ECON 304 MIDTERM EXAM ANSWERS

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Barbra Jenkins
6 years ago
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1 ECON 30 MIDTERM EXAM ANSWERS () The short questions: (a) Transitivity says that if y and y z, then z. Note the three bundles in diagram 0.. y because they are on the same indifference curve. y z because they are on the same indifference curve (albeit a different one from the first curve). So for transitivity z, but as you can tell from the diagram z lies above the indifference curve through, so z (in other words it is not true that z). (b) In the eample above,, y and z must each be assigned a different number, for eample, y so pick u() = 3 and u(y) = so that the utility numbers coincide with the preferences. Since y z we must then assign a number no higher than to z, say u(z) =. But then z which requires that we assign a number higher than 3 to z. We can t satisfy both. (c) The construction of the utility function that we used in our proof involved finding a point on the 5 o line that is on the same indifference curve as the point we are interested in. In Figure 0., suppose we want to find the utility level that is associate with the point. Notice that the coordinates of are labelled in the diagram 0 and. This point is y y z 0 Figure 0.. Indifference Curves can t cross

2 y Figure 0.. Construction of Indifference Curves below the 5 degree line, so it lies on an indifference curve that has slope. By high school algebra, the point on the 5 degree line that lies on a line through with slope is the point whose single coordinate z satisfies z z = Solving this for z gives the coordinates of this point to be 0, and this point is labelled in the diagram. That is the number we would like to assign to be the utility of. If the point is like on the other side of the 5 degree line, then the coordinates of the point we want are given by z = 3 z 0 or z = 3 0. The way to find the utility function for all is then to compute both these values for z, then observe that the one you want is always the lowest of the two. That is, u( 0, ) = min[ 0, 3 0 ] (d) Figure 0.3 shows the lottery you enjoy if you keep Door A. The probability you win is, the probability that the prize is placed behind door A. If you switch doors, then you lose if the prize is placed behind door A, but win otherwise.

3 Just switch the win and lose signs in the figure. The probability you win if you switch doors is also. Now suppose you pick door B even though you know that Door A,B or C B, W in C, W in B, Lose C, Lose Figure 0.3. You keep door A the probability that the prize is behind door B is only. Now Figure 0. describes the lottery you will enjoy if you switch doors when you are given the chance. Doors C, W in A, Lose C, Lose C, W in Figure 0.. You choose door initially, the switch doors () (a)maimize subject to y + y p + y W 0 y 0 The Lagrangian is y + y + λ (p + y W ) λ λ 3 y 3

4 The first order conditions are y ( + y) + λ p λ = 0 ( + y) + λ λ 3 = 0 λ 0; p + y W 0 λ 0; 0 λ 3 0; y 0 where the last three conditions hold with complementary slackness. Net check where the solutions will be. The marginal rate of substitution is y which is decreasing as increases and y falls. So, the indifference curves are conve and the solution will only be at the corners if the indifference curves are steeper than the budget line at ( W, 0) and flatter than the budget line p at (0, W ). The marginal rate of substitution at ( W, 0) is 0, so p indifference curves are perfectly flat, similarly at (0, W ) they are vertical. So the solution is always interior. By complementary slackness, this means that λ = λ 3 = 0. Since both goods have strictly positive marginal utility, the solution must be on the budget line. This reduces the first order conditions to y ( + y) + λ p = 0 ( + y) + λ = 0 p + y W = 0 λ 0 Solving the first three equations gives = W p+, y = W p p p+ p (which indicates that you have to use the positive root of p to get the right solution) and λ = (+ p). (b)the marginal rate of substitution is +y which is falling as increases and y falls, so the indifference curves are conve. The mrs at ( W, 0)is p. If this is larger than p, then the indifference p W curve is steeper than the budget line at the corner and ( W, 0) p is the solution. This occurs when p > p or W <. At (0, W ) W the indifference curve is vertical, so there are never solutions at (0, W ). Hence ( W, 0) is the solution when W <. Otherwise p ( W +, W ) is the solution. p

5 y Figure 0.5. (c)the solution to the problem is obvious here, = 5. trick is to find the multipliers. The Lagrangian is + λ ( + y W ) λ λ 3 y The and λ = 0 by complementary slackness. The first two first order conditions are + λ = 0 λ λ 3 = 0 from which λ =, and λ 3 =. (3) The picture is given in Figure 0.5: Since preferences are Cobb Douglas, one critical value for α is that value α such that the utility from consuming at the edge (0, 80) is eactly equal to the utility of consuming the Cobb Douglas demand on the lower segment of the budget line, i.e., α satisfies (α50) α (( α)00) ( α) = (0) α (80) ( α) (α 5 )α = ( 5 ( α) )( α) The other critical value for α is α 0 such that the Cobb Douglas demand for tennis along the upper budget line is eactly equal to 0, i.e. α00 = 0 or α =. If α is less than the consumer will consume fewer 5 5 than 0 hours of tennis and won t bother to join the club. If α is between and 5 α, the consumer will buy eactly 0 hours, 5

6 but won t join the club. If α > α the consumer will join the club and play for more than 0 hours. 6

Recitation #2 (August 31st, 2018)

Recitation #2 (August 31st, 2018) Recitation #2 (August 1st, 2018) 1. [Checking properties of the Cobb-Douglas utility function.] Consider the utility function u(x) = n i=1 xα i i, where x denotes a vector of n different goods x R n +,