1. RATIONAL CONSUMER CHOICE

2. Drawing on Chapter 3
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3. Rational Choice Model given 1. Possibilities 2. Preferences
opportunity set of quantities, bought with income M at prices P
2. Preferences
preference ordering (indifference curves) or utility function
and 3. Rationality,
seeking the best affordable bundle of quantities
we 4. Predict choices
of quantities
We’ll go through these building blocks in order, first using preference orderings. Then, after an application and a look at non-rational choice, we’ll go through them with utility functions.

4. What Choices? Consumer (chapters 3-5) Other
Two goods or services (“goods” or “products”)
One specific good and one composite good (expenditure on all other goods)
Current and future expenditure
Other
Leisure and a composite good (income or expenditure on all goods) (chapter 14)
Children and a composite good
A composite good and a “bad”
In each case, we assume all other choice problems are solved separately, so all else is equal while considering changes in these choices.

5. Opportunity Set and Budget Constraint
1. Possibilities
Opportunity Set and Budget Constraint
The set of all combinations of goods that can be bought (that are affordable) with income M at prices P
The constraint is the upper limit of the set
Corners at M/PR, M/PG, in Robin’s case
Hand out Exercise about Robin.
The prices are outside of the (small) consumer’s control. M has already been determined.
Opportunity set = Affordable set
Derive a general budget constraint equation, first expenditure not exceeding income, R Pr + G Pg <= M, then converted to slope intercept form at equality (using all income): … G = (-Pr/Pg)R + M/Pg.
Pr/Pg measures the (constant) rate of opportunity cost of meals, in foregone games/meal.
If M1=100, Pg=10, and Pr1=20, draw B1 and calculate and interpret Pr1/Pg.
If M2=200, Pg=10, and Pr1=20, draw B2.
If M1=100, Pg=10, and Pr2=10, draw B3 and calculate and interpret Pr2/Pg.

6. Figure 3-2: The Budget Constraint, or Budget Line
Recall here M=$100/wk, Pf=$10/lb., Ps=$5/sq.yd.
I’m showing this
- first, to note that the way that the slope is illustrated is misleading. A slope is a ratio of rise to run, not an angle. A right triangle along the budget constraint should be used, rather than an arc that suggests an angle, measured in degrees. In what units is the slope measured? They remind us that |slope| is a relative price, an opportunity cost.
- second, in the process to illustrate affordable and unaffordable bundles (point to each -- choices like E are unaffordable. D, L & K are affordable).
Derive and draw all three budget constraints for Robin’s case on the handout.
BEFORE the next slide, do experiments of:
Auffhammer IC, with 2 students and candy on desks (Snickers, Smarties); plot points
McCrimmon & Toda (1969 REStud), with all students and same candy, but given afterward; block out a sample student’s potential region of IC

7. 2. Preferences Preference Orderings Definition Assumed Properties
Personal ranking of all possible combinations (bundles) of goods
E.g., A is preferred to B, or vice versa, or the person is indifferent between them
Assumed Properties
Complete
More is better
Transitive
Convex (more variety or balance is better)
Completeness is plausible for familiar choices.
More is better for goods along the relevant range of quantities, and where disposal is free.
Transitivity is logical, and plausible where the number of choices is not too large.
Convexity is a matter of degree.
What do the experiments illustrate about your preferences? Discuss experimental results, focusing on more is better, possibly convexity, and difficulty of completeness.
“The letters e.g. stand for the Latin phrase exempli gratia, which means for example.” (

8. Figure 3-8: Generating Equally Preferred Bundles
Between W and Z there must be some bundle equally preferred to A. Repeat for each combination of bundles in the Better and Worse areas. This defines an IC including A.

9. Indifference Curves Definition Implied Properties
A set of equally preferred bundles of goods
i.e., the person is indifferent between them
Implied Properties
Every bundle is on one
Negative slope
Cannot cross
Diminishing |slope|
Each property is implied by the corresponding property of preference orderings.
Draw three ICs on Auffhammer experiment graph, and an IC into the McCrimmon & Toda experiment graph. Which properties are illustrated/contradicted?
Add points as necessary to illustrate each property, if possible. A few curves are understood as representative of a larger, possibly continuous, pattern.
“The Latin abbreviation i.e., which stands for id est, means that is, that is to say, or in other words.” (

