Econ501amidequationsW07 - The Ohio State University...

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The Ohio State University Department of Economics Econ 501.02—Prof. James Peck Equations for the Midterm Axiom 1: Preferences are complete: for any two bundles, A and B, exactly one of the following is true. (1) A is preferred to B, A P B , (2) B is preferred to A, B P A ,or(3)Aisind i f erent to B, A I B . Axiom 2: Preferences are re f exive: A I A . Axiom 3: Preferences are transitive: A P B and B P C A P C . Axiom 4: Preferences are continuous. If A P B ,andi f C is su ciently close to B ,then A P C . Axioms 1-4 allow preferences to be represented graphically by indi f erence curves , and by a utility function, u . A P B if and only if u ( x A ,y A ) >u ( x B B ) . Axiom 5: More is better. Starting with the bundle A =( x A A ) ,th en increasing any of the goods in A yields a new bundle that is preferred to A . Axiom 6: All indi f erence curves exhibit diminishing marginal rates of sub- stitution. The MRS is the absolute value of the slope of the indi f erence curve. The indi f erence curve f attens as you move along the indi f erence curve to the right. The marginal rate of substitution equals the ratio of marginal utilities: MRS yx = ∂u ∂x ∂y . The utility maximization problem is max u ( x, y ) subject to : p x x + p y y = M The Lagrangean approach transforms a constrained optimization prob- lem into an unconstrained problem of choosing x, y, and the Lagrange multiplier, λ, to maximize L = u ( x, y )+ λ [ M p x x p y y ] . By di f erentiating L with respect to x, y, and λ , and setting the derivatives equal to zero, the resulting F rst order conditions are: λp x =0 , λp y , and M p x x p y y . x ( p x ,p y ,M ) and y ( p x y ) are known as the generalized demand func- tions . 1
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The demand for, say, good y as a function of income, holding prices constant, is called the Engel Curve . Th isisrelatedtothe income-consumption curve ,the set of consumption bundles chosen as income varies, holding prices constant.
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Econ501amidequationsW07 - The Ohio State University...

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