T2: Constrained Optimisation

Satiation: why Ann's optimal bundle is (1,1) (image)

(2,2) is the maximum utility Ann obtains if she spends all her money. (1,1) maximises her utility (but she doesn't spend all her income)

Non-convexity: what else can go wrong? (2) (image)

two optimal consumption bundles if we want to know how consumption changes in response to change in prices, having two bundles is inconvenient: don't know which the agent would consume

marshallian demand fn (2) (image)

writing the soln as a fns of q1(p1,p2,Y) and q2(p1,p2,Y) also called uncompensated demand

strict convexity graphically (image)

z > x z > y

convexity: definition (image)

z >/= x z >/= y

constrained optimisation: main points (2) (image)

1) assumptions on individual preferences allow us to represent them by a continuous utility function 2) maximisation of a utility fn subject to budget constraint (via Lagrange) leads to Marshallian demand fns

non-convexity: definition (image)

as x >/= z and y >/= z these preferences are not convex

strictly convex preferences: proof (image)

by contradiction (slide 7: consumer maximisation problem)

types of goods: normal goods (1) (image)

consumption of good increases as income of individual increases

Giffen property vs. Giffen goods (image)

derivatives may change for different values (and therefore, the type of good can change)

budget constraint (2) (image)

good 1 and good 2 cost money, and an agent has a limited amount of it a consumer is restricted to choose a consumption bundle q = (q1,q2) such that p1q1 + p2q2 </= Y

solving the consumer maximisation problem (image)

i) find max u(q1, q2) ii) satisfy constraint

types of goods: luxury goods (1) (image)

individual spends a larger share of her income on the good as income rises

solns may not be unique with convexity (image)

interval of solns from s to S

solving consumer maximisation problem graphically (1) (image)

looking for the NE-most IC such that it touches the budget constraint (to guarantee feasibility of soln)

convexity and strict convexity: examples, perfect substitutes (image)

perfect substitutes: no difference b/w 2 objects

types of goods: giffen goods (1) (image)

price of good increases as consumption of the same good increases

constrained optimisation: lagrange's method example 2 (image)

produces negative quanitites, which is not the soln we are seeking

implication of monotonicity: proof (image)

proof by contradiction

shadow value of a constraint: example (2) (image)

shadow value of constraint is λ λ measures the "MU of income" (how much utility changes in response to change in income)

indifference curves (1) (image)

solutions to u(x,y) = u0, for different levels of u0

types of goods: inferior goods (1) (image)

the consumption of good decreases as individual's income increases

Giffen behaviour example (image)

there is a solution - corner solution

Satiation: constructing Ann's ICs (image)

Ann's utility is a circle with centre (1,1)

Satiation: why Ann's optimal bundle is (1,1) (graph) (image)

Any other bundle gives her lower utility

non-linear utility fns and Lagrange (image)

Applying a positive monotonic transformation can simplify the fn differentiated in the Lagrange method

implication of monotonicity: theorem (2) (image)

If preferences are monotonic, we can replace inequality </= with equality = in the consumer maximisation problem, so that p1q1 + p2q2 = Y Ann's preferences aren't monotonic.

constrained optimisation: lagrange's method example 2 (graph)

Lagrange is not aware that you cannot consume negative quantities

can Giffen goods be normal?

No. If consumption of good decreases when price decreases (so you have more money), then it must decreases while the price stays constant and you just have more money.

Satiation: what can go wrong? (1) (image)

Whilst Ann's utility is always negative, recall any positive monotonic transformation leads to a utility fn that represents the same preferences