{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture04

# Lecture04 - Lecture 4 Demand Perlo Chapter 4 Vladimir...

This preview shows pages 1–8. Sign up to view the full content.

Lecture 4: Demand Perlo/ Chapter 4 Vladimir Petkov VUW 12 March 2009 Vladimir Petkov (VUW) Lecture 4: Demand 12 March 2009 1 / 32

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Graphical Representation Of Duality A S R Vladimir Petkov (VUW) Lecture 4: Demand 12 March 2009 2 / 32
Duality De°ned Consider the utility maximization problem max q 1 , q 2 U ( q 1 , q 2 ) s.t. p 1 q 1 + p 2 q 2 = Y . Suppose that its solution is q ° 1 , q ° 2 . Now °x utility at ¯ U ° = U ( q ° 1 , q ° 2 ) and consider the expenditure minimization problem min q 1 , q 2 p 1 q 1 + p 2 q 2 s.t. U ( q 1 , q 2 ) = ¯ U ° . The solution to this problem will be exactly q ° 1 , q ° 2 ! Vladimir Petkov (VUW) Lecture 4: Demand 12 March 2009 3 / 32

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Duality (Continued) Similarly, suppose we solve the following expenditure minimization problem: min q 1 , q 2 p 1 q 1 + p 2 q 2 s.t. U ( q 1 , q 2 ) = ¯ U . Let the solution to this problem be q °° 1 , q °° 2 and °x consumer income at Y °° = p 1 q °° 1 + p 2 q °° 2 . Now consider the utility maximization problem max q 1 , q 2 U ( q 1 , q 2 ) s.t. p 1 q 1 + p 2 q 2 = Y °° . The solution to this problem will be exactly q °° 1 , q °° 2 ! So the expenditure minimization problem and the utility maximization problem are closely related: they are known as dual problems . Vladimir Petkov (VUW) Lecture 4: Demand 12 March 2009 4 / 32
Uncompensated Demand By de°nition demand tells us how much a consumer is willing to buy at a given price, holding constant other factors (such as tastes and preferences, income, and prices of complements and substitutes). Take the utility maximization problem analyzed in the previous lecture. We solved for the optimal quantities as functions of prices and income. That is, we solved for the consumer±s uncompensated demand functions for these goods: q 1 = Z ( p 1 , p 2 , Y ) q 2 = B ( p 1 , p 2 , Y ) . In the Cobb-Douglas example, q 1 = α Y / p 1 and q 2 = ( 1 ± α ) Y / p 2 . The two goods are neither complements nor substitutes: the demand depends only on the good±s own price. Vladimir Petkov (VUW) Lecture 4: Demand 12 March 2009 5 / 32

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Uncompensated Demand (Continued) Suppose that we double all prices and income. What will happen to uncompensated demand? The utility maximization problem becomes max q 1 , q 2 U ( q 1 , q 2 ) s.t. 2 p 1 q 1 + 2 p 2 q 2 = 2 Y . Note that " 2 " cancels out from the budget constraint. So this problem is equivalent to the original one. The two problems will have identical solutions. Thus, if we multiply all prices and income by some number, demand will not change! Consider our Cobb-Douglas example: q 1 = α 2 Y 2 p 1 = α Y p 1 , q 2 = ( 1 ± α ) 2 Y 2 p 2 = ( 1 ± α ) Y p 2 . Vladimir Petkov (VUW) Lecture 4: Demand 12 March 2009 6 / 32
Graphical Interpretation of Uncompensated Demand Now we use a graphical approach to construct the uncompensated demand curve. If we increase a good±s price while holding other prices, tastes and income constant, the budget constraint will rotate.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 32

Lecture04 - Lecture 4 Demand Perlo Chapter 4 Vladimir...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online