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Basie Preperties ef Utility and Preduetieu Funet inns (1E! paints) . {3 points} Consider the uti]it_r,r funetiun tulle] = r:_2 + 1', where

Basic Properties of Utility and Production Functions

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Screen Shot 2018-05-01 at 2.06.44 PM.png

(3 points) Consider the utility function u(c) = −c−2 + γ, where γ > 0.

  1. (a) Is marginal utility decreasing for this utility function? Show your work to prove
  2. your answer.
  3. (b) Find the coefficient of relative risk aversion.
  4. (c) Does this utility satisfy the first and second Inada conditions?

(2 points) Prove whether or not the following production functions exhibit constant returns to scale:

(a) F(K,L) = AKαL1−α, where α is between 0 and 1 and A > 0. (b) F(K,L) = AKL where A > 0.

(5 points) Consider a firm that uses capital K and land M as inputs to produce a final good, Y . Suppose that the technology for this firm is given by:

11 F(K,M)=2K2 +3M3

The firm must pay rental rate μ(1 + τK ) for each unit of capital rented, where τK > 0 represents a capital tax. It also must pay γ for each unit of land rented. Set up the firm's profit maximization problem and take FOCs. Find the optimal capital-land ratio. What happens when τK increases? Give some intuition. 

Screen Shot 2018-05-01 at 2.06.44 PM.png

Basie Preperties ef Utility and Preduetieu Funet inns
(1E! paints) . {3 points} Consider the uti]it_r,r funetiun tulle] = —r:_2 + 1', where "r 3:- I]-
{5] Is margins] nl:i]it_r,r decreasing far this utility tunetien? Show 3|qu wurlr. tu prmre _1.ruur answer-
[h] Find the msffieient oi relative risk euersiun- [e] Dues this litiiitjl' setisi'].r the first said ssmnd Ina-is. mnditiuns?‘
. {2 points} Prove whether DI net the Full-swing pruduetisun' fun-stems” exhibit mnstent
returns to scale: [5] FfK.LJ = AK“L]'“, where a: 'e hetwsenfl and 1 and A :1- I]. [h] F{K.LJ = AHL where A :- E3. . {5 paints] Ce-nsidernfirmthet useeespitle and lend M as inputs tnpreduee efinsi
Head, 1". Suppme that the teehnelugf far this Erin is given by: me: M] = 2rd + 3M1: Thefimnrustpeyrsntsiretefll+mj Sareeehunitefmpiteirented,whererfl 31-1]
represents empite] taI.Il:s.1su ninet ps3; Tfureeehnnit uflend reused. Set up the
firm's profit maximisation problem and take Fflfle Find theeptirusl sepitel—iend rstiu.
What happens when 1"}: hleresesls'? Give sures intuition.

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