A legislature consists of two political parties, L and S, who play an infinitely repeated game. In each round, a policy proposal, x ? [0, 1] is put forward. Both parties simultaneously vote whether to approve the proposal. If both parties approve, then the proposal passes. If at least one party rejects, then the proposal fails. If the proposal passes, the parties payoffs are uL = x and uS = 1?x. Thus, party L benefits more from "larger" policies, and party S benefits more from "smaller" policies. If the proposal fails, then both parties obtain a payoff of 1/4, associated with the current "default" policy. Thus, the stage game between the parties is represented by the following matrix

Party S

Approve Reject

Approve x, 1 ? x 14 , 14 Reject 1,1 1,1

Consider one round of the stage game (no repetition). The following parts are worth 2.5 points each.

(a) Suppose x ? [0, 1]. Is it a Nash equilibrium for both parties to reject? Explain in a sentence or two.

(b) Suppose x ? (1/4,3/4). Is it a Nash equilibrium for both parties to approve? Explain in a sentence or two.

(c) Suppose x ? (0,1/4). Is it a Nash equilibrium for both parties to approve? Explain in a sentence or two.

(d) Suppose x ? (3/4,1). Is it a Nash equilibrium for both parties to approve? Explain in a sentence or two.

Consider an infinitely repeated version of the previous game, with discount factor ? ? (0, 1). Suppose that at the beginning of period k, the policy to be considered in that period, denoted xk, is drawn (independently) from a uniform distribution on [0,1] (i.e. Pr(xk ? y) = y for y ? [0, 1]). Policy xk is observed by both parties in period k, but the parties do not know which policies will be considered in the future. Thus, the parties expect that on average, the expected value of future policies is 1/2.

We will analyze whether the following strategies are a Subgame Perfect Nash equilibrium of the infinitely repeated game.

Approve in period 1 for all x1 ? [0, 1].

Approve in period k for all xk ? [0, 1], as long as in all past periods the outcome

was (Approve, Approve).

Reject in period k if in any past period a policy was Rejected.

We will focus on the decision of party L (party S is symmetric). The following are worth 2.5 points each.

(e) Suppose that at some point in the past, a policy was rejected. In this case, the strategy specifies that both parties reject in every subsequent period. Is this credible? (Hint: see part (a))

(f) Suppose that in all past periods the outcome was (Approve, Approve). Given that today's policy proposal is x, what is party L's lifetime payoff of approving? (Hint: L gets x today, and in all future periods, it expects E[x] = 1/2).

(g) Suppose that in all past periods the outcome was (Approve, Approve). Given that today's policy proposal is x, what is party L's lifetime payoff of rejecting? (Hint: L gets 1/4 today and in all future periods).

(h) For which values of ? is the proposed strategy a Subgame Perfect Nash equilibrium? (Hint: it is most-tempting for party L to reject in period k if xk = 0)

4. Qualitative and quantitative variables Aa A3 E Do women wait longer than men to receive their orders at coffee shops? This question was studied by an economics major at Middlebury College. [Sourcez Myers, C.I(. (Sept. 2007). Ladies First: A Field Study of Discrimination in Coffee Shops. Mlddlebury College Economics DisCussion Paper No. 0711.] A researcher compiled the following information from a sample of 25 customers: 0 Wait Time {in seconds) 0 Gender 0 Age 0 Order [regular or fancy) - Height {in inches} The variable is a dichotomous variable. For each of the variables listed in the following table, select whether it is a classication variable or a quantitative variable. Variable Classification Quantitative Walt Time Ge nd er Height Order Age 00000 00000 Question 7 Not yet answered Pointsout of 1:00 7 Flagquestion Consider a concave mirror with focal length f, and an object at a distance 3fin front of it. Which of the following statements is true? Select one: O a. The image is smaller and inverted. b. The image is larger and upright. c. The image is the same size and inverted. d. The image is larger and inverted.QUESTION 2 Positive economics uses opinions to validate an economic argument while normative economics used facts to substantiate a claim. O True O False QUESTION 3 Scarcity is the condition in which human wants and needs exceed the availability of time, goods, and resources. True O False QUESTION 4 Another term meaning the same as scarce resources is unlimited resources. O True FalseQuestion 7 (2.5 points) For the variable listed, indicate whether it is quantitative or qualitative. If it is quantitative, indicate whether it is continuous or discrete. If it is qualitative, indicate whether it is ordinal, nominal or an identifier. Coat Size (S. M. L, XL): Oqualitative, nominal quantitative, discrete O qualitative, ordinal Oquantitative, continuousResearch conducted among employees of Tupperware, an age-old brand of plastic storage containers. found that their female employees of all ages thought that Tupperware was a hip and contemporary brand. An independent research group, however, conducted a large scale, national study and found that Tupperware's image was anything BUT hip and contemporary. What can we say for certain is true? 1} Tupperware didn't ask their employees the right question. 2} The independent research group was hired by the competitors and had an axe to grind. 3] The internal Tupperware research ndings lacked external yalidity. 4} The internal Tupperware research was not analyzed correctly. 5} 1 and 4 only