Question

Two firms produce a good q and receive a price p = 10 for the good. Firm 1 has marginal costs MC1 = q while firm 2 has marginal costs MC2 = 2q. The production of each unit causes marginal external damage of 2 monetary units. The government wants to limit production with a cap and trade system.

a) What is the optimal cap on total production?

b) The total cap is divided among firms as production quota. Each firm can only produce up to its production quota. However, they can trade quotas in order to expand production. What is the equilibrium quota price if the cap is set optimally?

Answer 1

ANSWER a)

This paper investigates the optimal production decisions of a self-pricing manufacturer and the optimal cap-setting decisions of a regulator under the cap-and-trade regulation. The objectives of the manufacturer and the regulator are to maximize profit and to maximize social welfare, respectively. We first derive the optimal production decisions and the corresponding total emissions of the manufacturer, with given parameters of the cap-and-trade regulation. Based on these results, we then solve the optimal cap-setting problem of the regulator. Furthermore, through sensitive analyses, we show that as the emissions intensity (i.e., the emissions generated from one unit of product) increases, both the optimal total emissions and the optimal cap first increase and then decrease.

ANSWER b)

we investigate the optimal operational decisions of a self-pricing manufacturer and the optimal cap of a regulator under the cap-and-trade regulation. To derive the optimal solutions, a Stackelberg game model is proposed. The regulator is the leader, who sets the emissions cap to the manufacturer for maximizing social welfare. The manufacturer is the follower, who makes the optimal production decisions to maximize profit with the given emissions cap. In our analysis, we first derive the manufacturer’s optimal production decisions and the corresponding total emissions (optimal total emissions for short) under varying cases and then solve the regulator’s optimal emissions cap to maximize social welfare. Moreover, some properties of both players’ optimal solutions with respect to production and regulation parameters (e.g., emissions intensity, emissions trading prices) are explored and some special results are found. In particular, in this paper, we assume that the permit’s purchasing price is not lower than the permit’s selling price, which plays a key role in deriving the optimal solutions. Although we call them prices, they actually represent the cost of purchasing one unit of additional permit and the revenue of selling one unit of redundant permit, respectively. In recent years, more and more research has been done allowing for the emissions trading prices (Drake et al, 2010; Hua et al, 2011; Zhang et al, 2011; Du et al, 2011; Benjaafar et al, 2013; Zhang and Xu, 2013; Gong and Zhou, 2013), but most of them assume that the permit’s purchasing and permit’s selling prices are equal.