10. Marginal Rate of Substitution (MRS)
The maximum rate at which a person is willing to give up one good to get another
The |slope| of the indifference curve
when substituting the good on the horizontal axis for the good on the vertical axis
where the curve is smooth (differentiable).
Its smoothness can be another assumption.
Often written as negative, the slope of the indifference curve
E.g., in Robin’s case, how would we express this? [Initially leave out the (RG), then return to it after the part about xy & yx.] MRS(RG) = |∆G/∆R|.
Economists often distinguish the MRS of good x for good y (MRSxy) from the MRS of y for x (MRSyx). If the quantity of x is measured on the horizontal axis of the graph, Frank’s MRS = MRSxy = the slope of an indifference curve. MRSyx = the reciprocal of the slope. This will be particularly relevant when considering the economic reasons for choosing a particular combination of goods, or changing a choice.

11. Diminishing Marginal Rate of Substitution
For goods with units that are not necessarily divisible, such as these, consider what an IC shows about the rate of willingness to give up the good on the vertical axis to get each additional whole unit of the good on the vertical axis. That rate matches the |slope| across an arc, between two points.
Students: calculate approximate MRS(RG) along I3 over intervals of R: 2-3, 4-5, 6-7. Write out units of measurement. For R:4-5, calculate corresponding MRSGR .
[During individual and small group work, sketch curve on board for large-group illustration of R:4-5 |slope| and calculation.]

12. Figure 3-12: The Marginal Rate of Substitution
With good that are close to perfectly divisible (can be consumed in small fractional increments), substitution can be over very small changes along an IC. The MRS is then effectively the |slope| of a line tangent to the curve at a point (bundle).
Frank’s example here blurs this distinction – between the slope at a point and across an arc – a bit inappropriately. Since he marks only one point for finely divisible goods, it should be the slope at a point – of a tangent line.
Even with finely divisible goods, if you lack an algebraic formula (function) for the indifference curve, or knowledge of calculus, the slope across an arc may be easier to approximate accurately than the slope of a tangent line.

13. Representing Different Preferences
Indifference curves with
Differences in slope (MRS) at a given bundle mean different relative preferences for the two goods
Different curvatures (rates at which MRS diminishes) mean different degrees of preference for variety
The link in “relative” is to an article from the Journal of School Health that says girls (at least in rural NE Ohio) tend to prefer fruits and vegetables, and boys prefer meat, fish and poultry.
Students: on one meat(x axis)-vegetable(y axis) graph, draw a separate IC for each gender, both passing through (3,6) [I sketch the axes and mark the point], and label one Igirl and one Iboy. Think about what the preference means for the MRS – the max. rate of willingness to give up veggies for meat.
Another way of thinking about the preference for variety is the willingness to substitute. Best illustrated with the extreme cases (next slide).

14. Extreme Preferences
If water is water, variety doesn’t matter. When two things must go together, anything but balance (not necessarily 1-to-1) is irrelevant.
These violate the strictest version of the assumption that preferences be convex. This version would require that any linear combination of two bundles be preferred to either bundle, without allowing the person to be indifferent between the bundles and their linear combination.

15. 3. Rationality Rational Choice
Choose more of a good if the benefit exceeds the cost and it is affordable
Benefit: the most you are willing to give up (MRS)
Cost: the relative price of the good
The best affordable bundle
Usually the only contact of indifference curve and budget constraint
If both choices are positive and if indifference curves are smooth, I and B are tangent (have the same slope): MRS = relative price
The relative price is the cost when the change is along the budget constraint. Changing from below the constraint to on it has no cost in foregone consumption of either good considered. Hence, a rational person will always choose a combination on the constraint.
Draw a picture of a best affordable bundle with positive quantities of both goods that is not a tangency of indifference curve and budget constraint. If someone asks why “usually,” draw an IC and BC that coincide entirely (if IC straight) or partly (if IC multi-kinked).

16. Figure 3-15: The Best Affordable Bundle
4. Predict
Figure 3-15: The Best Affordable Bundle
G is unaffordable, and from D, more is better. There are three ways to think through moving to F from A or E:
Compare MRS to Ps/Pf and change the choice accordingly. (MRS me
Go to the highest IC (I2) touching the BC.
Compare A or E with D and D with F.
All predict the same result. However, these last two ways require more global information about preferences; marginal analysis requires only local information about preferences, which is more plausible. Preferably focus on this.
Students: do the prediction part of the Robin exercise [either now – if much time left today – or for next class – if little time left today].

