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# Week 4 PAM 2000

Cornell

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This 3 page Class Notes was uploaded by Eunice on Saturday February 20, 2016. The Class Notes belongs to PAM 2000 at Cornell University taught by McDermott, E in Fall 2015. Since its upload, it has received 32 views. For similar materials see Intermediate Microeconomics in Political Science at Cornell University.

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Date Created: 02/20/16

PAM 2000 McDermott Spring 2016 February 18, 2016 tangency o consumers will maximize utility = choose a bundle om the “highest” indifference curve possible o the optimal curve will be where the IC and BC are tangent where the slope of the BC (-p /p x =yslope of the IC (MRS = -MU xMU ) y MRS is calculated by taking the partial derivative of the utility function in respect to x and then y (professor will give us the MU values) the ratio of marginal benefits is equal to the ratio of marginal costs at the optimal bundle o NOTE: just having the same slope does not mean that that the BC and IC are tangent. the point at which the slopes are the same must also be a point that satisfies the function of the BC.** o the algebra cobb-douglas utility function a 1-a u(x,y) = x y ensures that fractions of earnings go to each good, regardless of income/prices a + (1-a) = 1 (usually the case) typically 2/3 goes to capital and 1/3 goes to labor in the case that the cobb-douglas function is applied as a production function example given values M = 64, p = 2, p = 8, and u(x,y) = x y .25.75 x y then, given the marginal utilities,: o MU x .25x -.7y.75 .25 -.25 o MU y .75x y o -px/py= -MU /xU y o simplify and get an expression: p /p x (y/3x) y = 3xp /x y **this is the function of all points where the IC and BC have the same slope solve for x or y and plug it into the BC function to discover the actual point of tangency M = p xx+p (3yp /p x y = 4xp x x = M/4p x o NOTE: the constant is ¼, the value of a in the utility function o this means that the fractions in the exponent are the fractions of the income spent on each good (when the fractions sum to 1) x = 8 using x, solve for y (which = 6) so the point of tangency is (8, 6) cobb-douglas IC always smooth and will never touch the axes o u(x,y) = x y1-a o is x or y is zero then u = 0 o this assumes that the consumer will always want some of both o interior solution the optimal bundle lies in the interior of the first quadrant (often at the point of tangency between BC and IC) o exceptions being: Leontief and kinked BCs o Corner Solution (complement of Interior Solution) consumer is constrained from having a negative quantity of x or y*** the optimal bundle is on one of the axes (the IC touches the BC at the corner of the BC) it’s possible to have the IC tangent at the axes which is rare and would make it an interior solution mathematically, the consumer would want to purchase a negative units of a good (which is impossible so the solution is limited to the corner of the BC) ***consumer wants to be on the outside of the quadrant in order to calculate, plug in x=M/p xnd y=0, y=M/p and y=0 to find the optimal bundle o extreme preferences usually: people want a mix of both goods but ratio will change depending on prices Leontief: people will always want a specific ratio regardless of price linear preferences: goods are perfect substitutes u(x,y) = Ax + By o MU x A o MU y B ie, coke and pepsi even if coke is liked precisely twice as much as pepsi o u(c,p) = 2c + p o MU: easy to calculate in this form (not cobb- douglas); coke MU is 2 and pepsi MU is 1 MU are always constant and thus the MRS is always constant thus the slope of the IC is the negative of MRS so the IC is a straight line often the solution is a corner solution o given p /p and MU /MU x y x y if the two are not equal (one is greater than the other) then the solution is a corner one if BC’s slope is greater, then the consumer will purchase only good y if IC’s slope is greater, then the consumer will purchase only good x o an interior solution with linear preferences would mean that the BC has the same function as the IC in which case, any bundle on the BC would satisfy the consumer kinked BC: solution could be interior but not necessarily tangent (could be at the bend)

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