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201Spring2011Class3_to_Class4

201Spring2011Class3_to_Class4 - Class 3 Marginal Rate of...

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Class 3: Marginal Rate of Substitution The Budget Constraint De°nition. The budget constraint states that expenses ° income : Remark. In this course we examine only consumers who have no saving motive. The consumer thus spends all income so that expenses = income holds at the optimum. Formally : Budget constraint is p 1 x 1 + p 2 x 2 ° m; where p 1 and p 2 are the prices of goods 1 and 2 ; respectively, x 1 and x 2 are the quantities consumed of goods 1 and 2 ; respectively, and m is income. Remark . Good 2 may stand for the composite good "everything else except good 1 " De°nition . If the price of one good is set to p 2 = 1 it is called the numeraire good °all other prices are in terms of consumption of the numeraire good. Example : If good 2 is the numeraire good and p 1 = 3 then to consumer 1 unit of good 1 the consumer has to forgo 3 units of the numeraire good. De°nition . Budget set is all the combinations of x 1 and x 2 that satisfy the budget con- straint. 1
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De°nition . Budget line is the set of combinations of x 1 and x 2 that satisfy the budget constraint exactly: p 1 x 1 + p 2 x 2 = m: Rearrange budget line to get x 2 = m p 2 ± p 1 p 2 x 2 : Remark . The slope of the budget line is ± p 1 p 2 : This measures the opportunity cost of consuming good 1: To increase consumption of good 1 by one unit the consumer has to decrease the consumption of good 2 by p 1 p 2 units. FIGURE 3.1 HERE If income increases the budget line shifts outwards. FIGURE 3.2 HERE If price of good 1 increases the budget line becomes steeper. FIGURE 3.3 HERE 2
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Preferences De°nition . Each consumer±s preferences rank each possible consumption combination ( x 1 ; x 2 ) of goods 1 and 2 relative to any other possible consumption combinations ( y 1 ; y 2 ) of goods 1 and 2. Remark . Notation ( x 1 ; x 2 ) ² ( y 1 ; y 2 ) means that for the consumer the combination ( x 1 ; x 2 ) is strictly preferred to the combination ( y 1 ; y 2 ) i.e. consumer would choose ( x 1 ; x 2 ) over ( y 1 ; y 2 ) if they cost the same amount.
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