17. Figure 3-16: A Corner Solution
Sometimes both choices are not positive. Here preferences have all five properties we assumed, but because the MRS is always less than the relative price of shelter, the person decides not to buy any shelter. (This would make more sense for a less essential good.) The indifference curve is not tangent to the budget constraint at the best affordable bundle.

18. Choice among Perfect Substitutes
Suppose
Water budget M=$4.80/week
Prices P1A= $0.80/btl. and PD=$.60/btl.
What is the best affordable bundle?
What if P2A= $0.48/btl. and PD=$.60/btl.?
Bottles are 500 ml. Here preferences are not strictly convex – variety in sources of water does not matter to this person.
Students: draw B1. [from A=6 to D=8] What is the MRS(AD)? [1 btl.D/btl.A] What is the initial relative price PA/PD? [1.25 same units] How do they compare? What will the person choose? [MRS<PA/PD, the person is everywhere willing to give up less Dasani for an additional Aquafina (1) than s/he has to (1.25), so never sub A for D]
Students: repeat with new PA: [B2 A-intercept 10. PA/PD=.75, the reverse comparison and choice]
With such goods, fairly small price changes may lead to large changes in Q choices.

19. Choice among Perfect Complements
Suppose
Food budget M=$3/meal
Prices PB= $0.75/brat and PN=$.75/bun
What is the best affordable bundle?
What if the
Food budget doubled?
Prices were $1.00/brat and $.50/bun?
Here changes in income and relative prices do not change the proportions of the two goods consumed, although they may change the total amounts. The best affordable bundle is not a corner solution, but it is not a tangency (MRS=Pn/Pb) either.

20. Figure 3-5: Budget Constraint with a Composite Good
More Complex Cases
Figure 3-5: Budget Constraint with a Composite Good
We can represent choices between many goods by focusing on one good (here X) relative to all others. Lump those others into a “composite” good (Y) with its quantity measuring the expenditure on those goods, and the price set to 1. This price sounds arbitrary and rigid, but for changes in choices, the relative price (opportunity cost) is what matters, and that is simply and conveniently measured here by Px (=Px/Py, though in different units).
Students: draw B for hamburger and a composite good, with a budget of $10/week and Ph=$2/lb. [Note its |slope| and relative price of H.]
Although preferences should be defined over goods, not money, the units of Y are often equivalently labeled $/period.

21. Volume Discount Budget Constraint Linear in Segments
The link in “Volume” is to an exotic hardwood online retailer that offers volume discounts. The prices may be averages (“per unit”), but let’s treat them as marginal (“incremental”) to imitate Frank’s example, and carry the .
The link in “Discount” is to a page that shows prices and costs (max Y – actual Y in a BC graph) of several discounting systems, using the terms in parentheses above.

22. Figure 3-18: Food Stamp Program vs. Cash Grant Program
This model can shed light on a long-standing question of public policy (with which I had direct experience as a welfare caseworker). Should financial assistance to poor people be given in cash, or in kind? Many taxpayers are more comfortable supporting a program which requires that assistance be used to purchase more of an essential good such as food (or housing, or medical care). Others consider that an impingement on the individual liberty of recipients of the assistance. Economists have something to add to the debate.
Here a person receives $100 worth of food stamps per month, in addition to $400 of income. The effect is like that of receiving $100 of cash income (EDA), except that purchases of the composite good cannot be increased (without illegally selling the food stamps for cash), creating the kinked budget constraint (FDA). For a person with these strong preferences for food (X), the effect of in kind assistance is the same as that of cash assistance.

23. Figure 3-19: Where Food Stamps and Cash Grants Yield Different Outcomes
However, for someone with stronger preferences for other goods (flatter indifference curves), the best affordable bundle with cash assistance (L) could lie beyond the budget constraint – and on a higher indifference curve than the best bundle affordable – with in-kind assistance. This means that a rational person with these preferences would be better off with cash assistance (on I3) than with in-kind assistance (on I2) – though better off with either than with no assistance (I1). When we reasonably do not trust some (for example, drug-addicted) recipients of financial assistance to make good choices, we penalize the well-being of others who would rationally choose to spend more on other needs.

