Assignment
1. Consider two, goods ice cream and beer. At low levels of consumption both are goods (more is better) but after a threshold level each of them becomes a bad (more is worse). Draw an indifference curve for the case described and clearly identify the better than set.
2. Consider two goods x1 and x2, with M= $100, p1=1, and p2=2. Place x2 on the vertical axis and x1 on the horizontal axis. Label the budget lines as noted in parentheses.
a. Draw the budget line as noted above (1)
b. Draw a new budget line with M= $100, and p2=2
c. Draw a new budget line with prices as in part a., M = $200, and a coupon allowing consumption of 20 units of good 1. The coupon can be purchased for $5
3. Consider the utility functions U(x1,x2)=x1^ a + x2^b, V(x1,x2)=alnx1 + blnx2
a. Do U and V represent the same preferences? Prove your point mathematically.
4. Walt consumes strawberries and cream but only in the fixed ratio of three boxes of strawberries to two cartons of cream. At any other ratio, the excess goods are totally useless to him. That is, his utility function is given by U(S, C)=min (3S,2C). The cost of a box of strawberries is $10 and the cost of a carton of cream is $10. Walt's income is $200. What is Walt's demand for strawberries and cream?
5. a. Generate the cost function for the production function y=x1^1/4 x2^1/2 when W1=1 and w2=10. You can use the Lagrangean approach or make a solid argument about the optimality conditions. If you do the latter use a diagram to bolster your argument.
b. Using the first order conditions for profit maximization derive the supply function for all prices greater than zero.
c. What are firm profits if p=1? Does the firm choose to operate?
6. (i) Consider the production function y= f(K,L) = K^a L^b, where K and L represent capital and labor respectively. Show either analytically (using symbols only) r numerically that increasing returns to scale is associated with the condition a+b>1 and decreasing returns to scale with a+b<1.
(ii) What are the implications for cost and optimal output if the production function has increasing returns to scale. Use a diagram of your choice to demonstrate the implications of increasing returns to scale.
7. Consider the utility fuctionU(x1, x2) = x1^a x^b and the budget constraint m = p1x1 + p2x2. You can assume that a and b are between 0 and 1 and that a+b=1
a. Explain in plain English and with a graph why the utility maximizing choice occurs where the slope of the budget line and the slope of the indifference curve are equal.
b. Derive the optimality conditions associated with part a. for the utility function using the Lagrangean approach.
c. Solve the ordinary demands x1* and X2* using the budget constraint to pin down a specific level of demand. (Remember you should solve so that each good is expressed in terms of prices, income, and preference parameters, not the other x). Are your demands downward sloping?
d. Rearrange the expression you derived for x1* so that the preference parameters a, and b, are isolated on one side of the equality. Interpret the meaning of the expression defined by the preference parameters in terms of the other variables (x, p, and M).
8. Suppose that the optimality conditions in a consumer choice problem are not satisfied and the MRS is greater that the ratio of the good prices. Assume also that the consumer is consuming some of each good and has "well behaved" convex preferences. Would the consumer want to adjust their consumption by increasing x1 or x2? Why? Draw a diagram and show the direction adjustment.
9. Why is using the Lagrangian approach to solve the utility maximization problem inappropriate when goods are perfect substitutes? Use math, a graph and in plain English explain your reasoning.
10. Consider the case of concave preferences in which the "worse than" set is convex and the "better than" set is not. Draw an example of an indifference curve that is smooth that meets the requirements for concave preferences. Explain why the optimality conditions are relating the marginal rate of substitution to the price ratio are not helpful for determining optimal choice. (Hint: draw more than one indifference curve and a budget line).
11. A thirsty fox is walking down a path and sees a bundle of grabs hanging from a branch of grape tree. The fox tries repeatedly to reach the grapes but is unable to do so. As he walks away he thinks, "those grapes were probably sour anyway." Explain how his reasoning is inconsistent with the rational choice diagram.
Format your assignment according to the following formatting requirements:
1. The answer should be typed, double spaced, using Times New Roman font (size 12), with one-inch margins on all sides.
2. The response also include a cover page containing the title of the assignment, the student's name, the course title, and the date. The cover page is not included in the required page length.
3. Also Include a reference page. The Citations and references should follow APA format. The reference page is not included in the required page length.