am stuck

1. A profit maximizing firm operates in competitive input as well as output markets. The inputs are labor (L) and capital (K). The price of a unit of Labor is w while the price for a unit of capital is r. The output price is P. The production function of the firm is ?(?, ?) = ???(1 + ?) + ???(1 + ?) which means that L units of labor and K units of capital are needed to produce ?(?, ?) units of output. Determine the following: (a) The optimal quantities labor and capital. (b) The optimal output (c) The maximum profit (d) The comparative statics of the firm's maximum profit with respect to input prices and output price (e) The comparative statics of the firm's optimal input and output choices with respect to input prices and output prices (f) The equation for the firm's supply curve

2. Repeat Exercise 1. above with ?(?, ?) = ?? + ?L

Consider a competitive market for electricity production in which some firms are polluters:

Firm A (coal burning): C(x A ? )=10x A ? and the environment cost to society is E(x A ? )= 360 1 ? x A 3 ?

Firm B (wind power): C(x B ? )=40x B ? and the environment cost to society is E(x B ? )=0

i) Assume the market operates without any recognition of environmental costs. At what prices will each firm operate? Assume the market demand for the commodity is given by: X(p)= p 24000 ?

ii) Find the competitive market equilibrium price and quantity using the firm's supply responses in part i) and the market demand above. Are both firms operating? why?

iii) Suppose a planner uses the representative consumer's utility function to find the socially optimal allocation of resources to the two firms. Choose (x A B ,y) to solve maxU(x,y)=(24000)ln(x)+y subject to x=x A +x , x A ?0& x B ?0and C(x )+C(x )+E(x A )+y=50,000 What will be the aggregate production level and how much will each firm supply?

iv) Use a pair of sketches of the market and planner allocations (using the market demand/marginal benefit curve and cost curves of the firms) to compare the net benefit to society from the outcomes in part ii) and iv). Has the planner achieved an increase in net benefit?

(10) Consider the discrete time monetary-search model we saw in class. As in the baseline model, in the day time trade takes place in a decentralized market characterized by anonymity and bilateral meetings (call it the DM), and at night trade takes place in a Walrasian or centralized market (call it the CM). There are two types of agents, buyers and sellers, and the measure of both is normalized to the unit. The per period utility is u(q)+U(X)-H, for buyers, and -9+U(X)-H, for sellers, g is consumption of the DM good, X is consumption of the CM good (the numeraire), and H is hours worked in the CM In the CM, one hour of work delivers one unit of the numeraire. The functions u, U satisfy standard properties. What is important here is that there exists X>0 such that U'(X) = 1. Goods are non storable, but there exits a storable and recognizable object, called fiat money, that can serve as a means of payment. The supply of money, controlled by the monetary authority, follows the process M?+1 = (1+u)M,, and new money is introduced via lump-sum transfers to buyers in the CM. So far, this is just a description of the model we saw in class. What is different here is that only a fraction of buyers turn out to have a desire to consume the DM good in the current period; let us refer to these buyers as C-types (for consumption) and to the remaining 1- buyers as N-types (for no-consumption). The shock that determines each buyer's type in every period is id. A buyer learns her type after all CM has concluded but before the DM opens To make things interesting we will assume that between the CM and the DM there is a third market, where C-types and N-types can meet and trade "liquidity", i.e., money. Let us refer to this market as the loan market (LM). 4