Assurance Contracts to Support Multi-Unit Threshold Public Goods in Environmental Markets

Abstract

The free riding incentive has been a major obstacle to establishing markets and payment incentives for environmental goods. The use of monetary incentives to induce private provision of public goods has gained increasing support to help market ecosystem services. Using a series of lab experiments, we explore new ways to raise money from individuals to support private provision of multi-unit threshold public goods. In our proposed mechanisms, individuals receive an assurance contract that offers qualified contributors an assurance payment as compensation in the event that total contributions fail to achieve the threshold provision cost. Contributors qualify by contracting to support provision with a minimum contribution. Evidence from lab experiments shows that the provision probability, group demand revelation, and social welfare significantly increase when the assurance contract is present. Coordination is improved by the assurance payment especially for agents with values above the assurance level, leading to significantly higher aggregate contributions. A medium level of assurance payment used on units with medium and high value-cost ratios is observed to induce the largest improvement on social surplus. Our approach contributes to the private provision of environmental and other types of public goods.

Appendices

Appendices

1.1 Theoretical Framework and Proof

Proof for Proposition 4

When j units are provided, condition (a) ensures that the total bids on a provided unit just equal the cost, together with $\sum _ib_i^{j+1}<C$ we can conclude that only j units will be provided.

Individual i’s profit from providing j units is

$$\begin{aligned} \pi _i^{j}=\sum _{l=1}^{j}v_i^l-j\left( C-\sum _{k\ne i}b_k^{j}\right) +I(b_i^{j+1}\ge AP^{j+1})AP^{j+1}, \end{aligned}$$

(A.1)

where the term $\sum _{l=1}^{j}v_i^l-j(C-\sum _{k\ne i}b_k^{j})$ represents the profit from the public goods without the assurance payment, $I(b_i^{j+1}\ge AP^{j+1})AP^{j+1}$ is the potential assurance payment from the $j+1$ unit since the assurance payment is only available on the first unit not provided, which depends on individual i’s bid $b_i^{j+1}$ and the assurance payment level $AP^{j+1}$ with $I(\cdot )$ as the indicator function.

To eliminate the incentive to provide one more unit ($j+1$ units), we need to require no individual can obtain a higher profit by providing $j+1$ units. Individual i’s profit from providing $j+1$ units is

$$\begin{aligned} \pi _i^{j+1}=\sum _{l=1}^{j+1}v_i^l-(j+1)\left( C-\sum _{k\ne i}b_k^{j+1}\right) +I(b_i^{j+2}\ge AP^{j+2})AP^{j+2}, \end{aligned}$$

(A.2)

where the term $\sum _{l=1}^{j+1}v_i^l-(j+1)(C-\sum _{k\ne i}b_k^{j+1})$ represents the profit from the public goods without the assurance payment when $j+1$ units are provided. When providing j units are provided, we need $\pi _i^{j}\ge \pi _i^{j+1}, h>j$ so that no individual can be better by providing $j+1$ units unitarily, which implies,

$$\begin{aligned}&v_i^{j+1}+(j+1)\sum _{k\ne i}b_k^{j+1}-j\sum _{k\ne i}b_k^{j}-I(b_i^{j+1}\ge AP^{j+1})AP^{j+1}\nonumber \\&\quad +I(b_i^{j+2}\ge AP^{j+2})AP^{j+2}\le C, \forall i. \end{aligned}$$

(A.3)

To eliminate the incentive to provide $j-1$ units, we need to require no individual can obtain a higher profit by providing $j-1$ units. Individual i’s profit from providing $j-1$ units is

$$\begin{aligned} \pi _i^{j-1}=\sum _{l=1}^{j-1}v_i^l-(j-1)\left( C-\sum _{k\ne i}b_k^{j-1}\right) +I(b_i^{j}\ge AP^{j})AP^{j}, \end{aligned}$$

(A.4)

where the term $\sum _{l=1}^{j-1}v_i^l-(j-1)(C-\sum _{k\ne i}b_k^{j-1})$ represents the profit from the public goods without the assurance payment when $j-1$ units are provided. We need $\pi _i^{j}\ge \pi _i^{j-1}$ so that no individual can be better by providing $j-1$ units unitarily, which implies,

$$\begin{aligned} v_i^{j}+j\sum _{k\ne i}b_k^{j}-(j-1)\sum _{k\ne i}b_k^{j-1}+I(b_i^{j+1}\ge AP^{j+1})AP^{j+1}-I(b_i^{j}\ge AP^{j})AP^{j}\ge C, \forall i. \end{aligned}$$

