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Home , Neutral good, Normal good

1. Consider the following preferences over three : �~� �~� �~� � ≽ �

a. Are these preferences complete? Yes, we have relationship defined between x and y, y and z, and x and z. b. Are these preferences transitive? Yes, if �~� then � ≽ �. If �~� then � ≽ �. If �~� then �~� and � ≽ �. Thus the preferences are transitive. c. Are these preferences reflexive? No, we would need � ≽ � � ≽ �

2. Write a series of relations over x, y, and z that are reflexive and complete, but not transitive. � ≽ � � ≽ � � ≽ � � ≽ � � ≽ � � ≻ � We know this is not transitive if � ≽ � and � ≽ � then � ≽ �. But � ≻ �, which would contradict transitivity. 3. Illustrate graphically a set of indifference curves where x is a and y is a good that the person likes:

We know that this person finds x to be a neutral good because adding more x while keeping y constant (such as moving from bundle A to D, or from B to E), the person is indifferent between the new bundle with more x and the old bundle with less x.

We know this person likes y because adding more y while keeping x constant (such as moving from bundle A to B, or from D to E), the person is strictly prefers the new bundle with more y than the old bundle with less y.

4. Draw the contour map for a set of preferences when x and y are perfect substitutes. Are these well-behaved? Explain why or why not.

We know these are perfect substitutes because they are linear (the MRS is constant)

We know they are strictly monotonic because adding Y while keeping X constant (moving from bundle A to bundle B), leads to a strictly preferred bundle (� ≻ �).

However, they are not strictly convex because the bundle D which is the bundle containing the averages of bundles B and C is not strictly preferred to B and C.

Thus the preferences are not well behaved because they do not satisfy both montonicity and convexity.

5. Suppose the function over x, y, and z is: � �, �, � = ���

The price of x is 2, the price of y is 4, the price of z is 6. Income is 100.

a. Write the utility maximization problem for this consumer: ��� ��� �. �. �� + �� + �� = ��� We know this is the problem, as we are trying to maximize our utility subject to a budget constraint. b. Using the Lagrangian method, solve this problem for the optimal bundle. � �, �, � = ��� − �(2� + 4� + 6� − 100) The optimal bundle will satisfy the following first order conditions: 1. � = 0: � �� − 2� = 0 2. � = 0: �� � − 4� = 0 3. � = 0: ��� − 6� = 0 4. � = 0: 2� + 4� + 6� − 100 = 0

The best way to solve this is to use condition (1) and (2) to find y=f(x) and conditions (1) and (3) to find z=f(x). We want to do this so that we can substitute into condition (4) to solve for x.

Condition (1) becomes: 1 � �� = 2� 2 Condition (2) becomes: 1 �� � = 4� 4 Dividing (1) by (2) lets us cancel out z, so that what is left is a relationship between x and y.

. 5 � 2 = . 25 � 4 2 1 � = � = � 8 4

Dividing (1) by (3) lets us cancel out y, so that what is left is a relationship between z and x.

. 5 � 2 = . 25 � 6

2 1 � = � = � 12 6

Now we can use our last constraint to find x: 1 1 2� + 4 � + 6 � = 100 4 6 2� + � + � = 100 4� = 100 �∗ = 25 Then we can use our previous conditions to find �∗ and �∗: 1 1 �∗ = �∗ = 25 = 6.25 4 4 1 1 �∗ = �∗ = 25 = 4.17 6 6

The optimal bundle is �∗, �∗, �∗ = (25,6.25,4.17)

Note: you could find the for x, y, and z by replacing making prices unknown and income unknown.

6. Consider the following optimization problem: ��� 2� + 4� �. �. 4� + 2� = 8 a. Illustrate this problem graphically.

This is a case of perfect substitutes because U(x,y) = 2x+4y. All perfect substitutes have this linear form U(x,y) = ax+by.

Before I go further, I want to find the slope of the indifference curves (the MRS) so that I can plot them: � 2 ��� = − = − � 4 I also want to find the slope of the budget constraint: � 4 − = − � 2 Now, I can observe that the magnitude of the MRS is less than the magnitude of the slope of the budget constraint: � 1 4 � = < = � 2 2 � I’ll keep this in mind when I plot them.

First: the budget constraint. * The y-intercept is when we buy only y and zero x 4 0 + 2� = 8 8 � = = 4 2 * The x-intercept is when we buy only x and zero y 4� + 2(0) = 8 8 � = = 2 4 Now plot:

Now we need to add the contour map of the utility function over the top of the budget constraint. We know one thing: the curves are less steep than the budget constraint.

