Assignment:
1. In the voluntary-contribution model considered in the lectures, the public good was national defense, and z was measured by the total number of soldiers. This total was in turn equal to the sum of the soldiers hired by both consumers, so that z = z1 + z2. We saw that the consumers acting alone hired too few soldiers, so that z was less than z*. Consider now the issue of voluntary contributions to a different type of public good: flood prevention. Imagine that two farmers cultivate land in a narrow valley next to a river. Each farmer builds a dike (i.e., an embankment) along his part of the river in order to prevent flooding during the rainy season.
The height of farmer 1's part of the dike (measured in feet) is z1, while the height of farmer 2's part of the dike is z2. If farmer 1 wants to add extra height to his part of the dike, the cost is c for each extra foot of height added. The same cost applies to farmer 2. Since water comes over the dike at the lowest point, the flood prevention consumed by the farmers depends on the height of the lowest portion of the dike. Letting z denote flood prevention, it follows that z = min{z1, z2}. In other words, the prevention provided by the dike is equal to height of the lowest part. Obviously, this situation is quite different from case of the national defense.
Given these assumptions, farmer 1's demand curve for z1, which gives his valuation of an extra foot of height for his part of the dike, is given by
D1 = 10 - z1 if z1 ≤ z2
= 0 if z1 > z2
Note that an extra foot of height is worth something to farmer 1 as long as the extra foot improves flood protection (as long as z1 ≤ z2). Once z1 becomes larger than z2, further increases in z1 have no value. Similarly, farmer 2's demand curve for z2 is given by
D2 = 16 - z2 if z2 ≤ z1
= 0 if z2 > z1
Farmer 2 has the larger demand because his land extends farther back from the river, making his losses larger in the event of flooding.
a) The social planner recognizes that z1 and z2 should be set equal. The reason is that money spent to raise one of the dike heights above the other is wasted. Let z denote the common value of z1 and z2. Since z1 and z2 are equal, the first line of each of the above formulas is relevant, and the demand curves for the two farmers become D1 = 10 - z and D2 = 16 - z. Suppose in addition that c = 6. Using this information, compute the DΣ curve, and then compute the socially-optimal height of the dike. Note that since the dike must be built along the land of both farmers, the cost of an extra f equals 2c, not c.
b) Now consider what happens when the farmers make their own decisions. To start, suppose that farmer 1 knows that farmer 2 has a high demand. As a result, he expects farmer 2 to build a dike at least as high as his own. Therefore, farmer 1 uses the first line of his demand formula (D1 = 10 - z1) to decide on how tall a dike to build.
Remembering that farmer 1's individual cost is c = 6, what level of z1 does he choose?
c) Farmer 2 observes the choice made by farmer 1, and decides on the level of z2 accordingly. To understand his decision, graph his demand curve for z2 using both lines of the second formula above (graph the zero part as well as the formula in the first line, connecting them with a vertical line). Note that to draw the graph, you must plug in the z1 value that you computed in part (b). Look carefully at your graph, and think what it tells you about the best decision for farmer 2. What level of z2 does he choose?
d) Once farmer 2 has chosen, reconsider the decision of farmer 1. Given the level of z2, does the value of z1 originally chosen by farmer 1 remain optimal? To answer this question, draw farmer 1's demand curve, using both lines of the first formula above, as in part (c) (plugging in the chosen value of z2). Given farmer 2's choice, is farmer 1 happy with his original decision?
e) Using the above results, compare the socially-optimal level of z with the level that results from individual decision-making. Given an intuitive explanation of why the two levels differ.
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.
3. Suppose that the bureaucrats who run the Defense Department are budget maximizers. In dealing with Congress, the bureaucrats announce a price p per unit of national defense and Congress gets to choose the level of defense (denoted z). Assume that the DΣ curve, which represents Congress' demand for z, is given by DΣ = 16 - z.
a) Suppose the true cost per unit of national defense is 3 (i.e., c = 3). Find the sociallyoptimal level of defense. Then, find the value of p that is announced by the bureaucrats, and the resulting level of z chosen by Congress. Is national defense underprovided, overprovided, or optimally provided? Is part of the budget wasted by the bureaucrats? If so, how much is wasted?
b) Suppose instead that c = 12. Repeat part (a) for this case.
4. Suppose that the economy contains three consumers, whose demand curves for the public good z are given by
D1 = 150/z
D2 = 100/z
D3 = 50/z
The cost per unit of z is c = 60.
a) Find z*, the socially optimal level of z,.
b) Compute the Lindahl taxes, which generate unanimous agreement on z*. Verify that each consumer prefers z* when faced with these taxes.