A community lives near a forest. The forest provides wood, which the villagers can use for various purposes like heating,
building, or selling. However, the forest also acts as a carbon sink, absorbing carbon dioxide and providing fresh air, and
is home to various wildlife. If too many trees are cut down, the forest will degrade, leading to loss of these environmental
benefits.
There are N number of identical villagers. Each villager is endowed with $y, and must decide how much of their
endowment to invest in a communal fund for forest conservation and how much to keep for themselves. Let $g_i$ denote
each villager's contribution to the communal fund.
The total amount of dollars in the communal fund is multiplied by some constant factor R, where 1 < R < N. The
multiplied amount is then equally distributed among all villagers, irrespective of their individual contributions.
Each villager i's utility is as follows:
$$U_i = y - g_i + \frac{R}{N} \times G$$
$$= y - g_i + \frac{R}{N} \times (g_i + G - i),$$
where G is the total amount of dollars collected for the communal fund, and $G_i = G - g_i$.
a. Using calculus and graph, determine the equilibrium contribution ($g_i$) for each villager.
b. The forest manager, who is external to the community, aims to maximize the community's social welfare (SW).
Note that the social welfare is the sum of the individual villager's utility. Calculate the socially optimal level of $g_i$
that maximizes SW. How does this differ from the equilibrium contribution?
$$SW = U_1 + U_2 + ... + U_N$$
$$= N \times U_i$$
$$= N \times (y - g_i + \frac{R}{N} \times G)$$
for any villager i = 1, 2, ..., N
c. The forest manager decides to implement a matching grant program to encourage contributions to the forest
conservation fund. For every dollar contributed by a villager, the program will contribute an additional m dollars
(where 0 < m < 1). The utility function for each villager i now becomes:
$$U_i = y - g_i + \frac{R}{N} \times (1 + m) \times G$$
$$= y - g_i + \frac{R}{N} \times (1 + m) \times (g_i + G - i),$$
Determine the minimum value of m that would induce villagers to contribute the socially optimal level of $g_i$ found in
part (b). Show your work.
d. Using the minimum value of m found in part (c), calculate the total cost of implementing the matching grant
program to the forest manager. Assume all villagers contribute the socially optimal amount.
e. Calculate the change in social welfare ($\Delta SW = SW - SW$) with the introduction of the matching grant
program. Compare this change to the cost of the program calculated in part (d). Is the change in SW greater than,
equal to, or less than the cost of the program to the forest manager? Justify your answer analytically.
f. Discuss the practical implications and potential challenges of implementing such a matching grant program in the
context of community forest management.
