EconS 305 - Constrained Consumer Choice · Constrained Consumer Choice Now that we have –gured out where the tangency point is, we just need to use one more piece of information

EconS 305 - Constrained Consumer Choice

Eric Dunaway

Washington State University

[email protected]

September 21, 2015

Eric Dunaway (WSU) EconS 305 - Lecture 12 September 21, 2015 1 / 49

Introduction

We have analyzed what consumers prefer, in the form of utility.

We have also analyzed what consumers can actually buy, in the formof budget constraints.

It�s time to put them together and derive demand curves.


Constrained Consumer Choice

Given that consumers are constrained to choosing bundles that liewithin their opportunity set, the want to choose the bundle thatmaximizes their utility.



x

z

Opportunity Set

Budget Line



x

z



x

z



x

z



Intuitively, the consumer is going to choose the bundle on theirbudget line that is going to give them the highest utility.

This corresponds to the indi¤erence curve that touches the budget lineat only one point (a tangency point).

We can �nd this point mathematically.



A great feature of the tangency point is that the rate of change(slope) of the indi¤erence curve at that point is exactly the same asthe slope of the budget line.



x

z



x

z



We actually calculated the rate of change of the indi¤erence curvelast week. We called it the marginal rate of substitution.

MRS = �MUxMUz

where MUx is the marginal utility with respect to x and MUz is themarginal utility with respect to z . (Note that I changed the y into a zfrom last week. No change in its meaning).

Also, we calculated the slope of the budget line last week. We calledit the marginal rate of transformation.

MRT = �pxpz

where px is the price of good x and pz is the price of good z .



Thus, the point where the indi¤erence curve is tangent to the budgetline will be where the marginal rate of substitution equals themarginal rate of transformation

MRS = MRT

�MUxMUz

= �pxpz

MUxMUz

=pxpz



MUxMUz

=pxpz

This ratio is actually a representation of one of the most basic rules ofeconomics. To see this, let�s rearrange a few terms.

MUxpx

=MUzpz

Remember that the marginal utility of x is the amount of utility theconsumer gets for consuming one more unit of x . The price of x , pxis the price to consume one more unit of x .

Same thing for good z .

The consumer will want to consume until the relative gain to cost isthe same across all goods. If it weren�t the same, then they could getto a higher utility by switching the goods around.



Now that we have �gured out where the tangency point is, we justneed to use one more piece of information to solve for our demand.

We know that the tangency point happens on the budget line. Thistells us exactly what utility level we reach. Thus, our system of twoequations and two unknowns to �nd our solution are

MUxMUz

=pxpz

pxx + pzz = Y

Let�s look at an example.


Example

Consider the following utility function for two goods, x and z ,

U = x0.75z0.25

The price for good x is px = 3, the price for good z is pz = 4 and theconsumer has an income of Y = 64. Derive the optimal quantity of xand z .


Example

U = x0.75z0.25

Starting o¤, let�s calculate our marginal utilities using the power rule:

MUx = 0.75x�0.25z0.25

MUy = 0.25x0.75z�0.75

Next, the equation for our budget line is

pxx + pzz = Y

3x + 4z = 64


Example

Let�s �gure out our marginal rate of substitution

MRS = �MUxMUz

= �0.75x�0.25z0.25

0.25x0.75z�0.75= �3z

x

and our marginal rate of transformation

MRT = �pxpz= �3

4

Together, we can set up our tangency condition,

MRS = MRT

�3zx

= �34

3zx

=34


Example

3zx=34

Let�s rearrange some terms to make this a bit nicer.

x = 4z

Now, we just have this equation and the budget line to �nd a solution

3x + 4z = 64


Example

I can substitute x = 4z into the budget line to get

3(4z) + 4z = 64

16z = 64

z� = 4

Then, I put this value back into x = 4z to get my solution for x ,

x� = 4z� = 4(4) = 16


Example

x

z24

2421.3

16

16

4


Example

U = x0.75z0.25

3x + 4z = 64

x� = 16 z� = 4

Does this answer make sense?

Yes. From the utility function, we can see that the consumer gets a lotmore utility from consuming good x than from good z . Also, good x isrelatively cheaper than good z .We should expect in this case that x� should be higher than z�.Note: If x were more expensive than z , this might not be the case.


Corner Solution

A few notes:

When utility functions are well behaved, our solution will always liesomewhere along the budget line.

By this, I mean not at the end points.

If the curvature of the indi¤erence curve is small (a relatively straightcurve), we can get situations where the only intersection point is atan end point.