(A.5)

$\square$

Proof for Corollary 1

When providing j units is the equilibrium outcome and assurance is not available, according to Proposition 3, we have

$$\begin{aligned} \begin{aligned}&\sum _i\left( v_i^{j+1}+(j+1)\sum _{k\ne i}b_k^{j+1}-j\sum _{k\ne i}b_k^{j}\right) \le \sum _iC\\&\sum _iv_i^{j+1}+(N-1)(j+1)\sum _ib_i^{j+1}-(N-1)jC\le NC\\&\sum _ib_i^{j+1}\le \frac{C(N+j(N-1))-\sum _ib_i^{j+1}}{(N-1)(j+1)} \end{aligned} \end{aligned}$$

(A.6)

When assurance is available, according to Proposition 4, similarly we have

$$\begin{aligned} \sum _ib_i^{j+1}\le \frac{C(N+j(N-1))-\sum _ib_i^{j+1}+N^{j+1}_{+}AP^{j+1}-N^{j+2}_{+}AP^{j+2}}{(N-1)(j+1)} \end{aligned}$$

(A.7)

where we define that $N^{l}_{-}= \left| \left\{ i: b_i^l < AP^l \right\} \right|$ and $N^l_{+}= \left| \left\{ i: b_i^l \ge AP^l \right\} \right|$. Note that when the assurance payment is constant or decreasing, and since the number of individuals qualify for assurance payment becomes lower at a higher unit as the induced value decreases, the presence of assurance payment will increase the upper bound of the total group contribution in equilibrium when $N^{j+1}_{+}AP^{j+1}-N^{j+2}_{+}AP^{j+2}>0$. Also note that for $i\in N^{j+1}_{-}$, $C-\sum _{i \ne k}b_k^{j+1}\le AP^{j+1}$ since otherwise individual i can just contribute $AP^{j+1}$ on the $j+1$th unit to increase the profit by $AP^{j+1}$, therefore, we have

$$\begin{aligned} \begin{aligned} \sum _{i\in N^{j+1}_{-}}\left( C-\sum _{i \ne k}b_k^{j+1}\right)&\le \sum _{i\in N^{j+1}_{-}}AP^{j+1}\\ N^{j+1}_{-}C-\sum _{i\in N^{j+1}_{-}}\sum _{i \ne k}b_k^{j+1}&\le \sum _{i\in N^{j+1}_{-}}AP^{j+1}\\ N^{j+1}_{-}C-\left( N^{j+1}_{-}\sum _{i}b_i^{j+1}-\sum _{k\in N^{j+1}_{-}}b_k\right)&\le N^{j+1}_{-}AP^{j+1}\\ N^{j+1}_{-}(C-AP^{j+1})+\sum _{k\in N^{j+1}_{-}}b_k^{j+1}&\ge N^{j+1}_{-}\sum _{i}b_i^{j+1} \\ \sum _{i}b_i^{j+1}\ge C-AP^{j+1}+\frac{\sum _{k\in N^{j+1}_{-}}b_k^{j+1}}{N^{j+1}_{-}}&\ge C-AP^{j+1} \end{aligned} \end{aligned}$$

(A.8)

$\square$

Proof for Corollary 2

Here we impose a monotonic assumption so that individual i’s bid is not decreasing when induced value increases. As a result, individual i’s bid will be non-increasing when induced value is downward sloping as the unit number increases, or $b_i^{j}>b_i^{j+1}$. When equilibrium outcome is j units, we have $\sum _ib_i^{j+1}\le C$. If $b_i^{j+1}<AP^{j+1}$, then $AP^{j+1}\ge C-\sum _ib_i^{j+1}$ since otherwise individual i can just contribute $AP^{j+1}$ on the $j+1$th unit to increase the profit by $AP^{j+1}$. Also, since j is not provided, then individual i must receive a smaller profit providing $j+1$ units compared to when providing j units. Let ${\tilde{b}}_i^{j+1}<C-\sum _ib_i^{j+1}$ be the new bid needed to provide the $j+1$ units. Therefore, $v_i^{j+1}-(j+1){\tilde{b}}_i^{j+1}+jb_i^k< 0$, or $v_i^{j+1}< (j+1){\tilde{b}}_i^{j+1}-jb_i^k$. According to the monotonic constraint, we have $v_i^{j+1}< (j+1){\tilde{b}}_i^{j+1}-jb_i^k\le {\tilde{b}}_i^{j+1}=C-\sum _ib_i^{j+1}\le AP^{j+1}$.

