Econ 100A: Intermediate Microeconomics Notes on Consumer Theory

Size: px

Start display at page:

Download "Econ 100A: Intermediate Microeconomics Notes on Consumer Theory"

Russell Gibbs
7 years ago
Views:

1 Econ 100A: Interediate Microeconoics Notes on Consuer Theory Linh Bun Winter 2012 (UCSC 1. Consuer Theory Utility Functions 1.1. Types of Utility Functions The following are soe of the type of the utility functions that are iportant: Perfect Copleents Perfect Substitutes Cobb-Douglas Quasilinear 1.2. Perfect Copleents The Utility function: U(x, y Min {x, y} The Budget Constraint: x + y where is incoe, and and is the price of good x and y respectively. The graph of the indifference curves for Perfect Copletents is as follows: Y U U X Perfect copleents: L-shaped indifference curves This notes is prepared with soe help fro Aadil Nakhoda 1

2 Exaple: Right shoe and Left shoe: If we purchase one right shoe, we need to purchase one left shoe also. The cosuer axiization proble for Perfect Copleents: subject to Max U (x, y x + y The optial allocation, the consuption bundle that gives the highest utility (x, y, is when x y Replacing y with x into the budget constraint since x y, we have the following: x + x x ( + x y Perfect Substitutes Y Budget line U1 U2 X Consider the above graph: The slope of the budget line is Steeper than the slope of the indifference curves., The slope of the budge line in absolute value is px where px and is the price of good x and y respectively. The slope of the indifference curves in absolute value is MRS, where MRS is the Marginal Rate of Substitutions [ ] [ ] [ U(x,y ] Marginal Utility of Good x MUx Marginal Utility of Good y MU U(x,y y The slope of the budget line is Steeper than the slope of the indifference curves. This is equivalent to having the following: > MRS The Optial Allocation (x, y is (0, py. Equivalently, the quantity of good x and y deanded is (0, py. Exaple: Suppose we have two goods, Pepsi (x and Coke (y. Which good would you prefer to purchase? Spending all incoe on Coke (y, i.e. purchasing only Coke (y, will put you on the highest indifference curve given the budget constraint. The budget line is tangent to a higher indifference curve at the y Axis, than it is at the x Axis. 2

the budget constraint since x y, we have the following: x + x x ( + x y + 1.3.

3 Y Budget line U1 U2 X Consider the above graph: The slope of the budget line is Flatter than the slope of the indifference curves., The slope of the budge line in absolute value is px where px and is the price of good x and y respectively. The slope of the indifference curves in absolute value is MRS, where MRS is the Marginal Rate of Substitutions MRS Marginal Utility of Good x Marginal Utility of Good y MU x MU y U(x,y U(x,y The slope of the budget line is Flatter than the slope of the indifference curves. This is equivalent to having the following: < MRS ( The Optial Allocation (x, y is, 0. Equivalently, the quantity of good x and y deanded is (, 0. Exaple: Suppose we have two goods, Pepsi (x and Coke (y. Which good would you purchase? Spending all incoe on Pepsi (x, i.e. purchasing only Pepsi (x, will put you on the highest indifference curve given the budget constraint. The budget line is tangent to a higher indifference curve at the x axis, than it is at the y axis. Perfect Substitutes: Rules to follow: If the slope of the budget constraint is Steeper than the slope of the indifference curve, we consue the good on the y axis. In particular, > MRS Where and is the price of good x and y respectively. If the slope of the budget constraint is Flatter than the slope of the indifference curve, we consue the good on the x axis. In particular, < MRS Where and is the price of good x and y respectively. If we were to consue the good on the x axis, we represent it as: 3

The slope of the indifference curves in absolute value is MRS, where MRS is the Marginal Rate of Substitutions MRS Marginal Utility of Good x Marginal Utility of Good y MU x MU y U(x,y U(x,y The

4 Exaple: Suppose we have the utility function: U (x, y x + 5y and the budget constraint is as follows: 2x + 3y 10 The Marginal Rate of Substitution is as follows: 5 The MRS in absolute value is: MRS 5 5 The slope of the budget line is as follows: Slope (BC 2 3 The slope of the budget line in absolute value Slope (BC MRS > Slope (BC As a result, agents consue only goods x. The quantity of good x and y deanded, i.e. the Optial Allocation, is (x, y ( 10 2, 0 Note that when y 0 we have 2x 10 fro the budget constraint. x 5, y 0 The highest level of utility is 1.. Cobb-Douglas Utility Function The Cobb-Douglas utility function: U (5, 0 ( (5 + (5 (0 20 U(x, y x a y b, where a > 0 and b > 0 Alternatively, using onotonic transforation, Cobb-Douglas utility function could also be represented as follows: U (x, y a log (x + b log (y, where a > 0 and b > 0. This indifference curve will have a negative slope, which will incorporate the individual s willingness to ake tradeoffs between good x and y. How to solve for an Optial Bundle or Optial Allocation given a Cobb-Douglas function: ( px x a 1 y b bx a y b 1 ( y b x

