ECONOMICS

Answerallofthem

1.  Suppose  thatarisk-averse  individual  hasawealth  levelWo .  This  indi- vidual  hastheoption  ofinvesting  intherisk-freeassetwithareturnof

(1+rf)                                                        (1)

There  isanotherassetthatisriskywithareturnof(1+rh )inagoodstate withprobability1  and(1+rl )inabadstate  withprobability1.Theexpected

2

returnfromtheriskyassetis:

1

2

(1+rl )+

2

1

2 (1+rh )                                             (2)

Weassumethattheexpected  returnontheriskyassetishigher:

1

2 (1+rl )+

1

2 (1+rh )    >  (1+rf)                                (3)

1        1

2 rl+2 rh    >  rf                                            (4)

(a)  Showthatarisk-averse  person  willalways  invest  someamount  inthe riskyasset.

(b)What  willbetheamountinvested  intheriskyassetif1rl+1rh=rf?

2            2

2.   Consider  thefollowingPrisoner’s  dilemma  gameplayed  simultaneously:

•

Thefirstpayoffisthatofplayer  1andthesecondthatofplayer  2. Ifplayer  1movesfirstorweareinasequential  gamenow.

(a)What  isnormal  formofthegameinthesequentialversion? (b)What  aretheNashEquilibriainthesequential  version?

(c)Drawtheextensive  forminthesequential  version.

(d)What  isthesubgame  perfectNashEquilibriuminthesequential  ver- sion?

(e)Explain  whyitisthesubgameperfectNashEquilibriuminthesequen- tialversion?

3.Consider  thecostfunction  ofafirmforLRis:

C(q)=q3−20q2+110q                                     (5)

andthedemand  curveofthemarket  is:

QD=3,000−3P                                           (6)

(a)What  istheLRequilibrium  PandQ? (b)Howmany  firmswillexistintheLR?

4.Consider  twoindividuals  aandbwiththeutility  functions:

Ua      =   (xc).5(yc).5                                                    (7)

a        a

Ub     =   (xc).25(yc).75                                                 (8)

b             b

(xc,yc )aretheamountthatconsumersconsume.  Inordertopurchase  these

i     i

bundles,  theymustalsohavesomething  tosell.Inthiseconomy,theproduction

of goodshasalready  taken  place.  Eachconsumerhasthefollowingendowments ofgoods:

Ea    :   (xe,ye)=(500,500)                                       (9)

a     a

Eb      :     (xe,ye)=(200,800)                                    (10)

b       b

Theseendowmentscanbesoldinthe  market.The  total  amountofendow- mentsintheeconomycanbeexpressed  as:

X    =  xe  +xe  =700                                          (11)

a       b

Y      =  ye+ye  =1300                                         (12)

a       b

as:

(a)Find  theequilibrium  demand  forgoodX. (b)What  istheequilibrium   Px ?

y

[Hint:  Ifpricesofgoodsare:  Px ,Py,thebudget  constraintscanbewritten

Ia    =  Px xe  +Pyye

(13)

a            a

Ib    =   Px xe  +Pyye

(14)