Lecture 2- Constrained Optimisation

Constructing Ann's indifference curves ft budget line

(2,2) is where IC touches budget line

Non-convexity

z=ax+(1-a)y

Perfect complements, revisited

Preferences not strictly convex Do you expect solution is unique when goods are perfect complements u(x)=min(x1,x2) Answer: YES, solution is unique - Won't work if budget lines are horizontal or vertical but budget lines always slope down

Solutions not unique with convexity

All points between s and S are solutions

Example 4: what utility function looks like

Almost exactly linear- not surprising we can't find a solution Can't really maximise how we want to Corner solution- at either corner (again, depending on the slope of the lines)

More assumptions

Also assuming u(x) is differentiable- or twice differentiable Too technical tho!

Note on axioms/ assumptions, convexity

Alternatives Second 2 give 2 different definitions, not part of subject

What can go wrong? Satiation

Ann has preferences Utility always negative or zero. We don't care. Any positive monotonic transformation of utility function leads to a utility function representing same preferences, can add any constant to make Ann's utility positive

Example 4: What happened?

Answer: none of the above Corner solution- Lagrange doesn't give you a meaningful answer Budget line will be exactly sitting on same slope as IC

Example 3, Solution and Correct parameters

But q1 needs to be > b1 More complex/ less obvious that not within conditions so make sure you include them in answer!

What can go wrong? Satiation question

C is correct answer

Giffen goods

Consider good 1 Quantity with respect to price- if positive, Giffen Doesn't follow law of demand

Types of goods: inferior goods Why is this one not the share when the one before was?

Consider good 1 - Differentiate with respect to Y

Types of goods: normal goods

Consider good 2 Consumption of good 2 increases as the income of this individual increases These goods are called normal

Convexity and strict convexity: examples, perfect complements

Consider preferences such that y≥x if and only if min(y1+y2)≥min(x1+x2) --> (2,2)≥(100,1) --> (3,1)~(1,3) --> (3,3)~(5,3) What sort of convexity? Answer is CONVEX! - Can take two points on horizontal/vertical line

Convexity and strict convexity: examples, perfect substitutes

Consider preferences such that y≥x if and only if y1+y2≥x1+x2 --> (2,2)≥(3,0) --> (3,1)~(1,3) Perfect substitutes- describe situation where you see no difference between two objects What sort of convexity? Answer is convex!!

Strict convexity on a graph

Consumer indifferent between x and y z=ax+(1-a)y If utility increases in NE direction, whole straight line above indifference curve (more desirable) With strict convexity, unique solution to maximisation problem

What's going on...

Corner solution!

Can a Giffen good be normal?

Correct: No Even though you do separate derivatives, looks like it could be normal

Example 2 of Lagrange's method

FOC doesn't make sense What's happened - Didn't ask the right question

Budget contraint

Good 1 and 3 cost money, and agent has limited amount of it Consumer restricted to choose consumption bundle q=(q1,q2) such that p1q1+p2q2≤Y To graph: Imagine spending all money on each good, join line (since linear) All points within triangle are possible

The plan for constrained optimisation

Having continuous utility function is useful, not enough Budget set Utility maximisation problem (convexity) The Lagrange method Marshallian demand

Example 3, setup (transform)

I want to differentiate u() but that is messy af so let's do a positive monotonic transformation

Topic 1 refresher

If preferences satisfy completeness, transitivity, monotonicity and G-continuity, there exists utility function u(x) that represents preferences

Constructing Ann's indifference curves

Indifference curves look like circles

Shadow value of a constraint: an example

Interpreting lamda We sub in values for Y, p1 and p2 to get tow different shadow values of constraint - Change in utility when Y increases is almost equal to lamda times change in income - If you make the increase in income infinitesimally smaller, it would get even closer to lamda deltaY Can determine the marginal utility of income Basically shows how much the constraint restricts you - How much utility increases if I relax the constraint a little - Lamda close to zero, utility doesn't change --> doesn't bind you much - Lamda larger--> giving a lil money increases u a lot, restriction is quite binding

Constrained optimisation: Lagrange's method

Know how to solve unconstrained --> FOC, see if sol makes sense Lagrange method turns sth you don't know how to solve (constrained) into what you do --> introduced an auxiliary problem Define a Lagrangian...

Types of goods: luxury goods

Look at what share of income this individual spends on good 2 and how it changes with income Calculating derviative (how much money you spend on good 2, divided by total income) Harder to differentiate- need quotient rule Good 2 is luxury good, share is increasing with increase in income

Consumer maximisation problem

MAKE SURE you write decision variables under max!!!!!!!!!!!!! Can also write $5 problem from topic 1 as: max u(x,y) subject to x+y=5

Summary and what's next

Main points - Assumptions on individual preferences allows us to represent preferences by continuous utility function - Maximisation of utility function subject to budget constraint (using Lagrange method) leads to Marshallian demand functions --> Lagrange method: turn constrained (pseudo-)optimisation problem into unconstrained maximisation problem --> Solve unconstrained (pseudo-)optimisation maximisation problem using standard methods What's next - More difficult constrained max problem - Odd maximisation problem - Normal, inferior and Giffen goods

