Public Good Optimum © Allen C. Goodman 2008

Public Goods Most important factor is that everyone gets the same amount. We have to get some agreement as to how much we’ll want (we’ll discuss that a lot). We’ll have to get some agreement as to how to pay for it (we’ll discuss that a lot, also).

Consider a Town Trying to decide how many tennis courts should be provided in public parks. Comes out to be number per 10,000 people. Marginal benefit is in hundreds of dollars. Cost is in hundreds of dollars. Let’s assume MC is 32. We have people like Adam, Bert, and Charlie. Enough people to collect money to build what they might want. Query: Why, necessarily should tennis courts be provided publicly? Why shouldn’t people join private tennis clubs?

Look at Adam MB = 30 - Q What does this mean?

Look at Bert MB = Q

Look at Charlie MB = 20 – 0.5Q

Sum of Marginal Benefits – CalculationCalculation Efficient Amount at Q = 12 Why? At Q = 12, they are worth $1800 to Adam $0 to Bert $1400 to Charlie

What happens if demand changes? – CalculationCalculation Look at spreadsheetspreadsheet

Tricky Issue How do we pay for these. Why do it publicly? Suppose we say “How many tennis courts should we build?” assuming it costs $3,200 to build them. What will happen? A> No one here values a single court at $3,200, BUT collectively, they value 12!

Lindahl equilibrium With 12 courts –Adam values them at 18(00) –Bert values them at 0 –Charlie values them at 14(00) If we “know” these values, we can charge the people accordingly. This is sometimes called a “Lindahl” equilibrium.

Problems w/ Lindahl Eq’m 1.With private goods, people pay … it means that they value the goods at least that much. How do we get people to reveal preferences if we’re not withholding services from those who won’t pay? 2.What if MC is close to 0? If we charge where MB = MC, we get close to 0 price, and may not be able to afford the good. 3.It may be hard to exclude those who won’t pay, although for tennis court you could charge an hourly fee. Can’t do the same for police protection.

Who provides? Some public goods are provided at the national level. National defense for example (at least in US) – although in some other (generally less developed countries) you often have local militias. Others at the state, county, or lower levels. Some amounts of goods are also provided privately. Ford, or GM do not depend on public police to guard their property, for example.

Public Goods It's helpful to derive a public good equilibrium. My favorite way is a simple one where we have a social welfare function in which: W = W(U 1, U 2, U 3,...) = Weighted Σ U i for a community of individuals. We must decide how much public good to make. In the pure sense, the public good is nonexcludable and nonrival. U i = U i (x i, G) X =  x i = f (G) f' < 0.

Public Goods So, optimize W =  w i U i (x i, G) + [  x i - f(G)](1) w.r.t. x i  W/  x i = w i U i x + = 0.(2)  W/  G =  w i U i G - f' = 0(3) From (2) w i = - /U i x Insert into (3)  W/  G =  U i G /U i X + f' = 0(3) Factor out, and we get:  U i G /U i X + f' = 0   MRS GX = -f' = MRT Well-known Samuelson condition. w 1 = - /U 1 x w 2 = - /U 2 x w 3 = - /U 3 x Etc.

Let’s try one U 1 = 2x 1 1/2 + 2y 1/2 U 2 = 2x 2 1/2 + 2y 1/2 20 = x 1 + x 2 + y L = w 1 U 1 + w 2 U 2 + λ (20 - x 1 - x 2 – y) Let w 1 + w 2 = 1 Solve in as much detail as you can for x 1 *, x 2 *, and y*. Compare solutions for w 1 = w 2 = ½, with w 1 = ¼, w 2 = ¾. Explain differences.