Public Goods and Free-Riding

Everyone uses the same public good, but who pays to provide it? Public-good demands add up vertically, not horizontally.

Everyone consumes the same unit at the same time. So willingness to pay is stacked vertically. Where that sum meets the cost is the optimum. But each person reasons differently. Hoping others will pay, each pays nothing. Too little is collected, so supply falls short.

For example, 30 🧑 classmates chip in to buy a ⚽ ball for the classroom. The ball costs 30,000 won. If half think 'the others will pay' and pay nothing, the money falls short. Nobody gets a ball to play with.

∑iDi(G∗)=c(Samuelson)\sum_i D_i(G^*) = c \quad \text{(Samuelson)}
gi=max⁡(0,ai−cbi−gj)(Nash)g_i = \max\left(0, \frac{a_i - c}{b_i} - g_j\right) \quad \text{(Nash)}

Part of these learning paths

Examples & Exercises

For a public good with D1:P=10−GD_1: P = 10 - G, D2:P=8−GD_2: P = 8 - G and marginal cost c=4c = 4, find the socially optimal quantity G∗G^* from the Samuelson condition.

∑iMRSi=MRT⟺D1(G∗)+D2(G∗)=c\sum_i MRS_i = MRT \quad\Longleftrightarrow\quad D_1(G^*) + D_2(G^*) = c
The vertical sum meets MC=4MC = 4 at G∗=7G^* = 7.
Show proof

  1. Because the good is non-rival, demands sum vertically: ΣD(G)=(10−G)+(8−G)=18−2G\Sigma D(G) = (10 - G) + (8 - G) = 18 - 2G (where both are positive).

  2. The Samuelson condition ΣD(G∗)=c\Sigma D(G^*) = c gives 18−2G∗=418 - 2G^* = 4, so G∗=7G^* = 7. At G=7G = 7, D1=3>0D_1 = 3 > 0 and D2=1>0D_2 = 1 > 0, so we are not on the kinked part — the solution is valid.

In the same example, find the Nash equilibrium of voluntary contributions, identify the free-rider, and compute the deadweight loss of underprovision.

Private supply 66 vs social optimum 77 — consumer 2 free-rides.
Show proof

  1. Consumer ii's reaction function is gi=max⁡(0,(ai−c)−gj)g_i = \max(0, (a_i - c) - g_j) (slopes are 1). Desired totals are 66 for consumer 1 and 44 for consumer 2.

  2. Consumer 1, with the keener demand, provides g1=6g_1 = 6; consumer 2 free-rides with g2=max⁡(0,4−6)=0g_2 = \max(0, 4 - 6) = 0. No deviation pays: if consumer 2 added a little, marginal benefit 8−6=28 - 6 = 2 would fall short of cost 44.

    Total private supply is 6<7=G∗6 < 7 = G^*.

  3. Total welfare W(G)=18G−G2−4G=14G−G2W(G) = 18G - G^2 - 4G = 14G - G^2 gives W(7)=49W(7) = 49 and W(6)=48W(6) = 48, so the deadweight loss is 11 — the red region in the figure.

    DWL=W(G∗)−W(GNash)=49−48=1DWL = W(G^*) - W(G_{Nash}) = 49 - 48 = 1

10. Externalities and Public Goods