Coursenotes_ECON301

Mrs px py pxx pyy solving these two equations we get

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online