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Microeconomics Tutors at Yale

 

Problem Submitted by Intermediate Microeconomics Tutor at Yale

Consider the real-valued function ƒ defined over the interval​​ -3, 3​​ by:

 

fx=4-(x+2)2if x-1x2(2-x)if x>-1

 

 

Sketch​​ a​​ rough graph​​ of the function

  • Find​​ all​​ critical​​ points​​ (i.e.,​​ points​​ with​​ zero​​ first​​ derivative)​​ of​​ the​​ function.​​ Identify​​ the​​ interior​​ local​​ maxima​​ and​​ minima.

  • Is​​ the​​ function​​ discontinuous​​ anywhere?​​ Is​​ the​​ function​​ non-differentiable​​ anywhere?​​ If​​ your answer​​ to​​ either​​ of​​ these​​ questions​​ is​​ yes,​​ is​​ there​​ a​​ local​​ maximum​​ or​​ minimum​​ at​​ these​​ points?

  • What​​ is​​ the​​ function‘s​​ global​​ maximum​​ and​​ minimum.​​ [Hint:​​ Check​​ the​​ function​​ values​​ at​​ all​​ critical points​​ and​​ endpoints.]

 

Problem Submitted by Intermediate Microeconomics Tutor at Yale

You​​ have​​ two​​ midterm exams coming​​ up, and​​ have​​ to​​ decide​​ how​​ to allocate​​ your​​ study​​ time. After eating, sleeping, exercising,​​ and​​ maintaining​​ some​​ human​​ contact,​​ you​​ will​​ have​​ 10​​ hours​​ each​​ day ​​ in​​ which​​ to study for​​ your​​ exams.​​ You​​ have​​ figured​​ out that​​ your​​ GPA​​ (G)​​ from​​ your​​ two​​ courses, Economics​​ and​​ Sociology, takes​​ the​​ form

G=47(2 E+S)

where​​ E is the​​ number​​ of​​ hours​​ per​​ day​​ spent studying Economics​​ and S is the​​ number​​ of​​ hours​​ per​​ day​​ spent studying​​ Sociology.​​ You​​ only​​ care​​ about​​ your​​ GPA.

  • What​​ is​​ your​​ optimal​​ allocation of study​​ time?

  • If​​ you​​ follow​​ this​​ strategy,​​ what​​ will​​ your​​ GPA​​ be?

  • What​​ will​​ the​​ shadow​​ value,​​ measured​​ in​​ GPA​​ units,​​ of​​ the​​ study​​ time?​​ [Hint:​​ In​​ a​​ constrained​​ optimization​​ problem,​​ the​​ optimal​​ value​​ of​​ the​​ Lagrange​​ multiplier​​ is​​ the​​ shadow​​ value​​ of​​ the​​ constraint.]

 

 

Problem Submitted by Intermediate Microeconomics Tutor at New York University

John​​ is​​ a​​ stamp​​ (x1)​​ collector,​​ but​​ he​​ also​​ likes​​ fancy​​ clothes​​ (x2).​​ His​​ utility​​ function​​ is

Ux1,x2=x1+10x2-12x22

Each stamp costs​​ p1=1​​ and a piece of his favorite clothing costs​​ p2=2.

  • Assuming​​ that​​ his​​ total​​ income​​ is​​ $10,​​ find​​ his​​ optimal​​ x1​​ and​​ x2.​​ ​​ Is​​ this​​ solution​​ interior?

  • Suppose​​ now​​ George‘s​​ salary​​ doubles,​​ resulting​​ in​​ his​​ higher​​ income​​ $20.​​ ​​ Find​​ his​​ new​​ demanded​​ quantities​​ of​​ stamps​​ and​​ clothes.​​ Is this​​ solution​​ interior?

  • In​​ cases​​ (a)​​ and​​ (b),​​ what​​ is​​ the​​ marginal​​ utility​​ from​​ one​​ dollar​​ invested​​ in​​ stamps,​​ MU1p1,​​ and in clothing (at the optimal demand),​​ MU2p2,​​ ​​ . Are they equal? Discuss.

 

Problem Submitted by Intermediate Microeconomics Tutor at New York University

Mark​​ wants​​ to​​ buy​​ T-shirts​​ (good​​ x),​​ whose​​ (per​​ unit)​​ price​​ is​​ px,​​ and​​ notebooks​​ (good​​ y),​​ whose​​ (per​​ unit)​​ price​​ is​​ py.​​ ​​ His​​ utility​​ function​​ is​​ Cobb-Douglas​​ of​​ the​​ form​​ ux,y=x2y3​​ and​​ his​​ income​​ is​​ I.

  • Find the marginal utility of each good. Are the preferences monotone?

  • Define​​ an​​ indifference​​ curve​​ and​​ show​​ that (for​​ some positive​​ level​​ of​​ utility)​​ is​​ decreasing​​ and​​ convex.

  • Write​​ down​​ the​​ budget​​ constraint​​ of the​​ consumer.

  • Write​​ the​​ Lagrangian​​ for the​​ utility maximization​​ problem.

