2. In this problem, you will compare the level of a public good chosen under majority voting to the socially-optimal level under three different sets of circumstances. Suppose first that individual i's demand curve for z is given by αi/z, where αi is a positive parameter. Instead of being linear, this demand curve is a hyperbola. Suppose further that z costs $1 per unit to produce (c = 1) and that this cost is shared equally among consumers. Therefore, cost per person is 1/n per unit of z. Then consider the three sets of circumstances listed below. Each situation has a different number of consumers in the economy and different collections of α values for the consumers. The number of consumers is denoted n and the vector of α values is denoted A = (α1,α2,...,αn-1,αn). Case 1: n = 7, A = (4, 2, 12, 4, 5, 13, 8). Case 2: n = 5, A = (10, 6, 11, 14, 8). Case 3: n = 9, A = (6, 9, 10, 4.5, 12, 7, 13.5, 8, 11). Using this information, do the following: a)For each case, compute the preferred z level for each voter. Identify the median voter, and indicate the z level chosen in the voting equilibrium. b)For each case, compute the DΣ curve, and find the socially-optimal level of z. c)For each case, compare the z levels in the majority voting equilibrium and the social optimum. Give an intuitive explanation of the difference (if any) between the two z values.
2. In this problem, you will compare the level of a public good chosen under majority voting to the socially-optimal level under three different sets of circumstances. Suppose first that individual i's demand curve for z is given by αi/z, where αi is a positive parameter. Instead of being linear, this demand curve is a hyperbola. Suppose further that z costs $1 per unit to produce (c = 1) and that this cost is shared equally among consumers. Therefore, cost per person is 1/n per unit of z. Then consider the three sets of circumstances listed below. Each situation has a different number of consumers in the economy and different collections of α values for the consumers. The number of consumers is denoted n and the vector of α values is denoted A = (α1,α2,...,αn-1,αn). Case 1: n = 7, A = (4, 2, 12, 4, 5, 13, 8). Case 2: n = 5, A = (10, 6, 11, 14, 8). Case 3: n = 9, A = (6, 9, 10, 4.5, 12, 7, 13.5, 8, 11). Using this information, do the following: a)For each case, compute the preferred z level for each voter. Identify the median voter, and indicate the z level chosen in the voting equilibrium. b)For each case, compute the DΣ curve, and find the socially-optimal level of z. c)For each case, compare the z levels in the majority voting equilibrium and the social optimum. Give an intuitive explanation of the difference (if any) between the two z values.
Chapter12: The Partial Equilibrium Competitive Model
Section: Chapter Questions
Problem 12.11P
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2. In this problem, you will compare the level of a public good chosen under majority voting to the socially-optimal level under three different sets of circumstances. Suppose first that individual i's demand curve for z is given by αi/z, where αi is a positive parameter. Instead of being linear, this demand curve is a hyperbola. Suppose further that z costs $1 per unit to produce (c = 1) and that this cost is shared equally among consumers. Therefore, cost per person is 1/n per unit of z. Then consider the three sets of circumstances listed below. Each situation has a different number of consumers in the economy and different collections of α values for the consumers. The number of consumers is denoted n and the vector of α values is denoted A = (α1,α2,...,αn-1,αn). Case 1: n = 7, A = (4, 2, 12, 4, 5, 13, 8). Case 2: n = 5, A = (10, 6, 11, 14, 8). Case 3: n = 9, A = (6, 9, 10, 4.5, 12, 7, 13.5, 8, 11). Using this information, do the following:
a)For each case, compute the preferred z level for each voter. Identify the median voter, and indicate the z level chosen in the voting equilibrium.
b)For each case, compute the DΣ curve, and find the socially-optimal level of z.
c)For each case, compare the z levels in the majority voting equilibrium and the social optimum. Give an intuitive explanation of the difference (if any) between the two z values.
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