Question 1:

• u(x₁,x₂)=x₁x₂² is a Cobb-Douglas utility function. The MRS is
  MRS=-x₂/(2x₁).

  p₁=p₂=1. Therefore the MRS=-1. Hence 2x₁=x₂.
  Income m equals the value of the endowment. Thus, 2p₁+2p₂=4. Consequently, the budget line equation is x₁+x₂=4. Inserting 2x₁=x₂ therefore yields
  x₁=4/3, x₂=8/3.

• The consumer started out with 2 units of good 1 and ends up with 4/3 units. Thus the consumer is a net seller.
• Now p₁=4 and p₂=1. Therefore the MRS=-4. Hence -x₂/(2x₁)=-4, which implies x₂=8x₁.
  m=2p₁+2p₂=10. Consequently, the budget line equation is 4x₁+x₂=10. Inserting x₂=8x₁ therefore yields
  x₁=5/6, x₂=20/3.
• One can check that u(4/3,8/3)<u(5/6,20/3). Hence, the consumer is better off after the price increase. Intuitively, this is the case because the consumer is a net seller of good 1 and hence the price change increases the relative value of the good he/she sells.

Question 2: Note that if the consumer does not trade then (4,2) must be optimal. Since (4,2) is not a boundary optimum, the MRS must be equal to the slope of the budget line. The MRS for u(x₁,x₂))=x₁x₂² is MRS=-x₂/(2x₁). Thus, MRS(4,2)=-1/4. Thus, the price ratio must be -1/4. For example the prices (1,4) work (as usual only the price ratio but not the absolute price level is determined).

Finally, note that if we vary the price this corresponds exactly to the MRS at (4,2) is not equal to the slope of the budget left. Hence the consumer will be strictly better off.

Question 3. With Cobb-Douglas utility functions any example works as long as the optimal consumption choice before the price change does not coincide with the endowment and as long as the endowment of both goods is strictly positive.

The "counterintuitive" part of the result is clearly that the larger price change makes the person better off. Consider for example the situation of buyer, who faces the large price increase. As you can immediately see when running the program, the former buyer turns into a seller after the large price increase. In other words, the person become better off because he starts to sell a good he/she owns which became very valuable because of the price increase.

Question 4: If she sold all her kumquats, she would get 4 quinces for each of them. Therefore she would have receive 20 quinces in addition to the 20 she aready has.

If she sold all her quinces, she must sell 4 quinces to get one kumquat. Therefore she would have receive 5 kumquats in addition to the 5 she aready has.

Let p_q be the price of quinces and p_k the price of kumquats. The budget line equation is

p_q q + p_k k=p_q 20 + p_k 5 (the righthand-side is the value of the endowment).

p_k = 4 p_q (kumquats costs four times as much as quinces). If p_q=1 then p_k=4. Therefore the budget line equation can be written as q + 4 k= 20 + 20=40.

If p_q=2 then p_k=8 Therefore the budget line equation can be written as 2q + 8 k= 80. The set of consumption bundles that she can afford is the same.

If she sells 10 quinces, she can consume 7.5 kumquats.

The equation of the budget line is q+k=17.5

Question 5: In order to get the graph use the click here for the Java program on optimal choice with endowments.

In order to find the answer algebraically, note that the MRS=-b/a. The slope of the budget line is -1/2 Therefore, a=2b.
The budget line equation is a+2b=500 (500 is the value of the endowment). Substituting 2b for a in the budget line equation yields 4b=500 and b=125. Therefore a=250.

Net-demands are 150 of A, and -75 of B.

If the price ration is -1 then MRS=-b/a=-1, i.e., a=b.
The budget line equation is now a+b=300 (300 is the value of the endowment at the new prices). Therefore a-b=150.

In point (f) she will choose 250 units of each good. The answer is different, because her income decreases as the price is decreased (note that she is a net seller of good of good B. On the other hand, if she exchanges the goods for money before the price change, then her income is not affected.

Question 6: Note that the "budget line" equation is given by x+ws=24w, i.e., prices are 1 and w, respectively and the endowment is (0,24). The utility function is u(s,x)=sx². The MRS=-x/(2s). The slope of the budget line is w. Therefore x=2ws. Substituting 2ws for x in the budget line equation yields 3ws=24w. Therefore s=8, ndependent of w.

You can check the answer and view the graph by using the java program on optimal choice with endowments. How does your answer change if the person has some other sources of income M, which would result in a budget line ws+x=24w+M?