1 demand and consumer_s surplus.pdf - Intermediate...

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Intermediate MicroeconomicsLecture 1: Demand and Consumer SurplusDouglas GaleNew York University1The demand curveThe demand curve you studied in yourPrinciplescourse is a relationship between the priceof a good and the quantity demanded by one or more consumers.A typical example isillustrated in thefigure below.The demand curve is downward sloping—lower prices areassociated with higher quantities—and, in this case illustrated below, the relationship islinear—the slope of the curve is constant.01234567024681012QuantityPriceThe demand curveThis simple representation of the demand curve depends on some very special assumptions.In general, the quantity of the good demanded depends, not just on the price of this good, butalso on the consumer’s income and the prices of other goods. And the demand curve is notalways downward sloping.These complications are something we’ll postpone for anotherday.In this lecture, we are just going to derive the simple demand curve above as thesolution to autility maximization problem. In doing so, we will learn about the specialassumptions that lie hidden behind this innocuous-looking curve.1
1.1Quasi-linear utilityConsider the problem of a consumer deciding how much to spend on a single good andhow much to spend on everything else.For simplicity, we assume money and the goodare perfectly divisible, so their quantities can be represented by non-negative real numbers.Then the outome of the consumer’s decision is abundle( ), where0is the amountof the good and0is the amount of money spent on everything else.The consumer is assumed to have preferences that can be represented by autility func-tionthat assigns a real number( )to each bundle( )0. We say the consumerweakly prefersa bundle( )to a bundle(0 0)if( )(0 0)andstronglyprefers( )to(0 0)if( ) (0 0).We do not attach any meaning to the utility number assigned to a bundle( ); only thepreference ordering on the set of bundles matters.The same preferences can be representedby many different utility functions.For example, ifis any real number andis apositivereal number, then the utility function( )represents the same preferences as the utilityfunctionˆ( )+( ). This is because[+( ) +(0 0)]⇐⇒[( ) (0 0)]In fact, if()is anincreasingfunction of real variable, then(( ))represents thesame preference ordering as the utility function( ).To make things even simpler, we are going to focus on a particular type of utility functioncalled aquasi-linearutility function:( ) =() +where 0is a constant. The utility of the bundle( )is the sum of two components,(), which represents the utility ofunits of the good, and, which represents the utilityof theunits of money spent on everything else. The function is calledquasi-linearbecausethe second component is a linear function of. The parameter 0represents the utilityof an additional unit of money (the

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