When $j+1$th unit is not provided in the equilibrium and if $b_i^{j+1}<AP^{j+1}$, we find that $v_i^{j+1}<AP^{j+1}$, which implies that if $v_i^{j+1}\ge AP^{j+1}$, then $b_i^{j+1}\ge AP^{j+1}$. When $AP^{j+1}={\bar{v}}^{j+1}$, and the number of individuals with values greater than or equal to ${\bar{v}}^{j+1}$ is great than $C/v^{j+1}$, the contributions from the set of individuals with values greater than or equal to ${\bar{v}}^{j+1}$ would be at least C, contradicting with the non-provision condition when the $j+1$th unit is not provided. $\square$

1.2 Supplementary Tables and Figures

Table 6 The percentage of $B=C$ by AP and unit

Table 7 Two-factor random effects models of group contribution-value ratio for each unit after controlling for the group size

Table 8 Random effects tobit models of individual contribution by AP

Fig. 9

Distribution of group contribution by assurance scheme. This figure shows the cumulative probability distribution of group contributions normalized by the group size N at each unit under different assurance payment levels

1.3 Lab Experiment Instructions

This is an experiment in the economics of decision-making. During the experiment, you will be asked to make a series of decisions. If you follow the instructions and make careful decisions, you can earn a considerable amount of money.

1.3.1 Experiment Overview

• You will be asked to decide how much money to offer towards the cost of several public goods in discrete units. This cost of the public good is predetermined and known to you.

• You will be randomly assigned to one of the two groups at the beginning. Your group members will change after each decision period.

• All members of your group receive a benefit that depends on the number of units being provided. The number of units provided depends on your decision AND those decisions of the other people in your group.

• Earnings in each decision period are based on how much you are willing to invest, how much you earn (your benefits) if the good is provided and the investment decisions of the others in your group. In some of the treatments, you may also earn extra money by satisfying our assurance contract requirement.

1.3.2 How You Earn Money

At the beginning of each period, you will be told the individual values (benefits) you receive if that unit of public good is provided. The individual value for one unit of public good can be different across people; someone may have a higher value than you, while the others may have a lower value than you. Your individual value will change after each decision period. You will then be asked to make contributions according to our rules. There are six public good units available in total and the cost is same for each unit.

You will be working with experimental dollars. Your initial fund will be 250 experimental dollars, which represents your fee for showing up today. Your earnings for each period will be added to or subtract from this amount. After the experiment, we will convert your earning to cash with a ratio 50:1; that is, if your balance at the end of the experiment is $1000, you will receive $20 in cash. There are three treatments in the experiment. You will be paid as you leave.

1.3.3 Group

Your group is important because the moderator, using a computer program, will evaluate the combined decisions (i.e. contributions) from each member of your group to determine the outcome. In this way, the decisions of every person in your group may impact your profit. You do not know others’ contributions or benefits.

1.3.4 Communication

Communication is NOT allowed between participants once we begin today. If you have any questions during the treatments, please raise your hand.

1.3.5 Treatments, Periods

There are 15 decision periods in each treatment. We expect to finish the whole experiment within one hour and thirty minutes or so.

1.3.6 What you need to do?

Once the program is activated, please make a contribution for ${\underline{{\varvec{each}}}}$ public good unit. There are six units of public good available in total.

1.3.7 Your value

Your value on the first unit is randomly drawn from [15, 25], your value on the sixth (last) unit is randomly drawn from [5, 15]. Others’ values are also randomly drawn from the same intervals; thus, someone may have a higher value than you, while some may have a lower value than you. Your value decreases from the first unit to the last unit.

1.3.8 How is your profit calculated?

• Your profit$=$ Your benefit - Your cost.

• Your benefit$=$ sum of your values for all the units that are provided.

