1- [30 points] There are 10 producers, who work in a team to produce a product. Assume that the production function is , where q is the output,  is the effort level and 4 is the fixed cost of production which is independent of the number of team members. The net payoff of each team member is, where is the member i’s income. The sum of  is equal to q, i.e.

QUESTION

1- [30 points] There are 10 producers, who work in a team to produce a product. Assume that the production function is , where q is the output,  is the effort level and 4 is the fixed cost of production which is independent of the number of team members. The net payoff of each team member is, where is the member i’s income. The sum of  is equal to q, i.e.

1. a) [4 points] If a producer i decides to produce individually, . Why should a producer join the team and don’t produce individually?

1. b) [8 points] What is the efficient effort level of each team member?

1. c) [8 points] Suppose that the team members agree the divide the output, q equally, i.e. What will be the Nash equilibrium effort level by each team member?

1. d) [10 points] Now suppose that every team member agrees to get , where F is a fixed amount. What will be the Nash equilibrium effort level by each team member in this case? Find the value of F.

2- [16 points] Consider a hypothetical economy in which each worker has to decide whether to acquire education and become a high-skilled worker or remain low-skilled. Let  and  denote the incomes earned by a high- and low-skilled worker respectively. These incomes are defined as  and as  , where H and L are constants (H > L) and  is the fraction of the population that decides to become high skilled. The gain from education and acquiring skill is thus . However, education and acquiring skill carries a positive cost of C (a constant). Assume that all individuals simultaneously choose whether to become skilled or not.

1. a) [4 points] Does there exist any complementarity in acquiring skill? Describe it.

1. b) [12 points] Suppose that H − L < C < 2(H − L). Draw on the same graph the gain from acquiring skill versus f and cost of acquiring skill versus Is the situation in which only a fraction of the population becomes high-skilled equilibrium? Give an algebraic expression for this fraction. In case it is an equilibrium, is it stable or unstable? Find the other equilibrium values of f and determine which one is stable.

3- [14 points] Consider a group of farmers who want to build an irrigation project.  N: The number of farmers; n = The number that participate in the project. The payoff to each farmer depends on how many others participate in the project. If a farmer participates in the project, her payoff is equal to B(n) – C(n) & if the farmer shirks her payoff is B(n), where B(n) = 5n and C(n) = n-36. Show that if n < 40 then each farmer will participate. Draw the graph and determine the equilibria of this game in the graph. Determine whether the equilibria are stable or not. Is this game a Prisoner’s dilemma, assurance or chicken game? Why?

4- [25 points] Think about the following social dilemma: there are two classmates, Ana and Beatriz, who have to work on a joint assignment for their biology class. Each has to decide independently whether to spend the afternoon on the project or shirk. The table below shows the payoffs in each case.

1. a) [5 points] Assume Ana and Beatriz are self-interested. Do players have a dominant strategy? What is it? What is the equilibrium outcome? Is it efficient?

1. b) [10 points] Now suppose Ana and Beatriz are altruistic and get utility not only from their own payoff but from the other player’s payoff. Draw the indifference curves for Ana (label the graph properly). What is the possible dominant strategy for Ana & Beatriz in this case? In that case, what will be the equilibrium?

1. c) [10 points] Now suppose that Ana and Beatriz have utility functions based on self-interest and inequality aversion as follows: where , where  stands for payoff and if i stands for Ana, then j refers Beatriz; if i stands for Beatriz, then j refers Ana. For what values of  and  will both players work?

2. ANSWER

3.  A producer should join the team instead of producing individually because being part of a team allows for the sharing of fixed costs and the possibility of achieving higher output levels. When a producer decides to produce individually, their net payoff is . However, by joining the team, they can benefit from the combined effort of all team members, resulting in a higher output level and potentially higher net payoffs.

