Problem Set 1

Explain briefly the following terms. Positive and negative externalities (give some examples for both), Coase theorem, and transaction costs. How do you perceive the term of transaction costs in politics?

Bentham Welfare Function maximizes the sum of each individual’s utility while Nash Welfare Function the product. According to the Rawls Welfare Function, the individual that determines the overall welfare is the one with the lowest welfare value.

a) Assume that there are two individuals, 1 and 2, with the following utility functions. U1 = 10pX1 and U2 = X2. Using the Bentham’s as well as Nash’s social welfare functions, set up and solve the respective maximization problems. In both cases, for the budget constraint consider that there are only 100 units of the good X = X1 + X2 available.

BENTHAM

Bentham Welfare Function maximizes the sum of each individual’s utility while Nash Welfare Function the product. According to the Rawls Welfare Function, the individual that determines the overall welfare is the one with the lowest welfare value.

a) Assume that there are two individuals, 1 and 2, with the following utility functions. U1 = 10pX1 and U2 = X2. Using the Bentham’s as well as Nash’s social welfare functions, set up and solve the respective maximization problems. In both cases, for the budget constraint consider that there are only 100 units of the good X = X1 + X2 available.

NASH

Consider the Nash Welfare Function. There are two individuals with the following utilities. Uα(z1,z2) = z1αz2α and Uβ(z1,z2) = zβ1*zβ2/zα1,

with 0 < a < 1, 0 < β < 1, and z1 + z2 = ¯y > 0.

Solve the maximization problem and find the optimal z1∗(α,β, ¯y) and z2∗(α,β, ¯y)

Bentham Welfare Function maximizes the sum of each individual’s utility while Nash Welfare Function the product. According to the Rawls Welfare Function, the individual that determines the overall welfare is the one with the lowest welfare value.

b) Follow the same procedure using the Rawls’ social welfare function with Ui = pXi, i = A, B, given that XA + XB = 300

Use the following social welfare function. Ω = Pκj=1 wjUj with Uj = δjln(x j) + εjln(yj).

Here, there are two individuals (j = 1, 2) and two goods (x, y) with availability (x¯, ¯y) > (0,0), respectively.

a) Set up the maximization problem (budget constraint/s and Lagrangian function). Specify the endogenous and exogenous variables

Use the following social welfare function. Ω = Pκj=1 wjUj with Uj = δjln(x j) + εjln(yj).

Here, there are two individuals (j = 1, 2) and two goods (x, y) with availability (x¯, ¯y) > (0,0), respectively.

b) Solve the maximization problem from a).

Use the following social welfare function. Ω = Pκj=1 wjUj with Uj = δjln(x j) + εjln(yj).

Here, there are two individuals (j = 1, 2) and two goods (x, y) with availability (x¯, ¯y) > (0,0), respectively.

c) Given that w1 = w2 = w¯ and δ1 = δ2 = ε1 = ε2 = 1, what is the optimal solution?

Use the following social welfare function. Ω = Pκj=1 wjUj with Uj = δjln(x j) + εjln(yj).

Here, there are two individuals (j = 1, 2) and two goods (x, y) with availability (x¯, ¯y) > (0,0), respectively.

d) Given that w1 = w2 = 11w¯ ¯, δ1 = δ2 = ε1 = ε2 = 5, and x¯ = ¯y = 2021, compare the optimal solutions for two different random and strictly positive values of w¯ ¯. What do you notice?