Answer. the following attachments.

Suppose a market featuring two goods that differ in quality. Good 1 is of quality 51 and good 2 is of quality 52, with 32 > 31. There is a mass of n = l consumers, of different types, identied by a taste parameter 6 ~ Ullb]. Consumer's utility function is given by U (I?) _ 932 p if it buys one unit of good with quality 5 U o'terwise In what follows, assume 51 = 1 and 32 = 2- In other words, rms cannot choose their quality, this is given and does not change. Firms cost functions are given by: C;(q,-) 2 may for i = 1,2 where q,- is the output of fim't 1'. We will also conjecture that parameter values are such that there is full coverage. The budget constraint we considered thus far assumes only lump-sum taxes. In reality, lump-sum taxation is used very rarely. One of the reasons is that people are not identical and so the requirement that all people pay the same amount is not feasible. (Some people earn nothing and so are not able to pay any positive amount.) An alternative policy is a proportional labor income tax, where taxes are equal to a fixed fraction t, 0 > 1,playl, Vi = 1,2, ..., n. Prove or disprove that A has only non-zero eigenvalues, (1)Problem 2: marshallian demand (4 Pts.) Consider an agent with preferences u($1, 12) = 12- Her income is m > 0, price of good 1 is p, > 0, while price of good 2 is normalized to 1. a) Find the individual's marshallian demand for the two goods, I1(p1, m) and 12(p1, m) (Hint: carefully consider all situations) b) Fix m = 1. Write down the function that describes the agent's demand curve, and graph it (Hint:for the graph, it is easier to use the inverse demand). Is good 1 regular? c) Fix p] = 4. Find the function that describes the agent's Engel curve for good 2 and graph it. Is good 2 normal or inferior