Consider an economy with one firm and one consumer. The firm produces output y with a combination of coal c and gas g : y=(gc)1/2 The consumer derives utility from the good produced by the firm y, an outside good x (e.g. money). Moreover, every unit of coal and gas used by the firm in production creates pollution, which negatively impacts the utility of the consumer: u(y,x,g,c)=ln(y)+xggcc However, the consumer cannot influence the choice of gas and coal from the firm because they interact through a perfectly competitive market, so pollution creates an externality. The consumer has an endowment which has value normalized to 1. p=1.pg=2 and pc=1 are the prices of gas and coal, respectively. You can assume they are the world prices and do not change. a) For a given output quantity y, derive the cost-minimizing quantities of g and c that the firm will use in production. b) Derive the quantity demanded of good y from the consumer as function of the output price py. Use market clearing and profit maximization of the firm to find the optimal quantity of output y in the economy. c) Combine a) and b) to find the market allocation of gas g and coal c. d) Find the Pareto optimal quantity of coal c and gas g in this economy. e) Find the set of pigouvian taxes on coal c and gas g that achieve this pareto optimal allocation through the market. Consider an economy with one firm and one consumer. The firm produces output y with a combination of coal c and gas g : y=(gc)1/2 The consumer derives utility from the good produced by the firm y, an outside good x (e.g. money). Moreover, every unit of coal and gas used by the firm in production creates pollution, which negatively impacts the utility of the consumer: u(y,x,g,c)=ln(y)+xggcc However, the consumer cannot influence the choice of gas and coal from the firm because they interact through a perfectly competitive market, so pollution creates an externality. The consumer has an endowment which has value normalized to 1. p=1.pg=2 and pc=1 are the prices of gas and coal, respectively. You can assume they are the world prices and do not change. a) For a given output quantity y, derive the cost-minimizing quantities of g and c that the firm will use in production. b) Derive the quantity demanded of good y from the consumer as function of the output price py. Use market clearing and profit maximization of the firm to find the optimal quantity of output y in the economy. c) Combine a) and b) to find the market allocation of gas g and coal c. d) Find the Pareto optimal quantity of coal c and gas g in this economy. e) Find the set of pigouvian taxes on coal c and gas g that achieve this pareto optimal allocation through the market
