Coursenotes_ECON301

# To do this we must take the partial derivatives of

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Unformatted text preview: UX = Y(X) X MUX = (XY) X MUX = U X and... MUY = U Y MUY = X(Y-1) MUY = X(Y) Y 22 MUY = (XY) Y MUY = U Y Now we can calculate the general form of the MRS for a Cobb-Douglas utility function. Recall that MRS is defined as: MRS = MUX MUY Using the marginal utilities we just derived for the general form of the CobbDouglas utility function we have: MRS = MUX MUY MRS = U __X__ U Y Y X MRS = Let's do an example, assume you are given that = 1,969, = 0.35, and = 0.65. Then we have the following Cobb-Douglas utility function: U(x,y) = XY = 1,969X0.35Y0.65 and the marginal rate of substitution (using our shortcut) is... MRS = MRS = MRS = Y X 0.35Y 0.65X 7Y 13X Let's check that this is correct using the long way around... 23 MRS = MUX MUY MRS = 1969(0.35)X-0.65Y0.65 1969(0.65) X0.35Y-0.35 MRS = (0.35)Y0.35Y0.65 (0.65) X0.35X0.65 MRS = 0.35Y 0.65X MRS = 7Y 13X Does this result hold if the Cobb-Douglas utility function is NOT constant returns to scale? Assume you are given that = 1,969, = 0.65, and = 0.65. Then we have the following Cobb-Douglas utility function: U(x,y) = XY = 1,969X0.65Y0.65 and the marginal rate of substitution (using our shortcut) is... MRS = MRS = MRS = Y X 0.65Y 0.65X Y X Let's check that this is correct using the long way around... MRS = MUX MUY MRS = 1969(0.65)X-0.35Y0.65 1969(0.65) X0.65Y-0.35 24 MRS = (0.65)Y0.35Y0.65 (0.65) X0.65X0.35 MRS = 0.65Y 0.65X MRS = Y X This result DOES, in fact, hold for IRS functions (or DRS functions). LIONTIEF FUNCTIONS The general form of a Liontief utility function is defined as: U(x,y) = min{X , Y} meaning, U(x,y) = {X {Y if X &lt; Y if Y &lt; X This is interpreted as meaning that the utility is equal to the smaller of the two values X and Y. In the special case where X = Y, the utility is equal to either one of the values (since they are the same). We know what a Liontief indifference curve looks like, but let's investigate why this is. As an example, let's draw the unit indifference curve (the indifference curve that gives the consumer one unit of happiness) of the following Li...
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## This note was uploaded on 05/25/2010 for the course ECON 301 taught by Professor Sning during the Spring '10 term at University of Warsaw.

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