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( ), where≥0is the amountof the good and≥0is 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