Raising Money for a Streetlight through a “Subscription Campaign”: Sometimes, a civil society institution’s goal can be clearly articulated in terms of a dollar value that is needed. Consider, for instance, the problem you and I face when we want to fund a streetlight on our dark cul-de-sac. We know the total cost of the light will be C —and so we know exactly how much money we need to raise. One way we can raise the money is through what is known as a subscription campaign. Here is how a subscription campaign would work: We put a money “pledge jar” in between our two houses, and you begin by pledging an amount xY1. We then agree that we will alternate putting a pledge for a contribution into the jar on a daily basis — with me putting in a pledge xM2 the second day, then you putting in a pledge xY3 the third day, me putting in xM4 the fourth day, etc. When enough money is pledged to cover the cost C of the street light, we pay for the light — with you writing a check equal to the total that you have pledged and me writing a check for the total I have pledged.
A: Suppose you and I each value the light at $1,000 but the light costs $1,750. We are both incredibly impatient people— with $1 tomorrow valued by us at only $50 cents today. For simplicity, assume the light can be put up the day it is paid for.
(a) Suppose it ends up taking T days for us to raise enough pledges to fund the light. Let xiT be the last pledge that is made before we reach the goal. What does sub game perfection imply xiT is?
(b) Next, consider person j whose turn it is to pledge on day (T −1). What is xj T−1? (Hint: Person j knows that, unless he gives the amount necessary for i to finish off the required pledges on day T, he will end up having to give again (an amount equal to what you calculated for xiT ) on day (T +1) and have the light delayed by one day.)
(c) Continue working backwards. How many days will it take to collect enough pledges?
(d) How much does each of us have to pay for the streetlight (assuming you go first)?
(e) How much would each of us be willing to pay the government to tax us an amount equal to what we end up contributing—but to do so today and thus put up the light today?
(f) What is the remaining source of inefficiency in the subscription campaign?
(g) Why might a subscription campaign be a good way for a pastor of a church to raise money for a new building but not for the American Cancer Association to raise money for funding cancer research?
B: Now consider the more general case where you and I both value the street light at $V, it costs $C, and $1 tomorrow is worth $δ < 1 today. Assume throughout that the equilibrium is sub game perfect.
(a) Suppose, as in a (a), that we will have collected enough pledges on day T when individual I put in the last pledge. What is xiT in terms of δ and V?
(b) What is xj T−1? What about xiT −2?
(c) From your answers to (b), can you infer the pledge amount xT−t for t ranging from 1 to (T −1)?
(d) What is the amount pledged today—i.e. in period 0?
(e) What is the highest that C can be in order for (T +1) pledges—i.e. pledges starting on day 0 and ending on day T —to cover the full cost of the light.
(f) Recalling that ∑∞t =0 δt = 1/(1−δ), what is the greatest amount that a subscription campaign can raise if it goes on sufficiently long such that we can approximate the period of the campaign as an infinite number of days?
(g) True or False: A subscription campaign will eventually succeed in raising the necessary funds so long as it is efficient for us to build the street light.
(h) True or False: In subscription campaigns, we should expect initial pledges to be small—and the campaign to “show increasing momentum” as time passes, with pledges increasing as we near the goal.