Microeconomics question

Welfare Consider a perfectly competitive market with 11. identical consumers and m identical rms. The rm's production function is given by q(k,l) = m and the consumer's preference are given by u(:s, y) = maylC' where x is the good under c0nsideration and y is simply the rest of the goods that the consumer wants to buy. The price of labor and capital are l and r respectively, and are assumed to be given. (a) (b) (C) (d) (6) (f) Find the short-run equilibrium price and quantity. Show the effect on the shortrun equilibrium price of: (i) increase in the number of consumers, (ii) a reduction of the number of rms, (iii) an increase in w, (iv) an increase in r, (v) a reduction of consumer's income, and, (vi) an increase in the preferences for 1'. Find the long-run equilibrium price, total industry production and number of rms. Show the change in the consumer and producer surplus of an as a function of an income tax rate, where the inc0me tax takes the form of tM, where M is the consumer's income and t 6 (0,1) is the income tax rate. Are consumers and rms better off? (Hint: The consumer surplus has to be calculated with the new demand equation.) Can you model the effect of a sales tax on consumer's and producer's surplus? (For this part I'm giving you some liberty in how to proceed doing the analysis, feel free to add extra assumptions if necessary.) Imagine that the good being traded in this market pollutes the environment and the government found that per every unit produced, the society lose '7 dollars in the form of reduced life expectancy and increased medical expenditures. (i) Is our calculation of welfare using the consumer/ producer surplus in the previous parts an overestimation or an underestimation of the actual welfare created by this market to the society? (ii) In theory, can you nd a way in which, via taxation, we can extract some value from the market to cover the damage caused by it?, and, (iii) By modifying the utility function and / or the production function, can you make the economic agents solve the pollution problem by themselves (i.e. without taxation)