To show the process of utility maximization on a graph, we will begin by illustrat- ing how to draw an individual’sbudget constraint. This constraint shows which combinations of goods are affordable. It is from among these combinations that a person can choose the bundle that provides the most utility.
The Budget Constraint
Figure 2.6 shows the combinations of two goods (which we will call simplyXand Y) that a person with a fixed amount of money to spend can afford. If all available income is spent on goodX, the number of units that can be purchased is recorded as Xmaxin the figure. If all available income is spent onY,Ymaxis the amount that can be bought. The line joiningXmaxtoYmaxrepresents the various mixed bundles of goodsXandYthat can be purchased using all the available funds. Combinations of
F I G U R E 2 . 6
Ind i v i du a l ’s Bu dg e t C ons tr ai n t fo r T w o Go od s Quantity of Y
per week
Ymax
Not affordable Income
Affordable
Quantity of X per week
0 Xmax
Those combinations ofXandYthat the individual can afford are shown in the shaded triangle. If, as we usually assume, the individual prefers more than less of every good, the outer boundary of this triangle is the relevant constraint where all of the available funds are spent on eitherXorY. The slope of this straight boundary is given byPX/PY.
Budget constraint The limit that income places on the
combinations of goods that an individual can buy.
goods in the shaded area below the budget line are also affordable, but these leave some portion of funds unspent, so these points will not be chosen.
The downward slope of the budget line shows that any person can afford to buy moreXonly ifYpurchases are cut back. The precise slope of this relationship depends on the prices of the two goods. IfYis expensive andXis cheap, the line will be relatively flat because choosing to consume one lessYwill permit the purchasing of many units ofX(an individual who decides not to purchase a new designer suit can instead choose to purchase many pairs of socks). Alternately, ifYis relatively cheap per unit and X is expensive, the budget line will be steep. Reducing Y consumption does not permit very much more of goodX to be bought. All of these relationships can be made more precise by using a bit of algebra.
Budget-Constraint Algebra
Suppose that a person hasIdollars to spend on either goodXor goodY. Suppose also thatPXrepresents the price of goodXandPYthe price of goodY. The total amount spent onXis given by the price ofXtimes the amount purchased (PXÆX).
Similarly,PYÆYrepresents total spending on goodY. Because the available income must be spent on eitherXorY, we have
Amount spent onXþAmount spent onY¼I or
PX ·XþPY ·Y ¼I (2.3)
Equation 2.3 is an algebraic statement of the budget line shown in Figure 2.6. To study the features of this constraint, we can solve this equation forYso that the budget line has the standard form for a linear equationðY ¼aþbXÞ. This solution gives
Y¼ PX
PY Xþ I
PY (2.4)
Although Equations 2.3 and 2.4 say exactly the same thing, the relationship between Equation 2.4 and its graph is a bit easier to describe. First, notice that the Y-intercept of the budget constraint is given byI/PY. This shows that ifX¼0, the maximum amount ofYthat can be bought is determined by the income this person has and by the price ofY. For example, ifI¼$100, and each unit ofYcosts $5, the maximum amount that can be bought is 20ð¼I=PY¼$100=$5Þ.
Now consider the slope of the budget contraint in Equation 2.4, which is PX/PY. This slope shows the opportunity cost (in terms of goodY) of buying one more unit of goodX. The slope is negative because this opportunity cost is negative—because this person’s choices are constrained by his or her available budget, buying moreXmeans that lessYcan be bought. The precise value of this opportunity cost depends on the prices of the goods. IfPX¼$4 andPY¼$1, the slope of the budget constraint is4ð¼ PX=PY ¼ $4=$1Þ—every additional unit ofXbought requires thatYpurchases be reduced by 4 units. With different prices, this opportunity cost would be different. For example, ifPX¼$3 andPY¼$4, the slope of the budget constraint is$3=$4¼ 0:75. That is, with these prices, the opportunity cost of one more unit of goodXis now0.75 units of goodY.
