Budget Constraints

 

Budget Constraints

0. A bundle is a collection of goods (e.g., 2 apples, 3 green beans). In an economy with n goods, a bundle has n elements, some of which may be zero.

1. Let y be income, Pb the price of the first good, bread, B the quantity of Bread, Pw the price of wine and W the quantity of wine.

2. The equation y = Pb B + Pw W defines all the (B,W) pairs that cost exactly y. The equation is called the budget constraint. Any (B,W) pair that lies below the budget constraint is affordable. All pairs (bundles) that lie on or below the budget constraint are the budget set. Any pair that lies above can't be purchased with y.

3. Put W on the vertical and B on the horizontal axis of a graph. The budget constraint can also be written: W = y/Pw - Pb/Pw B. The vertical intercept is y/Pw, the slope is -Pb/Pw, and the horizontal intercept is y/Pb.

 

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In the picture y/Pw = 100.  The slope of the budget line is -5, which equals - Pb/Pw.  The prices cannot be learned from the graph as the slope gives only their ratio and the intercept the ratio of income to price of wine.  Clearly doubling all prices and income will leave the budget constraint the same.  By choosing Pw = 1, however, one gets that Pb = 5 and Y = 100.

4. Show what happens if price of bread increases to twice its previous value. Does the vertical intercept change? the horizontal intercept, the slope? has the budget set become larger or smaller.

5.  Now show what happens if income increases from 100 to 120.  What happens to the vertical intercept, the horizontal intercept and the slope.  Has the budget set become smaller or larger?

6.  For completeness, suppose the price of wine doubles.  What are the intercepts now?

7.  A more advanced exercise.  Suppose that bread is put on sale so that once 10 loaves are bought at the original price all additional loves can be bought at half price.  What is the budget constraint now?

   To answer questions of this sort it is best to create some new variables:   Let EB be expensive bread, that bread purchased at the original price of 5.   Cheap bread, CB, is bought at a price of 2.5.  B = CB + EB.  Until 10 units of bread are purchased, the budget constraint is unaltered, it is just as is given in paragraph 3 above.  When B > 10 however the budget constaint will be flatter.   For B> 10, B = CB + 10.  The income remaining after 10 units of bread is purchased is Y - Pb 10.  It is available for purchasing CB or W, so Y - Pb 10 =  CB Pb / 2  + Pw W = (B-10) Pb / 2  + Pw W.  One solves this to get:  Y- 5 Pb = Y - 25= 75 =  B Pb / 2  + Pw W for B > 10.  Now the budget constraint is kinked, with the kink at B= 10.   For B <= 10,  Y =  B Pb   + Pw W while for B > 10,  Y - 25 =  B Pb / 2  + Pw W.   There is a question like this on your homework.

8. Food Stamps. Pay $A to buy $C worth of stamps that can only be used for food. Call this food stamp food FB and call food bought with cash OB. Thus total food B = OB +FB >= $C/Pb = FB, food bought with food stamps. Now after the purchase of stamps there is y - $A left to spend so y - $A= Pb OB + Pw W, where OB >= 0. Subsitute B - FB for OB and $C/Pb for FB, to get the new budget constraint, y - $A +$C = Pb B + Pw W , where B >= $C/Pb

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