The quantity of good x and y deanded, i.e. the Optial Allocation, is (x, y ( 10 2, 0 Note that when y 0 we have 2x 10 fro the budget constraint. x 5, y 0 The highest level of utility is 1.

5 ( y b x To solve for y, we can rewrite the above as: b y x Fro the budget constraint, we have the following: b y + y b + 1 y ( y b ( ( b y And, now to solve for x Substitute y above into the budget constraint: [( ( ] b x + Cancelling in above equation, we have the following: ( b x + The Optial Bundle is: ( b x x (x, y ( ( 1 b ( b x x ( ( b, This should be siilar to the case presented in the textbook. ( 5

constraint: [( ( ] b x + Cancelling in above equation, we have the following: ( b x + The Optial

6 1.5. Quasilinear Utility Functions A Quasilinear utility function is as follows: U(x, y ln (x + y where ln (x is the natural logarith. The function U(x, y is linear in y. Let us solve for the above function: At the Optial Bundle (x, y, we have the following equations ( px i.e. the slope of the indifference curve at the Optial Bundle is equal to the slope of the budget line. The Marginal Rate of Substitutions MRS is as follows: [ ] [ ] [ U(x,y ] Marginal Utility of Good x MUx Marginal Utility of Good y MU U(x,y y U (x, y [ln (x + y] ln (x 1 x Recalling the rule for differentiating a natural logarith function ln (x : d [ln (x] dx 1 x U (x, y [ln (x + y] 1 ( 1 x ( px ( 1 x 1 1 x x Substitute the above into the following budget constraint: x + y ( + y So our optial bundle is: y 1 (x, y (, 1 6

The Marginal Rate of Substitutions MRS is as follows: [ ] [ ] [ U(x,y ] Marginal Utility of Good x MUx Marginal Utility of Good y MU U(x,y y U (x, y [ln (x + y] ln (x 1 x Recalling the rule for

7 2. Practice Probles: 2.1. Copleents: Utility function is U (x 1, x 2 Min(x 1, 2x 2 Suppose an accountant needs 1 eraser for every 2 pencils he/she uses. Any ore pencils will not be useful as the accountant will not be able to erase the caluclations. Any ore erasers will also not serve his purpose also. Therefore, x 1 is pencil and x 2 is eraser. To solve for axiization proble: where The budget constraint: Max U (x 1, x 2 U (x 1, x 2 Min(x 1, 2x 2 p 1 x 2 + p 2 x 2 where p 1 and p 2 is the price of good x 1 and x 2 respectively. With x 1 2x 2 2p 1 x 2 + p 2 x 1 x 2 2p 1 + p 2 And for x 1 : p 1 x p 2x 1 x 1 p p Substitutes: U 3 (Coke + 6 (P epsi The price of Pepsi is $2, and the price of Coke is $0.8. What is the Optial Consuption Bundle for the individual? First assue, Coke to be on y axis ( ( slope of budget line ( 2 5 ( We know that the slope of the budget line is greater than the MRS, the slope of the indifference curve. So what good will the individual consue and why? 2.3. Cobb-Douglas: Find the Optial Consuption Bundles of x and y for the following utility functions: U (x, y x 2 3 y 5 U (x, y x 2 + y Where the price of good x is $2 and the price of good y is $1, and incoe is $10. 7

To solve for axiization proble: where The budget constraint: Max U (x 1, x 2 U (x 1, x 2 Min(x 1, 2x 2 p 1 x 2 + p 2 x 2 where p 1 and p 2 is the price of good x 1 and x 2 respectively.

Problem Set 2: Solutions ECON 301: Intermediate Microeconomics Prof. Marek Weretka. Problem 1 (Marginal Rate of Substitution)

Problem Set 2: Solutions ECON 301: Intermediate Microeconomics Prof. Marek Weretka. Problem 1 (Marginal Rate of Substitution) Proble Set 2: Solutions ECON 30: Interediate Microeconoics Prof. Marek Weretka Proble (Marginal Rate of Substitution) (a) For the third colun, recall that by definition MRS(x, x 2 ) = ( ) U x ( U ). x