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Unformatted text preview: e objective function, we get the indirect utility function as a function of output prices PX , PY , and income level, . Objective: Lowest budget line Constraint: Fixed indifference curve As before, we need to satisfy two conditions of consumer equilibrium. MRS = _PX_ PY U(X , Y) = Solving these two equations, we get consumer demands for X & Y as functions of prices PX , PY , & utility level, . X = X(PX , PY , ) Y = Y(PX , PY , ) If we substitute these demands into the objective function, we get the expenditure function as a function of output prices PX , PY , and utility level, . 44 V = U(X(PX , PY , ), Y(PX , PY , )) V = V(PX , PY , ) CONSUMER'S DUALITY EXAMPLE E = PXX(PX,PY,) + PYY(PX,PY,) E = E(PX , PY , ) As we mentioned above, duality theory provides us with a procedure to construct an expenditure function from only two pieces of information, namely, a utility function and the prices of the goods. For example, suppose we have prices PX = 1 and PY = 2 and the following Cobb-Douglas utility function U = X3/4Y1/4 U = X0.75Y0.25 Using our "short cut" from last class, we get the marginal rate of substitution MRS = 0.75Y 0.25X = 3Y X We solve the problem using both the primal and dual formulations as follows: Utility Maximization (Primal) Using the primal formulation of utility maximization, we have the following two equations of consumer equilibrium: MRS = _PX_ = 3Y PY X PXX + PYY = Solving these two equations, we get the demands for X & Y 3Y = _1_ X 2 3Y = _1_ X 2 6Y = X 6Y = X Y = 1/6 X 8Y = 4/3 X = Y = 0.125 X = 0.75 X + 2Y = 45 Substituting these demands into the utility function, we get the optimal utility as a function of income (expenditure). U = X0.75Y0.25 = (0.75)0.75(0.125)0.25 = (0.75)0.75(0.125)0.25 0.75 + 0.25 = (0.805927448)(0.594603557) = 0.479207327 Rearranging the result so that the expenditure term is on the left hand side, we get the expenditure term as a function of the utility level. = _____U_____ 0.479207327 = 2.086779445 U W...
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- Spring '10