When this happens, our tangency condition does not hold any more.


Corner Solution

x

z


Corner Solution

x

z


Corner Solution

If we let the budget line extend to the negative axis (either z isnegative or x is negative), we would be able to �nd a tangency point.

But we don�t consume negative quantities of goods.

Therefore, if you�re trying to solve a problem and you get that one ofthe goods is negative, set it to zero and solve for the other good.


Corner Solution

Lastly, let�s talk about an extreme example: perfect substitutes.

When we have perfect substitutes, the indi¤erence curves are straightlines.

Remember that their marginal rate of substitution is constant nomatter where on the line the bundle falls.

Since it�s constant, it�s almost impossible for a tangency point to befound.

This means that when we�re dealing with perfect substitutes, we almostalways have a corner solutionThe only exception is when the slope of the indi¤erence curve is thesame as the slope of the budget line; then any bundle can be a solution!

When dealing with perfect substitutes, people only buy the good thatgives the highest utility.


Deriving the Demand Curve

And that�s equilibrium.

But what about deriving the demand curve?

This is actually pretty easy from here. All we need to do is startchanging prices, and the curve will develop.

Let�s look back at our �gures.



x

z



x

z



x

z



x

z

x

px



x

z

x

px



x

z

x

px



x

px



x

px



Mathematically, we can derive the demand curve by not pickingvalues for prices or income.

This should make sense, since usually these are factors that determinethe demand for an item.

Let�s go back to our example.


Example

Recall the utility function from our earlier example

U = x0.75z0.25

Now, we don�t specify any prices or income and we leave the budgetline in a general form

pxx + pzz = Y


Example

Our marginal rate of substitution hasn�t changed, it is still

MRS = �MUxMUz

= �0.75x�0.25z0.25

0.25x0.75z�0.75= �3z

x

We can also just express our marginal rate of transformation ingeneral terms

MRT = �pxpz


Example

Now, we �nd our tangency point

MRS = MRT

�3zx

= �pxpz

3zx

=pxpz

Let�s rearrange this for easier use

pxx = 3pzz


Example

pxx = 3pzz

Now, we use our tangency condition, along with the budget line

pxx + pzz = Y

and we can solve it for x and z

Remember that the while we haven�t given them numbers, we assumethat we know what the prices and income are. We just need to solvefor x and z as a function of those values.


Example

pxx = 3pzz

pxx + pzz = Y

Let�s subtract both sides of the �rst equation by pxx

�pxx + 3pzz = 0

pxx + pzz = Y

and add the equations together (getting rid of pxx)

�pxx + 3pzz + pxx + pzz = 0+ Y4pzz = Y

z� =Y4pz

which is the demand curve for zEric Dunaway (WSU) EconS 305 - Lecture 12 September 21, 2015 42 / 49

Example

z� =Y4pz

Like before, we can plug this value back into our tangency point toget the demand for x

pxx = 3pzz

pxx = 3pz

�Y4pz

�pxx =

3Y4

x� =3Y4px

which is the demand curve for x .


Example

x� =3Y4px

z� =Y4pz

These demand curves probably look strange.

They�re non-linear. It�s actually really hard to pick well behaved utilityfunctions that give linear demand curves.

The important thing is that when their own price increases, thequantity demanded falls (Law of Demand).

We can see this because px is in the denominator. As it gets bigger, x�

gets smaller. The same thing happens with pz and z�.


Example

x

px


Example

x� =3Y4px

z� =Y4pz

When we plug in our original values, px = 3, pz = 4 and Y = 64, weget the same solution from the �rst example

x� =3(64)4(3)

= 16 z� =644(4)

= 4


Summary

Combining utility functions with budget constraints allows us to �gureout solve the constrained consumer choice problem, which in turnallows us to derive demand curves.


Preview for Wednesday

We are going to analyze what happens to the constrained consumerchoice problem when prices and incomes vary.


Assignment 3-1 (1 of 1)

1. Consider the following market with two goods, x and z . Theconsumer�s utility function is

U = x0.2z0.8

a. Derive the demand curves for x and z . (Remember to include px , pzand Y in there as unknowns. Look at the last example if you needhelp.)

b. Let px = 2, pz = 4 and I = 50. Find the equilibrium quantitiesdemanded of x and z .


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EconS 305 - Constrained Consumer Choice · Constrained Consumer Choice Now that we have –gured out where the tangency point is, we just need to use one more piece of information