What if no monotonicity

Monotonicity is technical assumption: easier to solve problem if assumption is imposed, but not doomed if it doesn't hold More sophisticated methods to find a solution. Withe quality, we can use Lagrange method. With inequalities (i.e. no monotonicity), we need Kuhn-Ticker method

Implication of monotonicity: proof

Proof by contradiction: suppose we cannot replace ≤ by =: in other words, optimal consumption bundle (p1',q1') is such that p1q1'+p2q2'=Y'<Y The ageny has (Y-Y')>0 income left over, which can be spent on good 1 and good 2 (e.g. equally) to buy bindle (q1,q2) Note that (q1,q2) >> (p1',q1'). Since we assumed monotonicity, (q1,q2)> (p1',q1'). therefore, better with (q1,q2) Note that by construction, (q1,q2) is affordable Therefore, (q1,q2) is in budget set and preferred to (q1',q2')- hence, (q1',q2') not optimal, contradiction to initial supposition

Marshallian demand function

Revisit solution in example 1 Depends on p1,p2 and income Y We can write q1(p1,p2,Y) and q2(p1,p2,Y) Marshallian/ uncompensated demand posints

Giffen property vs Giffen goods

Should actually be called Giffen property Depends on the parameters! q1 isn't a Giffen good, but does have Giffen property under some parameters!

Example of solving- not caring abt lambda

Showing how much you'll spend on each Spend half income on q1 and q2

Summary

Skills - Lagrange method - Cases where Lagrange doesn't produce meaningful results or doesn't work - Impose conditions that makes sol make sense - Necessity to watch for corner solutions Economics - Marshallian demand, formal definitions of different types of goods

Diversion: complete constrained problem

Solve problem, see solution, if solution makes no sense, we know solution is likely to be last point in the range Solve more visually If outside of range, check corner solutions and calculate utilities

Example 3, FOC

Solving for q1 and q2 in absolutely standard way

What else can go wrong? Non-convexity

Suppose IC look like this 2 optimal consumption bundles- not that big problem if funding optimal bundle is ultimate goal But if we want to know how consumption changes in response to change in prices, having 2 bundles inconvenient- don't know which one consumes Want to rule out this situation (ensuring convexity)

Convexity- formal definition

Suppose a consumer indifferent between x and y Consider bundle z=ax+(1-a)y for any a between 0 and 1 - Preferences convex if z≥x and z≥y - Strictly convex if z>x and z>y Convex preferences and convex functions are 2 different notions

Example 4: Setup

Talk more about Giffen behaviour Another bad thing that can happen without considering what happens...

What can go wrong? Saturation 4

The maximum utility is at point (1,1) --> she can afford to buy bundle If Ann cannot get utility higher than 0, must be optimum consumption bundle given the budget set

Solution is unique with strictly convex preferences

Theorem Suppose that ≥ are strictly convex. Then solution is unique Proof By contradiction: suppose two solutions to problem on slide 7 which aren't equal x≠y - Consider z=0.5x+(0.5)y - Since both x and y in budget set, z also in budget set - Since preferences strictly convex z>x - Thus agent can afford a better bundle z, we arrive to the contradiction with x≠y and conclude that x=y

Solving the consumer maximisation problem

This is constrained maximisation because (i) Need to find max utility (ii) satisfies your constraint Problem (i) alone is unconstrained optimisation problem How to solved constrained - Sometimes you can eye it off and see solution - Substitute into budget constraint - Lagrange method --> directly use a solution which comes from this method

Constraint

Typical agent wants stuff, if 5th TV in room doesn't make agent happier, soething else would If agent to face meaningful problem, where solution isn't giving an infinite amount of something, we need a constraint We see budget limit e.g. $5 that you need to allocate as a constraint, not the most typical but will see many similar constraints Budget constraint is a typical constraint- income and prices But there's other constraints!

Convexity in chocolate and candy

Usual assumption is that MU is decreasing: love first choc, not that happy about second May not be willing to trade your first chocolate bar for a candy, but you may be willing to trade second This naturally leads to convecity: if you're indifferent between x: 3 choc, 1 candy y: 3 candies, 1 choc You must prefer 2 bars of choc and 2 candies to x and y Does always hold? Not necessarily: both aspirin good against headache (so you are indifferent) but you won't want 1/2 tab aspirin and 1/2 paracetamol Unlikely to be violated with broadly defined goods, at same time- need to think about whether assumption satisfied

Satiation- why do we find (2,2)

We have fed the question into our math machine "What is max utility Ann can obtain if she must spend all the money?" Then correct answer would be (2,2) but this isn't what we're interested in

Solving consumer maximisation problem graphically

We look for North-East most IC such that it touches the budget constraint (so solution is feasible)

Why all these differences

Weak and simple where it's at

Example 4: FOC

Wha? q1 drops out of expression

Implication of monotonicity: theorem

With monotonicity, such a situation (satiation) cannot happen Theorem: if preferences are monotonic, we can replace inequality ≤ with equality = in the consumer maximisation problem so that p1q1+p2q2=Y

Indifference curves

With utility function, can define indifference curves as u(x,y)=u_0 for different levels of u_0