a little more detail.

First, let us see what the condition Y − p 2b 2 > 0 implies in terms of theamount of good 1 consumed. The demand for good 1 is 2b 1 − Y

− p 2b 2p 1 , so

if Y − p 2b 2 > 0, then q 1 < 2b 1. That is, ∂ q 1(p 1 ,p 2 ,Y )

∂ p 1 > 0 only when theconsumption of good 1 is low.


Types of goods

Giffen property vs Giffen goods

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Giffen property vs. Giffen goods

∂ q 1(p 1, p 2, Y )∂ p 1

> 0 if Y − p 2b 2 > 0

∂ q 1(p 1, p 2, Y )∂ p 1

< 0 otherwise

We see that derivative may change for different values of Y , p 2 and b 2.A more proper terminology should be a good that “exhibits Giffenproperty” in the range b 1 < q 1 < 2b 1.13

All other goods can be change their “properties” depending on particularprices and income; for example, a good can be normal over a range of prices and incomes and inferior over a different range.

13

The condition b

1 < q

1 comes from slide (39).ECON30010 Topic 2: Constrained optimisation 9 - 16 March 44 / 48

Types of goods

Another example of constrained optimisation

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Another example of constrained optimisation

We will talk more about Giffen behaviour; before we turn to that, we willrevisit our utility function that gave Giffen behaviour and will change justone coefficient.

Old:

u (q 1, q 2) = ln( q 1 − b 1) − 2 ln(b 2 − q 2)

New:

u (q 1, q 2) = ln( q 1 − b 1) − ln(b 2 − q 2)


Types of goods

Solving problem 2: FOC

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Solving problem 2: FOC

∂ L

∂ q 1=

1

q 1 − b 1− λp 1 = 0

∂ L∂ q 2

= 2

b 2 − q 2− λp 2 = 0

∂ L∂λ

= Y − p 1q 1 − p 2q 2 = 0

1q 1 − b 1 = λp 1

2b 2 − q 2

= λp 2

p 1q 1 + p 2q 2 = Y

b 2 − q 2

q 1 − b 1=

p 1

p 2p 1q 1 + p 2q 2 = Y .

q 2 = b 2 − p 1p 2

(q 1 − b 1)

Y = p 1q 1 + p 2q 2 = p 1q 1 + p 2(b 2 − p 1p 2

(q 1 − b 1))

= p 1q 1 + p 2b 2 − p 1(q 1 − b 1) = p 2b 2 − p 1b 1

Where is q 1?Or q 2?What if p 2b 2 − p 1b 1 is not equal Y ?


Types of goods

How a new utility function looks like?

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How a new utility function looks like?

Indifference curves are indifference lines, like with perfect complements! Itis not surprising that we cannot nd a solution (it is akin to maximisinglinear function f (x ) = x using rst order conditions: you’ve triedsomething like this in Tutorial 1).


Types of goods

Summary

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Summary

Skills: Lagrange method of optimisation. Cases when Lagrange method does not produce meaningful results ordoes not work.

Necessity to watch for corner solutions / impose conditions to ensurethat your solution makes sense.

Economics:

Marshallian demand, formal denitions of different types of goods.


ECON30010-L2-2016

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