# Class Notes 10 (1)

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1 Fun tions of One Variable Notes 10 1.1 Introduction ã Functions arise routinely in the study of economic problems. ã For example, a demand function relates quantity demanded to price. ã Need to study properties of such functions, such as slope ã The slope and elasticity measure responsiveness of demand to price variations. ã A function, f , from a set X to a set Y is a rule associating with each element of a set X some element in the set Y . ã This is written f : X → Y . ã In words, to each
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1   Notes 10 1.1 Introduction ã Functions arise routinely in the study of economic problems.ã For example, a demand function relates quantity demanded to price.ã Need to study properties of such functions, such as slopeã The slope and elasticity measure responsiveness of demand to price vari-ations.ã A   function ,  f  , from a set  X   to a set  Y   is a rule associating with eachelement of a set  X   some element in the set  Y  .ã This is written  f   :  X   → Y  .ã In words, to each point in  X  ,  f   associates a point,  y =  f  (  x ) in  Y  . 1  E255-Matrices E255-Matrices  ã The set  X   is called the domain and the set  Y   is called the range.ã The inverse demand function, relates price to quantity demanded,  p ( Q ). 1.2 Some Examples. DEMANDã write,  D (  p ), for the aggregate demand as a function of own price.ã Writing   Q  for the quantity demanded,  Q =  D (  p ).ãã With this function, we can consider the impact of price changes on quan-tity demanded.ã Consider a price  p ′ =  p  with corresponding demand  Q ′ =  D (  p ′ )ã the variation in quantity resulting from the price variation is Q ′ − Q =  D (  p ′ ) −  D (  p )ã The change in aggregate quantity demanded per unit change in price is: Q ′ − Q p ′ −  p  =  D (  p ′ ) −  D (  p )  p ′ −  p ã This gives the slope of the demand function. ***  2   E255-Matrices E255-Matrices  ã Similarly, the percentage change in quantity resulting from a percentagechange in price is given by: Q ′ − QQ p ′ −  p p =  D (  p ′ ) −  D (  p )  D (  p )  p ′ −  p )  p =  p D (  p )(  D (  p ′ ) −  D (  p )(  p ′ −  p )The negative of this term gives the elasticity of demand.ã Note that the elasticity of demand is computed at a point on the demandcurve: at different points, the value of the elasticity will vary.ã Thus, we may write: ǫ (  p ) =−   p D (  p )  (  D (  p ′ ) −  D (  p ))(  p ′ −  p )  ***  3   E255-Matrices E255-Matrices  P RESENT  V   ALUE ã The present value of a cash ﬂow may be viewed as a function of the in-terest rate.ã A bond is characterized by a face value,  P , an interest rate,  ρ , a maturitydate  T  ,ã It pays the holder a ﬁxed payment,  f   = ρ  P , each time period (such asevery six months or every year), and the face value  P  at the end of the T  th period.ã Thus, the cash ﬂow generated by the bond is:(  f  ,  f  ,  f  ,...,  f  ,  P )where  f   is paid at the end of periods 1, 2, up to  T  − 1.ã What is the value of this cash ﬂow?ã If the market interest rate is  r ,  x  dollars invested today for one period isworth  y =  x (1 + r ) at the end of the period.ã To have  y  dollars at the end of the period,  x  =  y (1 + r )  dollars should beinvested now.ã How much should one pay for a payoff of   y  dollars in one period fromnow? ***  4

Jul 23, 2017

#### Numerical probability

Jul 23, 2017
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