Similarly if there are 52 weeks in a year the weekly

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Similarly, if there are 52 weeks in a year, the weekly demand curve is: Weekly demand : P = 78 – 13/6 Q D . If P = 0, then Q D = 36 kg/week, and the slope of the demand curve expressed on a weekly basis is 52 times as steep as that of the annual demand curve. The moral of the story is that whenever someone presents you with a demand curve, you are entitled to ask, “How long is the period in relation to which this demand curve is defined?” The corollary to the moral is that when you are using demand and supply curves, you should also know the answer to the question.

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Example 6: Why the Slope of the Budget Line is ∆Y/∆X = –P X /P Y . There are 2 puzzles in this relationship for most students just beginning their study of economics. First, why isn’t there a negative sign in front of ∆Y/∆X ? The answer here is simple. The negative sign is built into the ∆ notation, since if the budget line is negatively sloped, then ∆Y and ∆X will have opposite signs: a decrease in Y is associated with an increase in X , and vice versa. The second question is the trickier one. Why is the variable on the vertical axis ( Y ) in the numerator for quantity changes, while its price is in the denominator when the slope is expressed in terms of relative prices? There are in fact a number of different but equiv- alent ways of seeing this relationship, and you should know and understand them all. But being clear on economic units is one of the easiest ways of seeing how it works. In the specific numerical example depicted in Figure M.4-2, the consumer has money income M = \$48/week, P X = \$2/kg of X , P Y = \$6/case of Y , X is in kg/week and Y is in cases/week. (As you work through the following equations, plug in these numbers to ground the equations.) We can therefore write the equation of the budget line as fol- lows, with the units directly underneath: M = P X X + P Y Y (M.4.1) \$/wk = ( k \$ g )( w kg k ) + ( ca \$ se )( c w as k es ) = \$/wk + \$/wk Since the “kgs” and “cases” cancel, the units for both terms on the right-hand side of the equation are both in \$/week, which makes sense, since P X X is the weekly expen- diture (in \$) on X and P Y Y is the weekly expenditure (in \$) on Y. Rearranging, we have: Y = M / P Y ( P X / P Y ) X (M.4.2) (cases/wk) = ( w \$ k )/( ca \$ se ) ( k \$ g )/( ca \$ se ) ( w kg k ) = (cases/wk) (cases/wk) Once we cancel the “\$s” and “kgs,” both terms on the right-hand side of the equa- tion are in cases/week , like the left side. In particular, the term – ( P X / P Y ) X indicates the reduction in cases/week that results from a 1-kg/week increase in consumption of X . The slope of the budget line, ∆ Y/ X , is measured in (cases/week)/(kg/week), or simply cases/kg , and as we saw earlier, it is negative. Checking the units of the other M4-4 MATH MODULE 4: USING ECONOMIC UNITS Y X (cases/ week) (kg/week) 8 0 24 Slope = – = 1 3 Y X 2 6 cases kg ( ( ) ) P x P y \$/kg \$/case = = – FIGURE M.4-2
MATH MODULE 4: USING ECONOMIC UNITS M4-5 expression for the slope, – ( P X / P Y ), we see that they are (\$/kg) ÷ (\$/case) = (\$/kg) (cases/\$) = cases/kg , once we have cancelled the \$s. In the numerical example of Figure M.4-2, if we reduce our purchases of Y by 1 case per week, we save \$6/week.

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