A Numerical Example
Suppose that a person has $30 to spend on hamburgers (X) and soft drinks (Y) and suppose also that PX¼$3,PY¼$1.50. This person’s budget constraint would then be:
PXXþPYY¼3Xþ1:5Y ¼I¼30 (2.5) Solving this equation for Y yields:
1:5Y¼303X or Y¼202X (2.6)
Notice that this equation again shows that this person can buy 20 soft drinks with his or her $30 income because each drink costs $1.50. The equation also shows that the opportunity cost of buying one more hamburger is two soft drinks.
Utility Maximization
A person can afford all bundles of X and Y that satisfy his or her budget constraint. From among these, he or she will choose the one that offers the greatest utility. The budget constraint can be used together with the individual’s indifference curve map to show this utility maximization process. Figure 2.7 illustrates the procedure. This person would be irrational to choose a point such asA; he or she can get to a higher utility level (that
is, higher than U1) just by spending some of the unspent portion of his or her income. Similarly, by reallocating expenditures he or she can do better than pointB. This is a case in which the MRS and the price ratio differ, and this person can move to a higher indifference curve (say,U2) by choosing to consume lessYand moreX. PointDis out of the question because income is not large enough to permit the purchase of that combination of goods. It is clear that the position of maximum utility will be at point Cwhere the combination X*, Y* is chosen. This is the only point on indif- ference curveU2that can be bought withIdollars, and no higher utility level can be bought.Cis the
KEEPinMIND
Memorizing Formulas Leads to Mistakes
When encountering algebra in economics for the first time, it is common for students to think that they have to memorize formulas. That can lead to disaster. For example, if you were to try to memorize that the slope of the budget contraint isPX/PY, there is a significant likelihood that you could confuse which good is which. You will be much better off to remember to write the budget constraint in the form of Equation 2.5, then solve for the quantity of one of the goods. As long as you remember to put the good you have solved for on the vertical (Y) axis, you will avoid much trouble.
M i c r o Q u i z 2 . 3
Suppose a person has $100 to spend on Frisbees and beach balls.
1. Graph this person’s budget constraint if Frisbees cost $20 and beach balls cost $10.
2. How would your graph change if this person decided to spend $200 (rather than $100) on these two items?
3. How would your graph change if Frisbee prices rose to $25 but total spending returned to $100?
single point of tangency between the budget con- straint and the indifference curve. Therefore all funds are spent and
Slope of budget constraint
= Slope of indifference curve or (neglecting the fact that both slopes are negative)
(2.7)
PX=PY ¼MRS (2.8)
The intuitive example we started with is proved as a general result. For a utility maximum, the MRS should equal the ratio of the prices of the goods.
The diagram shows that if this condition is not fulfilled, this person could be made better off by
F I G U R E 2 . 7
G r a ph i c De m o n s t r a t i o n o f U t i l i t y Ma x i m iz a t i o n Hamburgers
per week
Y*
B
A C
D
U3 Income
U2 U1
Soft drinks per week
0 X*
PointCrepresents the highest utility that can be reached by this individual, given the budget constraint. The combinationX*,Y* is therefore the rational way for this person to use the available purchasing power. Only for this combination of goods will two condi- tions hold: All available funds will be spent, and the individual’s psychic rate of trade-off (marginal rate of substitution) will be equal to the rate at which the goods can be traded in the market (PX/PY).
M i c r o Q u i z 2 . 4
Simple utility maximization requires MRS¼PX=PY:
1. Why does the price ratioPX/PYshow the rate at which any person can tradeYforXin ‘‘the market’’? Illustrate this principle for the case of music CDs (which cost $10 each) and movie DVDs (which cost $17 each).
2. If an individual’s current stock of CDs and DVDs yields him or her an MRS of 2-to-1 (that is, he or she is willing to trade two CDs for one DVD), how should consumption pat- terns be changed to increase utility?
reallocating expenditures.5You may wish to try several other combinations ofX andYthat this person can afford to show that all of them provide a lower utility level than does combinationC. That is whyCis a point of tangency—it is the only affordable combination that allows this person to reachU2. For a point of non- tangency (sayB), a person can always get more utility because the budget constraint passes through the indifference curve (seeU1 in the figure). In Application 2.4:
Ticket Scalping, we examine a case in which people do not have such complete freedom in how they spend their incomes.