24. 2. Preferences (again) Utility Functions
Assign a number (indexing utility or satisfaction) to each bundle of goods
Assume positive but diminishing marginal utilities (slopes relative to each good’s quantity)
Measure utility cardinally, but are better interpreted ordinally
Imply indifference curves, each showing bundles providing a given level of utility
The units of measurement of utility and its starting value are arbitrary and need not affect its use in predicting consumer choices.
Although a utility function may say that one combination provides a particular amount (or %) more utility than another (cardinal measurement), the important implication of that is simply that the first combination is preferred to the second (ordinal ranking). By how much it is preferred is irrelevant here.

25. Analogy to Topographic Maps
Each brown curve shows the combinations of longitude and latitude for which altitude is constant. From lower left to the top of Blue Mound, they look like “typical” indifference curves.
Show styrofoam 3-D model: from side, then from above to see “contour lines”. Note that the layers shown are only samples; there would be many in between, making the hill smoother, like the quasi-3-D one on the next slide.
References:
U.S. Geological Service.

26. Figure A3-3: A Three-Dimensional Utility Surface
For the visually-oriented among us …

27. Figure A3-4: Indifference Curves as Projections

28. Figure A3-1: Indifference Curves for the Utility Function U=FS
Cobb-Douglas function in 3-D
Here we assume a particular mathematical formula linking F and S to U, and choose a few levels of U to plot. E.g., for U=2, solve for S and create table of
F S=2/F 2 1
4 1/2
Does this simple formula (function) obey all of our previous assumptions about preferences? Yes.
Does it also involve diminishing marginal utility of each good? No, e.g., for S=1,
F U MU=∆U/∆F
0 0 Before calculating MUs, graph U(F,S=1)
1 >1
2 >1
3 >1
4 >1
Economists often consider diminishing MU a useful additional assumption, which requires more complicated functions.
This is a special case of a more complicated function, named Cobb-Douglas after its inventors, which can permit diminishing MU. Write on board U =aFbSc . Both constants a,b,c & variables F,S >0. In this special case, a=b=c=1. The linked web page shows graphs of several cases (this one is the fourth), distinguished by their values of a,b,c (or what are called alpha,a,b in the page – adapt to changes in notation between sources), which are constant within a case as x1,x2 vary.
Constant-U slices project down ICs in each case.
Constant-x2 slices project back U(x1) [cet.par.] curves that,
In the 4th case, show constant MU(x1)
When the exponent on x1<1, e.g., the 1st case, show diminishing MU(x1)

29. Figure A3-2: Utility Along an Indifference Curve Remains Constant
Utility lost in moving from K to L must equal utility gained (in absolute value) for a person to be indifferent between them. Since ∆S<0, ∆F>0 and both MUs>0, the negative sign is needed, as most authors show.

30. Rational Choice with Utility Functions
3. Rationality (again)
Rational Choice with Utility Functions
Suppose utility from two divisible goods: U(X,Y)
Using more of X and less of Y along an indifference curve, the gain in utility must equal the loss in utility
MUX ∆X = -MUY ∆Y
So -MUX /MUY = ∆Y/∆X = MRS
At the best affordable bundle, this equals the slope of the budget constraint:
-MUX /MUY = -PX /PY
So MUX /PX = MUY /PY
Marginal benefit equals marginal opportunity cost per unit of X or dollar spent on X.
The S,F case has been helpfully somewhat concrete, but more abstract notation is needed to signal the generality of these insights. Sketch Y-X axes with IC and ∆Y,∆X triangle.
Where – on both sides of =, drop it.
Recall that MRS measures benefit of more X in rate of Y willing to give up per unit of X, and Px/Py measures its rate of opportunity cost.
It is equivalent to compare benefits and costs in terms of: MU gained and foregone per $ spent (on X). [Show units of measurement of both.]
If the marginal utility per dollar spent is greater for good X at the current choice, how can the choice be changed to increase utility?

31. Simple numerical example
4. Predict (again)
Simple numerical example
Additively separable utility
Table rather than formula
Compares benefits and costs in MU/P along budget constraint
Shows that the scale of the utility function does not affect the rational choice
Activities: 03 Example - additively separable utility & consumer choice.xlsx
I use this in the principles of economics-microeconomics course.
Additive separability is less realistic: does how much you like popcorn depend on how much you can drink with it?

