Documents

Class Notes 10 (1)

Description
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
Categories
Published
of 20
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
  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 

Test

Jul 23, 2017
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks