MECON PS4 sol - Microeconomics Problem Set 4 Solutions Due...

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Microeconomics Problem Set 4 Solutions Due February 10 th , 2010 Winter 2010 ECON 100A Professor Michael Noel University of California San Diego 1. Now, let’s see if you can do the general perfect complements problem. Matt gets utility from X and Z. Matt always consumes α X with β Z. a. What type of utility function represents Matt’s preferences? Write down an expression for Matt’s utility. U=min{( 1 / α) X, ( 1 / β) Z} b. What is Matt’s utility maximizing combination of X*(P x , P z , Y) and Z*(P x , P z , Y)? Use the facts that (1) that there will be no excess X or Z (so 1 / α X*= 1 / β Z*) and (2) the utility maximizing bundle must be on the budget constraint (PxX*+PzZ* =Y). Substitute the expression for Z*= β / α X* from (1) into the budget constraint in (2) to get: Px(X*)+ Pz( β / α X*)=Y X*(Px+ β / α Pz) = Y and then solve for X*= z x p p Y α β + Finally, substitute this expression for X* into Z*= β / α X* to solve for Z*: Z*= ) ( z x p p Y α β α β + c. What is the Matt’s indirect utility function, i.e., V(P x , P z , Y)=U(X*(P x , P z , Y), Z*(P x , P z , Y))? U*(px, pz, Y)=min{ 1 / α X*, 1 / β Z* }=min{ ) ( , ) ( z x z x p p Y p p Y α β α α β α + + } d. What share of his budget does Matt spend on X*? What share of his budget does Matt spend on Z*? (Note: check that these two shares sum to one.) Share of Budget spent on X; PxX*/Y= z x x z x x p p p p p Y Y p α β α β + = + Share of Budget spent on Z; PzZ*/Y= z x z z x z p p p p p Y Y p α β α β α β α β + = + ) (
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Sum of the two budget shares? 1 = + + = + + + z x z x z x z z x x p p p p p p p p p p α β α β α β α β α β 2. Now, let’s see if you can do the general perfect substitutes problem. Toni gets utility from X and Z. Toni thinks that α units of X are exactly as good as β units of Z. a. What type of utility function represents Toni’s preferences? Write down an expression for Toni’s utility. U= β X + α Z b. What are Toni’s ordinary demand functions, X*(P x , P z , Y) and Z*(P x , P z , Y)? MRS= β/α X*=Y/Px if β/α> Px/Pz =0 if β/α< Px/Pz Z*=0 if β/α> Px/Pz = Y/Pz if β/α< Px/Pz c. What is the Toni’s indirect utility function, i.e., V(P x , P z , Y)=U(X*(P x , P z , Y), Z*(P x , P z , Y))? U*(px, pz, Y)= β Y/Px if β/α> Px/Pz = α Y/Pz if β/α< Px/Pz d. What share of her budget does Toni spend on X*? What share of her budget does Toni spend on Z*? If β/α> Px/Pz then her budget share on X* is 1 and Z* is 0 If β/α< Px/Pz then her budget share on X* is 0 and Z* is 1 3. Consider the utility maximization problem subject to a budget constraint with the following utility function: z x z x U z x 2 ) , ( max , + = a. Are the commodities x and z goods? Prove your answer. What does this imply about the indifference curves for U? X is a good since utility increases as you get more X (holding Z constant), 0 1 ) , ( > = x z x U ; Z is a good since utility increases as you get more Z (holding X constant), 0 1 ) , ( > = z z z x U ; Since both X and Z are goods, indifference curves are negatively sloped. Also utility is always increasing as X and Z increases so this consumer never reaches a bliss point (monotonicity).
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