MICROECONOMICS: CONSTRAINED OPTIMISATION, INDIRECT UTILITY, EXPENDITURE FUNCTION

OPTIMISATION

When faced with a constraint on spending, i.e., an INCOME, optimisation can be achieved via lagrange:

L = u(X,Y) + n[M – Px.X – Py.Y] Where you are maximising X and Y when

faced with a constraint ‘M’. Utility will only be maximised IF INCOME IS

EXHAUSTED. Thus Px.X + Py.Y = M To solve Lagrange, Partially differentiate w.r.t

x first then y then ‘n’.

OPTIMISATION

Pi

MU

Py

PxMRS

MUy

MUx

YPyXPxML

PyMUL

PxMUL

YPyXPxMYXuL

i

yy

xx

)2/()1(

)3(..

)2(0

)1(0

]..[),(

OPTIMISATION Therefore, Utility will be

optimised when the rate at which goods can be substituted for one another (MRS) is equal to the rate of exchange of those goods in the marketplace (Relative Price)

In addition to Income being exhausted.

Graphically, the optimal point is the point of tangency between the indifference curve and budget constraint, shown to the right.

ALWAYS CHECK SECOND ORDER CONDITIONS TO ENSURE A MAXIMUM.

The Lagrangian Multiplier (LAMBDA) is the change in utility from a marginal change in the constraint – change in utility from an extra £1 of income.

Otherwise, it can be interpreted in the form on the previous slide: At the margin, the Price reflects people’s WILLINGNESS TO PAY FOR ONE MORE UNIT.

Y

X

U

Optimal

Budget Constraint

OPTIMISATION – CORNER SOLUTION May be the scenario where

the optimal level of consumption of good ‘i’ is zero – vegetarians + hamburgers.

Maths is slightly different: For lagrange, FOCs are maximised by setting ≤ 0, not =0.

If the FOC <0, then the optimal quantity for that good consumed is 0.

The Lagrange Multiplier also becomes:

Pi > (MUi/n) where n=Lagrange Multiplier.

Thus the price exceeds the marginal value to the customer, and will not be purchased.

Y

X

U

Optimal

Budget Constraint

INDIRECT UTILITY

Standard Utility function is a function of number of goods consumed: U = u(x,y)

Solving the constrained optimisation problem gives you the optimal level of x and y x*, y*

x* = x(Px, Py, M); y* = y(Px, Py, M) Therefore, we can write the standard utility

function as a function of prices and income; Utility is INDIRECTLY determined by prices

and income: Max. Utility = V(Px, Py, M) In addition: dV/dM = λ

DUALITY – TO THE EXPENDITURE FUNCTION

Duality in this context means that there are TWO EQUIVALENT methods of solving the consumer choice problem:

A - MAXIMISE UTILITY SUBJECT TO GIVEN EXPENDITURE.

B – MINIMISE EXPENDITURE GIVEN A DESIRED UTILITY LEVEL.

Both approaches will lead to the same conclusion. Discussion thus far has been on the former (path A).

The expenditure minimisation problem is described on the next slide.

EXPENDITURE FUNCTION

Minimise total expenditure: E = P1X1 + P2X2 + ... PnXn

Constrained to fixed Utility Level: U = u(X1, X2, ... , Xn)

Set up the Lagrange: L = Px.X + Py.Y + j[U – u(X,Y)] Lx = Px + j.MUx = 0 (1) Ly = Py + j.MUy = 0 (2) U – u(X,Y) = 0 (3) (1) / (2) = Px/Py = MRS = MUx / MUy Which we know to be equivalent to the result gained

from utility maximisation.

EXPENDITURE FUNCTION Therefore: UTILITY Function = U(X, Y) INDIRECT UTILITY Function = V(Px, Py, M) EXPENDITURE Function = E(Px, Py, U) E(Px, Py, U) = 1 / [V(Px, Py, M)]

Properties of expenditure Functions: I – HOMOGENOUS OF DEGREE 1 (2x prices,

therefore 2x expenditure) II – NONDECREASING IN PRICES (dE/dPi > 0) III – CONCAVE IN PRICES (As price changes,

consumption pattern will also likely change) IV – INCREASING IN ‘U’ (Higher Utility requires

higher expenditure)