The utility of consuming only y is U(0,4) = 16. The utility of consuming only x is U(2,0)=4 The best bundle is to consume only y at bundle A.

Note: this example walks through all the steps. An answer with just the graph is perfectly acceptable. b. Illustrate the income offer curve for this problem. The income offer curve maps out all the best bundles in the (X,Y) coordinate plane as m changes.

Lets start with the best bundle and from our last problem:

I am going to decrease m and see what happens:

When income decreases, the budget constraint shifts to the left. The new best bundle will be be bundle B.

If we keep doing this, we will see that each bundle follows the y-axis. Connecting the dots, we get the I-O curve:

c. Illustrate the Engel curve for x and y. As income decreases, x doesn’t change. As income decreases, y decreases.

7. Suppose that good x is an and good y is a . Illustrate an income offer curve which would show this. If income goes up, we would expect x to go down. If income goes up, we would expect y to go up:

As income goes up, we move from B1 to B2 to B3 to B4. The best bundles go from A to B to C to D. The income offer curve is down ward sloping when x is inferior and y is normal.

8. Explain what a is. A Giffen good is a good where a person more of it when the price of it goes up. The for a Giffen good is upward sloping.

9. Illustrate the income, substitution, and total effects when x is a Giffen good and y is a normal good. Let the price of x decrease. The budget constraint under the original higher price is B1. The budget constraint under the new lower price is B2. If X is a Giffen good, then we know that the total effect will be a movement from A to C (x3-x1). The substitution effect is when we hold purchasing power constant and rotate budget curve B1 around bundle A to reflect the new price ratio. The new best bundle is bundle B and the substitution effect is x2-x1. The income effect occurs when purchasing power increases from this price decrease. The new bundle would be bundle C and the income effect would be x3-x2. The income effect here is sufficiently negative so that x3

10. Suppose we have a utility function � �, � = min {��, ��}. a. Set up a utility maximization problem for this consumer. ��� ��� ��, �� �. �. ��� + ��� = �

b. Find the demand for good x and good y. This is the case of perfect complemnets. We know what the utility function of perfect complements always takes the form U(x,y) = min{ax,by}. They are “L- shaped” utility curves with the joint where ax=by. We CANNOT use the Lagrange method here because this minimum function is not differentiable. Instead we are going to use the fact that the best bundle always lands on the joint, so �� = �� � � = � � We can substitute this into the budget constraint: � � � + � � = � � � � � � + � � = � � � �� + �� � = � � � �(�, �, �) = � �� + �� And we can use the fact that y = (a/b)x to solve for y: � � � � � �, �, � = � = � = � � � �� + �� �� + �� c. Let � = � = 1. Suppose that the initial price of x is � = 2, the price of y is � = 1 and income is � = 12. Find the income and substitution effects if price increases to � = 3. First lets find the initial demand for x: 1 � � , � , � = � 2,1,12 = 12 = 4 2 + 1 If price were to increase, then income would have to increase to compensate for this and make x=4 still affordable. We can find how much income has to increase by looking at: ∆� = ∆� ∗ � �, �, � = � − � � �, �, � = 3 − 2 4 = 4 Income would have to go up by 4. So the compensated income would be: � = � + ∆� = 12 + 4 = 16 The quantity demanded at the new price and compensated income is 1 � � , � , � = 16 = 4 3 + 1 So the demand did not change. This is not surprising because the optimal bundle was on the joint. If we keep the bundle affordable by compensating income, then all that happens is that the budget constraint rotates. Because of the shape of these indifference curves, this wouldn’t change the best bundle (see graph at end).

The quantity demanded at the new price and original income is 1 � � , � , � = 12 = 3 3 + 1 The substitution effect is: ∆� = � �, �, � − � �, �, � = 4 − 4 = 0 The income effect is: ∆� = � �, �, � − � �, �, � = 3 − 4 = −1

The best bundle under lower prices was bundle A. When price goes up, the budget constraint rotated to B2 and the new best bundle was bundle B. Keeping A affordable by compensating income after the price change (rotating around A to new price ratio and the budget constraint becomes B1’) did not lead to a change in quantity demanded of x, so the substitution effect was zero. When our purchasing power decreased due to the price increase, we decrease quantity demanded of x. This means that X is a normal good.