pls tackle all questions

1. Consumer 1 has expenditure function ,(p. p2, wi) = wjypipz and consumer 2 has utility function us($1, ry) = (a) What are Marshallian (market) demand functions for each of the goods by each of the consumers? Denote the income of consumer 1 by my and the income of consumer 2 by my. (b) For what value(s) of the parameter a will there exists an aggregate demand functions that is independent of the distribution of income? 2. Suppose that utility maximization problems and expenditure minimization problems are well defined and utility and expenditure functions satisfy all necessary "nice" properties. (a) Show (prove) that utility maximization implies expenditure minimization and vice versa. (b) List all relevant identities that are result of a. (c) Derive Roy's identity. (d) Derive Slutsky equation. 3. An economy has two kinds of consumers and two goods. Type A and type B consumers have the following utility functions UN(11, 12) = 41 - -+ 1. U"(1,1) = 2n - -+1 Consumers can only consume nounegative quantities. The price of good 2 is 1 and all consumers have incomes of 100. There are A type A consumers and NV type B consumers. (a) Suppose that a monopolist can produce good 1 at a constant unit cost of c per unit and cannot engage in any kind of price discrimination. Find its optimal choice of price and quantity. For what values of c will it be true that it chooses to sell to both types of consumers? (b) Suppose that the monopolist uses a "two-part tariff" where a consumer must pay a lump sum & in order to be able to buy as much as he likes at a price p per unit purchased. Consumers are not able to resell good 1. For p 0. Output can be converted one-to-one into either investment in physical capital or human capital. Saving directed towards investment into physical capital is given by 0 0 (b) (8 points) What will happen to your conditional demand for labor if there is an increase in the wage rate, assuming that r and @ remain the same? Explain in one sentence why your answer makes intuitive sense. (c) (5 points) Use your answers from (a) to write down an expression for your total cost function TC(r, w. Q). Is this function "homogeneous of degree one" in w and r; that is, does TC(tr. tu, () = . TC(r, w. Q)?Consider a variation to the baseline Solow growth model without population or technological progress. The per-capita production function is given by yt = f(kt) = kta, where 0