Download presentation

Presentation is loading. Please wait.

Published byWhitney Matthews Modified over 2 years ago

1
Appendix to Chapter 4 Demand Theory: A Mathematical Treatment

2
Consumer Maximization Maximize U(X,Y) subject to the constraint that all income is spent on the two goods PxX + PyY = Income (I) Use technique of constrained optimization: – Describes the conditions of utility maximization

3
Lagrangian Method Used to maximize or minimize a function subject to a constraint Lagrangian is the function to be maximized or minimized λ = lagrangian multiplier Take the utility function to be maximized and subtract the lagrangain multiplier multiplied by the constraint as a sum equal to zero

4
Lagrangian Method U(X, Y) – λ (PxX + PyY – I) If we choose values of X that satisfy the budget constraint, the sum of the last term will be zero Differentiate this function three times with respect to X, Y and λ and equate them to zero This will give us the three necessary conditions for maximization

5
Lagrangian Method We will end up with the following three conditions: – MUx – λPx = 0 – MUy – λPy = 0 – PxX + PyY – I = 0 What do these mean? – MUx = λPx: Marginal Utility from consuming one more X = a multiple (λ) of its price – MUy = λPy: Marginal Utility….

6
Lagrangian Method If we combine the first two equations (the third is the budget constraint), we get: – λ = MUx/Px = MUy/Py – This is the equal marginal principal from chapter three – To optimize (maximize utility subject to a budget constraint), the consumer MUST GET THE SAME UTILITY FROM THE LAST DOLLAR SPENT ON BOTH X AND Y

7
Marginal Utility of Income λ = MU of income, or marginal utility of adding one dollar to the budget We will see in an example how this works, but for now: – If λ = 1/100 – Then if Income increases by $1, Utility will increase by 1/100

8
Example: Cobb-Douglas Utility Function U(X, Y) = X a Y 1-a We can express this function as linear in logs: alog(X) + (1-a)log(Y) These two are equivalent in that they yield identical demand functions for X and Y

9
Lagrangian Set-up alog(X) + (1-a)logY – λ(PxX +PyY – I) Differentiating with respect to X, Y and λ, and setting equal to zero gives three necessary conditions for a maximum X: a/X – λPx = 0 Y: (1-a)/Y – λPy = 0 λ: PxX + PyY – I = 0 Solve for PxX and PyY and substitute into the third equation

10
Lagrangian Set-up Solving for PxX and PyY gives: – PxX = a/λ – PyY = (1-a)/λ Now: substituting these back into the budget constraint gives: – a/λ + (1-a)/λ – I = 0 – And solving for λ gives: λ = 1/Income (I)

11
Lagrangian If λ = 1/I then we can use λ as a function of Income to solve for X and Y using the two original conditions Recall: – PxX = a/λ and PyY = (1-a)/λ – Now: PxX = a/(1/I) = Ia – And: PyY = (1-a)I – So: X = Ia/Px and Y = I(1-a)/Py

12
Lagrangian Notice that the demand for X is dependent on Income and the price of X, while the demand for Y is dependent on Income and the price of Y Demand for X, Y, NOT dependent on the price of the other good Cross-price elasticity is equal to zero

13
Meaning of Lagrangian Multiplier λ = Marginal Utility of an additional dollar of Income If λ = 1/100, then if income increases by $1, utility should increase by 1/100

14
Duality Optimization decision is either a maximization decision OR a minimization decision We can use a Lagrangian to: – Maximize utility subject to the budget constraint, OR – Minimize the budget constraint subject to a given level of utility

15
Duality and Minimization Lagrangian problem would be: – Minimize PxX + PyY subject to U(X,Y)=U* Formal set up would look like this: – PxX + PyY – μ(U(X,Y) – U*) – Where U* = a fixed, given level of utility just the same as Income was fixed in the maximization problem This method will yield the same demand functions as the maximization approach

Similar presentations

OK

WHAT YOU WILL LEARN IN THIS CHAPTER chapter: 10 >> Krugman/Wells Economics ©2009 Worth Publishers The Rational Consumer.

WHAT YOU WILL LEARN IN THIS CHAPTER chapter: 10 >> Krugman/Wells Economics ©2009 Worth Publishers The Rational Consumer.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on 21st century skills for teachers Ppt on itc group of hotels Ppt on swine flu Ppt on social etiquettes tips Ppt on global warming and its effects Ppt on formal education crossword Ppt on 5 star chocolate chip Ppt on social networking good or bad Ppt on panel discussion rubric Ppt on electricity generation by walking