   When producing individually, the fixed cost of production, denoted as 4, is incurred solely by the individual producer. This cost remains the same regardless of the level of output. However, when joining a team, the fixed cost is shared among all team members, reducing the burden on each individual. By sharing the fixed cost, the team members can achieve a higher level of output for the same total cost, increasing their net payoffs compared to producing individually.

   b) The efficient effort level of each team member can be determined by maximizing the net payoff function. The net payoff for each team member is given by . To find the efficient effort level, we need to maximize this net payoff function with respect to the effort level .

   Taking the derivative of the net payoff function with respect to and setting it equal to zero, we have:

   Simplifying the equation, we get:

   Solving for , we find the efficient effort level of each team member to be .

   c) If the team members agree to divide the output equally, i.e., , we can determine the Nash equilibrium effort level by considering the individual team member’s best response to the effort choices of others. In the Nash equilibrium, no team member has an incentive to deviate from their chosen effort level given the effort levels chosen by others.

   Since the team members agree to divide the output equally, each member’s income is given by . To find the Nash equilibrium effort level, we need to maximize this income function for each team member while considering the effort choices of others.

   Taking the derivative of the income function with respect to and setting it equal to zero, we have:

   Simplifying the equation, we get:

   Solving for , we find the Nash equilibrium effort level for each team member to be .

   d) If every team member agrees to receive a fixed amount , where F is a fixed amount, we can again determine the Nash equilibrium effort level by considering the individual team member’s best response to the effort choices of others. In this case, the net payoff for each team member is given by .

   To find the Nash equilibrium effort level, we need to maximize this net payoff function for each team member while considering the effort choices of others.

   Taking the derivative of the net payoff function with respect to and setting it equal to zero, we have:

   Simplifying the equation, we get:

   Solving for , we find the Nash equilibrium effort level for each team member to be .

   To find the value of F, we substitute the Nash equilibrium effort level back into the net payoff function:

   Simplifying the equation, we get:

   Therefore, the value of F in the Nash equilibrium is .

   2a) Yes, there is complementarity in acquiring skill. The gain from acquiring skill is not just the direct increase in income but also the positive effect it has on the gain itself. Acquiring skill increases the potential gain, and this increased gain further incentivizes individuals to acquire skill. In other words, the more people decide to become high skilled, the higher the gain from acquiring skill becomes, which encourages even more individuals to choose the high-skilled path.

   b) Given that H – L < C < 2(H – L), let’s analyze the situation graphically. On the x-axis, we have the fraction of the population that decides to become high skilled (f), and on the y-axis, we have the gain from acquiring skill (G) and the

   cost of acquiring skill (C).

   The gain from acquiring skill is given by G = Hf – L(1 – f), and the cost of acquiring skill is C. Since H > L, the gain from acquiring skill increases linearly with f, while the cost remains constant.

   We can plot the gain from acquiring skill versus f and the cost of acquiring skill versus f on the same graph. The gain curve will start at zero when f is zero and increase linearly, while the cost curve will be a horizontal line at the level of C.

   If only a fraction of the population becomes high skilled, there will be an equilibrium when the gain from acquiring skill equals the cost of acquiring skill. Algebraically, we can express this as Hf – L(1 – f) = C.

   To find the equilibrium fraction, we solve the equation for f:

   Hf – L + Lf = C
   (H + L)f – L = C
   f = (C + L) / (H + L)

   In this case, the equilibrium fraction is given by (C + L) / (H + L). To determine whether it is stable or unstable, we need to compare the slope of the gain curve to the slope of the cost curve at the equilibrium point.

   If the slope of the gain curve is greater than the slope of the cost curve at the equilibrium point, the equilibrium is stable. If the slope of the gain curve is smaller than the slope of the cost curve at the equilibrium point, the equilibrium is unstable.

   3) To analyze the irrigation project game, let’s consider the payoff to each farmer when deciding to participate (P) or shirk (S). If a farmer participates in the project, their payoff is given by P = B(n) – C(n), and if the farmer shirks, their payoff is given by S = B(n), where B(n) = 5n and C(n) = n – 36.

