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Unformatted text preview: Y = X1/2 Y1/2 COBBDOUGLAS UTILITY U(X,Y) = X Y where , are parameters for shares and is for scaling. CES UTILITY U(X,Y) = [X + Y]1/ where , are parameters for shares, is for curvature and is for scaling. The production side of the economy can also be represented equivalently in these functional forms. To do this, simply make all of the Xs into Ks and all of the Ys into Ls. Let's go through a couple of examples where we will find the consumer demands using the Lagrange technique and our new found knowledge about exponents... 10 1. U(X,Y) = (X1/3)(X1/3)[(Y2/3)/(Y1/3)] subject to 9X + 3Y = 1500.
First we need to set up the Lagrange...but what exponents do we have on X and Y? L: Max X2/3Y1/3 + (1500  9X  3Y) X,Y Taking the First Order Conditions, we get: L / X = FOCx L / Y = FOCy 2Y1/3 / 3 X1/3  9 = 0 X2/3 / 3 Y2/3 3 = 0 (1) (2) Rearrange (1) and (2) to isolate on the LHS... = 2Y1/3 / 27 X1/3 = X2/3 / 9 Y2/3 Equate (3) and (4) to get, X2/3 / 9 Y2/3 = 2 Y1/3 / 27 X1/3 or simply, 27X = 18Y 2Y = 3X Y = 3/2 X X = 2/3 Y (5) (6) from (1) from (2) (3) (4) or Substitute (5) into the Budget Constraint to get X: 3(3/2 X) + 9X = 1500 27/2 X = 1500 27X = 3000 X = 111.111 Substitute (6) into the Budget Constraint to get Y: 3Y + 9(2/3 Y) = 1500 27/3 Y = 1500 9 Y = 1500 Y = 166.6667 Note: The share of income spent on good X is: Px X = 9(111.111) = 1000 = 2 = exponent on X in the Utility Function M 1500 1500 3 11 Similarly, the share of income spent on good Y is: Py Y = 3(166.666) = 500 = 1 = exponent on Y in the Utility Function M 1500 1500 3 2. U(x,y) = [(X1/5) / (X2/5)]5 + 2 {[(Y2/3) (Y1/3)] / (Y3/4)}2 subject to 5X + 2Y = 40.
Once again, we need to set up the Lagrange...but what exponents do we have on X and Y? L: Max X + 2Y1/2 + (40  5X  2Y) X,Y Taking the First Order Conditions, we get: L / X = FOCx L / Y = FOCy 1  5 = 0 1 / Y1/2 2 = 0 (1) (2) Rearrange (1) and (2) to isolate on the LHS... = 1/5 = 1 / 2Y1/2 Equate (3) and (4) to get, 1 / 2Y1/2 = 1/5 or simply, 2/5 Y1/2 = 1 Y...
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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.
 Spring '10
 sning
 Economics

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