Micro

Lecture 15: Public Goods II

Larbi Alaoui

June 6, 2012

Public good example

Two flatmates, 1 and 2, who can purchase some public good, e.g. a TV for their living room. Agent 1 and agent 2 have wealth w1 and w2 , respectively. Choose to contribute: g1 and g2 to purchasing the TV. The TV costs c, so they need to contribute a least that amount in total, if they decide to purchase it: g1 + g2 ≥ c.

They’re left with x1 , x2 : x1 = w 1 − g1 x2 = w2 − g2 .

Public good example: reservation price

What’s each agent’s willingness to pay? An agent’s reservation price here is the maximum price that she is willing to pay for the good. Suppose that the agent’s utility is a function of her money and the TV. That is, suppose that the agent 1’s utility function is u1 (x1 , TV ), (x1 is how much money he has left), and TV = {0, 1} is 0 if they have a TV, 1 if they don’t. Similarly, agent 2’s utility function is u2 (x2 , TV ). Note that it’s not TV1 for agent 1 and TV2 for agent 2. There’s one TV for everyone.

Public good example: reservation price

So to find the reservation price for agent 1, the maximum price she’s willing to pay, it’s when she’s indifferent between having a TV and not having one. If they don’t get a TV, her utility is u1 (w1 , 0). If they do, it’s u1 (w1 − g1 , 1) (or u1 (x1 , 1)). So the reservation price, denoted r1 (so substitute r1 for g1 ) is such that u1 (w1 − r1 , 1) = u1 (w1 , 0). Similarly, u2 (w2 − r2 , 1) = u2 (w2 , 0).

Public good example: reservation price

So what? Consider a payment scheme (g1 , g2 ) which covers the cost of the TV, so let’s say g1 + g2 = c. Each agent will only agree to it if it is individually rational for him to do so: u1 (w1 − g1 , 1) ≥ u1 (w1 , 0) u2 (w2 − g2 , 1) ≥ u2 (w2 , 0) Compare this to how we determined the reservation prices: u1 (w1 − r1 , 1) = u1 (w1 , 0), u2 (w2 − r2 , 1) = u2 (w2 , 0). What this means is that r1 ≥ g1 , and r2 ≥ g2 . This isn’t a big revelation: the reservation price is their...