32. Figure A3-6: The Optimal Bundle when U=XY, Px=4, Py=2, and M=40
See in 3-D
How can we determine this? One way is by drawing the budget constraint and experimenting with different levels of utility, plotting as indifference curves the combinations of x and y that produce each level of utility, until an apparent tangency is found. The tangency can be confirmed by calculation.
Visualization can help: follow “3-D” link to same page of Cobb-Douglas function graphs. Under the heading (>80% of the way down) “Utility maximization: utility function with decreasing returns to scale” and the next heading there are pictures like this one, as well ones showing constrained maximization visually, with the budget constraint slicing the utility hill. Looking parallel to the x2 axis, the resulting cross section has a maximum height at some value of x1 … and x2,U.
A method more certain to work with anything but an example set up for convenience is to use calculus (OPTIONAL! but recommended). What is the slope of a function, such as utility, of one variable when it attains its maximum value? (Illustrate with a graph of y=f(x) as inverted U.) How is the slope related to the function, in calculus?
We can convert this two-variable function and constraint equation into a one-variable function (workable w/ Calc I methods) by substituting the budget constraint (in slope-intercept form) into the utility function. This gives us the values of the utility function along the budget constraint as a function of x.
Substituting y=-2x+20 into U=xy, we get U=x(-2x+20)=-2x2+20x.
What is the slope of U with respect to x? It is the derivative of U( ) with respect to x.
(Following the link at “Optimal” and clicking on the link at “derivative,” we can see its definition. The rules [zoom browser in] have been discovered from that definition.)
Following the “scalar multiple,” “sum” and “power” rules of derivatives on the page linked to “Optimal,” we get dU/dx = -4x Set this equal to zero (where utility is maximized, given our assumptions on preferences-ICs-U) and solve for x: -4x + 20 = = 4x. X = 5.
Then U = -2* *5 = = 50. Also y = -2* = 10.
Finally, for U = 50, along the indifference curve, Y = 50/x = 50x-1. Applying the “power” rule, the slope of the indifference curve is -50x-2.
At x=5, the slope is -50*5-2 =-50/25 = -2, which matches the slope of the budget constraint: tangency!
For those without knowledge of calculus, I will sometimes give you formulas for MU() functions that are not obvious. Then you can derive the tangency condition and budget constraint and combine them by substitution to derive the rational choice. That’s equivalent to what we just did, as shown on pp

33. More Examples of Utility Functions
Relatively simple, with assumed properties
Activities: 03 Examples - utility functions.xlsx
We’ll skim through these examples, from simpler to more complex. Explore afterward from the portal, return with questions, and use them if interested.
As we develop functions that better approximate likely properties of consumer preferences, we must add complexity that makes it harder to work with the functions. That’s a tradeoff, so even economists with extensive knowledge of mathematics usually stick to relatively simple functions in working on price theory.
1-good: like ECO 212 beginnings, but based on a formula
Quasi-linear: with this, price changes have no income effects; more likely to lead to a corner solution. What will happen if b=1? [Write function] Diminishing MU of each good? What relationship between goods x and y?
Additively separable: like ECO 212 example, but based on a formula
U=aRG: can scale U up and down like ECO 212 example, but based on a formula
Cobb-Douglass: often simplified like this to constant returns to scale, which we’ll study with production. If 0<b<1, diminishing MU as well as other assumed properties.
Constant elasticity of substitution (CES): allows wide range of substitutability between goods,
linear (perfect substitutes) when r=1 [Point out how],
becomes like perfect complements as r-∞, and in between
becomes like C-D as r0.
[Optional: Cobb-Douglas restricts the elasticity of substitution (between x and y, %∆(y/x)/%∆MRSxy.) to 1; CES allows it to be any constant.]
These are not the only ways to use relatively simple algebraic formulas to approximate likely properties of consumer preferences. We may encounter other functions later when dealing with utility as a function of one variable, wealth, under conditions of risk and uncertainty. For example, the natural logarithm function may be used instead of exponents to generate different degrees of convexity (substitutability) and diminishing marginal utility.

34. Review Predict Rational choice of most Preferred Possibility.
Adapts to different Possibilities and Preferences, not to non-rational choice.
Ordinal ranking is less restrictive.
Utility functions are mathematically convenient, and not misleading if interpreted ordinally.
I.e., predict choice of the best affordable bundle, whether consumption possibilities are ordinally ranked or their utility cardinally measured.