Question 1: u₁(x₁,x₂)=1 and u₂(x₁,x₂)=1/x₂. Hence MRS=-x₂. Thus, MRS=-x₂=-p₁/p₂ and hence, Thus, x₂=p₁/p₂, and we already have the first demand function. The second one can be found by inserting this demand function in the equation of the budget line p₁x₁+p₂x₂=m. In particular, p₁x₁+p₁=m. Thus, x₁=(m-p₁)/p₁= m/p₁-1. This is the second demand function.

Note that for p₁>m, x₁<0. This indicates that we have a boundary optimum in those cases. In fact, the correct demand functions for p₁>m are x₁=0 and x₂=m/p₂.

Question 2:

• Note that MRS=-1/3. The price ration is -p₁/p₂=-2/3. Thus, MRS=-p₁/p₂ never holds. Thus, the first part of the procedure to find an optimum did not work and we must have a boundary optimum). The two intercepts are given by (6,0) and (0,4). Their utilities are 6 and 12, respectively. Thus, (0,4) is optimal.
• Now MRS=-3. The price ration is -p₁/p₂=-2/3. Again, MRS=-p₁/p₂ never holds. and we have a boundary optimum). The two intercepts are again given by (6,0) and (0,4). Their utilities are 18 and 4, respectively. Thus, (6,0) is optimal.
• Now MRS=-2/3. The price ration is -p₁/p₂=-2/3. Thus, MRS=-p₁/p₂ holds for all consumption choices. As a consequence, all consumption choices on the budget line 2x₁+3x₂=12 are optimal. Geometrically, the budget line coincides with one indifference curve.

Question 3: The partial derivative are given by u₁(x₁,x₂)=2/x₁, and u₂(x₁,x₂)=1/x₂. Thus, MRS=-2x₂/x₁=-p₁/p₂. Consequently, p₁x₁=2p₂x₂ or (1/2)p₁x₁=p₂x₂. The equation of the budget line is p₁x₁+p₂x₂=m. Substituting (1/2)p₁x₁ for p₂x₂ in this equation yields (3/2)p₁x₁=m, and consequently, x₁=2m/(3p₁). Similarly, it follows that x₂=m/(3p₂).

Note that this is the same computation as for the Cobb-Douglas utility function u(x₁,x₂)=x₁²x₂ in your class notes.

Question 4: u₁(x₁,x₂)=1 and u₂(x₁,x₂)=1/x₂^1/2. Hence MRS=-x₂^1/2. Thus, MRS=-x₂^1/2=-p₁/p₂. If we take the square on both sides of the equation we therefore get x₂=p₁²/p₂². This is already the demand function for good 2. In order to get the demand function for the first good, we insert this in the equation of the budget line p₁x₁+p₂x₂=m. Thus, p₁x₁ + p₁²/p₂ = m. Consequently, p₁x₁ = m - p₁²/p₂. Dividing both sides by p₁ gives the demand function for good 1. Specifically, x₁ = (m/p₁) - p₁/p₂.

Note that for m/p₁< p₁/p₂ we get x₁<0 which means that we have a boundary optimum. You can check that in this cases the boundary optimum is given by x₁=0 and x₂=m/p₂.

Question 5: Note that the demand for good 2 decreases as income is increased (indicated in the graph on the left). Hence good 2 is an inferior good. Similarly, you can see in the graph on the right that a price increase of good 2 can result in an increase of demand for good 2.

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