Your benefit depends on the number of units that your group collectively supports. You will receive your value for each unit supported. For example, if your group supported the first three units, and your values for the first three units are $20, $15, $10, your benefit is $20+$15+$10=$45.

• Your cost = contribution on the last unit provided $\times$ number of units provided.

Your cost also depends on the number of units that your group could collectively support. Your cost is your contribution on the last unit provided times the number of units provided. For example, if your group supported the first three units, your contribution on the third unit is $5, then your cost $5 × 3 = $15.

• Under this situation, your profit = $45−$15 = $30.

1.3.9 How to decide if a unit can be provided?

• We will compare the total contribution of your group for each unit with the public good cost for that unit, starting from the first unit. If the group’s total contribution on the first unit is higher or equal to the cost for the first unit, we continue to compare the contribution on the second unit with the cost of that unit, and so on.

• We will stop when the total group contribution for a unit is smaller than the unit cost.

All the numbers used in examples serve only illustrative purpose; please do not try to use these examples to guess what would actually happen in the experiment.

1.4 Assurance

In this treatment, we offer an assurance contract for the first three units. Your total profit is whatever you can get from providing the public good, plus any assurance payment whenever applicable. We try to protect you from getting zero benefits when you were willing to contribute above a certain level. That is, if your contribution on the first three units is higher or equals to a certain level, and if that unit is the first unit not provided, then we will compensate you an amount equal to the level we set with the table below, which we call the minimum contribution for assurance. Below is the minimum contribution for assurance you need to reach in order to get an assurance payment in case of provision failure; you may decide to contribute less if you wish, but then you will not be eligible to receive the assurance payment.

Unit	Minimum Contribution	Your Compensation
1	14 experimental dollars	14 experimental dollars
2	14 experimental dollars	14 experimental dollars
3	14 experimental dollars	14 experimental dollars

You receive assurance payment only for the first unit not provided. For example, if your values on the six units are $\{ 20, 15, 10, 5, 4, 3\}$ , and contributions on the six units are $\{15, 14, 10, 5, 4, 2\}$ , with the assurance,

• If 0 units are provided, and we fail to provide Unit 1: Since your contribution on Unit 1 is $15, which is higher than $14, the minimum contribution, you will get a compensation from our assurance, $14. Thus, your total profit is $14. However, if you contributed lower than $14, say $10, your profit is $0.

• If 1 unit is provided, and we fail to provide Unit 2. Since your contribution on Unit 2 is $14, which equals $10, the minimum contribution, you will get a compensation from our assurance, $14. Thus, your total profit is your profit from providing 1 unit, $20-$15$=$$5, plus the compensation from our assurance, $14; therefore, your total profit is $5$+$$14$=$$19. However, if you contributed lower than $14 on the unit 2, say $8, your profit is $5.

• If 2 units are provided, and we fail to provide Unit 3. Since your contribution on Unit 3 is $10, which is smaller than $14, the minimum contribution, you will NOT get a compensation from our assurance. Thus, your total profit is your profit from providing 2 units, $20$+$$15-2*$14$=$$7. However, if you contributed more than (or equal to) $14 on Unit 3, say $15, then your profit is $7$+$$14$=$$21, since you get the assurance $14.

• We only provide assurance for the first three units.

1.4.1 Quiz (4 mins)

• 1. If your contributions on the first 4 units are $15, $10, $9, $6, respectively, and your group provides 2 units, what’s your cost in this case?

• 2. If your contributions on the first 4 units are $15, $10, $9, $6, respectively, your benefits on the first 4 units are $20, $10, $5, $3, and if your group provides 2 units, what’s your profit in this case? What’s your profit if your group provided 1 unit? What’s your profit if your group provided 0 units?

• 3. If there are five people in your group, their values are the same for the first five units which are {20, 18, 16, 14, 12}; their contributions on the first unit are {15, 15, 15, 15, 15}, their contributions on the second unit are {13, 13, 13, 13, 13}, their contributions on the third unit are {9, 9, 9, 9, 9}, their contributions on the fourth unit are {5, 5, 5, 5, 5}, and if the provision cost is 50. How many units are provided in total? What’s the profit of one people? If only one unit is provided, what’s the profit of one people?

At the end of the experiment, your earnings will be totaled across all periods and converted from experimental dollars to real dollars. You will be paid as you leave.

Now please make your decisions!