1 / 15

Appendix to Chapter 4

Appendix to Chapter 4. Demand Theory: A Mathematical Treatment. 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.

etta
Download Presentation

Appendix to Chapter 4

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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) = XaY1-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

More Related