  • Solve the first-order conditions to find the demand functions for both​​ x and y. [Hint: Your​​ results​​ should​​ be​​ functions​​ of​​ px,py and I].

  • Check the second order conditions for maximum.

  • In​​ the​​ optimal​​ consumption​​ bundle,​​ how​​ much​​ money​​ is​​ spent​​ on​​  x​​ and​​ how​​ much​​ on​​  y?​​ Express​​ your answer​​ in proportion to the total​​ income​​ I.

  • Suppose​​ now​​ that​​ the​​ price​​ of​​ notebooks​​ doubles​​ and​​ Mark​​ has​​ to​​ pay​​ 2py per​​ unit.​​ How​​ does the​​ optimal consumption bundle​​ change?​​ Does the spending proportion​​ change?

 

Problem Submitted by Intermediate Microeconomics Tutor at New York University

John​​ (from​​ last​​ week‘s​​ problem​​ set)​​ is​​ wearing​​ his​​ fancy​​ clothes​​ and​​ is​​ going​​ to​​ a​​ cinema​​ to​​ watch​​ "Destroyer"​​ (Nicole​​ Kidman​​ in​​ the​​ leading​​ role)​​ with​​ some​​ friends.​​ Once​​ he​​ is​​ arriving​​ at​​ the​​ cinema,​​ he​​ wants​​ to buy​​ something​​ to eat. In the​​ cinema,​​ he can buy​​ chocolate bars,​​ z,​​ and popcorn,​​ y ​​ (measured​​ in​​ cups).​​ He​​ loves​​ chocolate​​ and popcorn.​​ His​​ utility function​​ is uz,u=z2+y2.​​ The budget​​ constraint​​ ispzz+pyy=I.

  • Define​​ an​​ indifference​​ curve.​​ Write​​ down​​ the​​ function​​ for​​ the​​ indifference​​ curve​​ for​​ some​​ positive​​ level​​ of​​ utility,​​ u-.​​ Draw​​ the​​ indifference​​ curve.

  • Solve​​ the first-order conditions of the utility maximization problem and find the critical (not necessarily​​ optimal!)​​ quantities​​ for​​ chocolate​​ bars,​​ z*,​​ and​​ cups​​ of​​ popcorn,​​ y*.

  • Derive​​ the​​ Hessian​​ matrix​​ of​​ the​​ utility​​ function.​​ Then,​​ write​​ down​​ the​​ bordered​​ Hessian matrices,​​ and​​ calculate​​ their​​ determinants.​​ (Hint:​​ It​​ is​​ almost​​ always​​ simpler​​ to​​ calculate​​ if​​ you​​ put the​​ Lagrange​​ multiplier​​ as​​ the​​ first​​ variable​​ in​​ the​​ Hessian​​ matrices.)

  • Check​​ the second order conditions. Is​​ your answer​​ to part (b) a constrained​​ maximum?

  • Go​​ back​​ to​​ your​​ graph​​ of​​ the​​ indifference​​ curve​​ and​​ use​​ intuition​​ to​​ explain​​ where​​ one​​ might​​ find​​ the​​ true​​ optimal​​ consumption​​ bundles.​​ Consider​​ each​​ of​​ the​​ following​​ three​​ cases:i pz<py   ii pz>py   iii pz=py.

 

Problem Submitted by Intermediate Microeconomics Tutor at Columbia University

Sophia​​ likes​​ donuts,​​ x,​​ and​​ milkshakes,​​ y​​ .​​ ​​ The​​ prices​​ are​​ px and py,​​ respectively.​​ ​​ Her​​ utility​​ function​​ ux,y=x12+y.

  • Derive​​ the​​ marginal​​ rate​​ of​​ substitution​​ at​​ an​​ arbitrary​​ point​​ x-, y-, where x-, y->0.

  • Suppose​​ that​​ Sophia‘s​​ income​​ is​​ ​​ I.​​ Use​​ the​​ Lagrangian​​ method​​ to​​ obtain​​ the​​ optimal​​ consumption​​ bundle​​ of​​ the​​ consumer​​ (as​​ functions​​ of I,px,py).​​ ​​ [Hint:​​ ​​ Make​​ sure​​ you​​ cover​​ cases​​ in​​ which​​ I​​ ​​ is​​ small,​​ as​​ well​​ as​​ cases​​ in​​ which​​ ​​ I​​ is​​ high;​​ consumptions​​ cannot​​ be​​ negative!]

  • Suppose​​ Sophias​​ total​​ income​​ is​​ I=$10 ,​​ the​​ price​​ of​​ a​​ donut​​ is​​ px=$0.05,​​ while​​ the​​ price​​ of​​ a​​ milkshake​​ is​​ py=$1.​​ What​​ is​​ Sophia‘s​​ optimal​​ consumption​​ bundle?​​ Is​​ this​​ an​​ interior​​ solution?

  • Now​​ suppose​​ Sophias​​ total​​ income​​ triples,​​ so​​ that​​ ​​ I=$30.​​ What​​ is​​ her​​ optimal​​ consumption? Is this new demand​​ interior?

 

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Intermediate Microeconomics tutors for Yale

 

 

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