   When n < 40, each farmer will participate because the payoff from participating is greater than the payoff from shirking. Substituting the values into the payoff functions, we have P = 5n – (n – 36) = 4n + 36 and S = 5n.

   To illustrate this in a graph, we can plot the payoff (P or S) on the y-axis and the number of participating farmers (n) on the x-axis. When n < 40, the payoff from participation (P) is above the payoff from shirking (S) for all values of n. Therefore, each farmer will choose to participate.

   The equilibrium point(s) on the graph will be where the payoff curves intersect. In this case, there will be a single equilibrium point when n < 40, where all farmers participate. This equilibrium is stable because no farmer has an incentive to deviate from participating since the payoff from shirking is always lower.

   This game can be classified as a Prisoner’s dilemma because if all farmers acted in their self-interest, the individual rational choice would be to shirk since it provides a higher payoff. However, when all farmers shirk, they collectively lose the benefits of the irrigation project, resulting in a suboptimal outcome. By participating, they could achieve a higher collective payoff, but the temptation to shirk individually makes it challenging to coordinate and reach the cooperative outcome.

   4a) Assuming Ana and Beatriz are self-interested, we can examine the payoffs in the table. A dominant strategy is a strategy that yields the highest payoff regardless of the other player’s choice.

   Looking at the table, we can see that there is no dominant strategy for either player. If Ana chooses to work, Beatriz’s best response is to shirk, but if

   Ana chooses to shirk, Beatriz’s best response is to work. Similarly, if Beatriz chooses to work, Ana’s best response is to shirk, but if Beatriz chooses to shirk, Ana’s best response is to work.

   In this case, the equilibrium outcome will depend on the strategies chosen by Ana and Beatriz. Without additional information, we cannot determine the specific outcome or whether it is efficient.

   b) Now, let’s consider the case where Ana and Beatriz are altruistic and derive utility not only from their own payoff but also from the other player’s payoff. We can draw indifference curves for Ana to represent her preferences.

   On the graph, the x-axis represents Ana’s payoff, and the y-axis represents Beatriz’s payoff. We can plot several indifference curves, which represent the combinations of payoffs that provide Ana with the same level of utility.

   Given that Ana is altruistic and values Beatriz’s payoff, the indifference curves will be upward-sloping, indicating that Ana prefers higher payoffs for both herself and Beatriz. The specific shape of the indifference curves will depend on Ana’s preferences.

   In this case, the possible dominant strategy for both Ana and Beatriz is to work together on the project. By working together, they can achieve the highest combined payoff, maximizing their utility.

   The equilibrium outcome will be when both Ana and Beatriz choose to work. This outcome is efficient because it maximizes the total payoff or utility for both players.

   c) In this case, let’s consider the utility functions based on self-interest and inequality aversion. The utility function for Ana can be defined as U(A, B) = A – β(A – B)^2, and the utility function for Beatriz can be defined as U(B, A) = B – α(B – A)^2, where α and β are positive constants.

   To determine the values of α and β for which both players will choose to work, we need to find the equilibrium point(s) where both players maximize their utility.

   Ana’s utility-maximizing condition is to choose the level of effort (work or shirk) that maximizes her utility, given Beatriz’s choice. Taking the derivative of Ana’s utility function with respect to her payoff (A) and setting it equal to zero, we have:

   dU(A, B) / dA = 1 – 2β(A – B) = 0

   Simplifying the equation, we get:

   2β(A – B) = 1

   Similarly, Beatriz’s utility-maximizing condition is:

   dU(B, A) / dB = 1 – 2α(B – A) = 0

   Simplifying the equation, we get:

   2α(B – A) = 1

   To find the values of α and β for which both players will choose to work, we need to solve these two equations simultaneously.

   By substituting the expression for A – B from the first equation into the second equation, we can solve for the values of α and β:

   2α(1 / (2β)) = 1
   α / β = 1

   Therefore, the values of α and β for which both players will choose to work are α = β.