(a) Provide a formal rigorous-proof of Sen's theorem that Liberalism and Pareto Optimality are incompatible. (b) Consider an economy with 15 consumers and 2 goods. Consumer 3 has utility function U³ = X₁ + X₂ At a certain Pareto efficient allocation, consumer 3 has an allocation (10, 0). What can you say about the competitive prices that support this Pareto Optimal allocation? Explain your answer.

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An investment in a new handling equipment would cost $84,248 and save $20,000 per year for 5 years. What is the nominal annual interest rate earned in this investment?

True/False? Briefly explain. Assume good x is a normal good. In terms of consumer behavior, a cash grant and a small (relative to the initial consumption of x) limited non-matching grant on good x (with the limit that is equal to the cash grant) are essentially equivalent.

2(b)There are three people and two goods in an economy. The utility functions and the initial bundles are given below: U1 = X11X 12 U = 2x 21+ X 22 U3 = X 31 X 3z+X 32 W; = (0,8) W;= (2,2) Wy= (8,0) Suppose it is suggested that the three traders move to the allocation x given by x;= (1,2), x2= (8,4), X3= (1,4) b. Show that x makes no one worse off than the original allocation.

Grapes and bananas are perfect complements for Jane and Betty. Jane eats one grape per three bananas.; Betty eats three grapes per one banana. Once they go to sports for hours, Jane had 20 grapes and 10 bananas and Betty had 10 grapes and 20 bananas. (assume that grapes and bananas can be divided into half) 1.Show all Pareto optimal allocations of grapes and bananas between Jane and Betty in the Edgeworth Box. Draw a bunch of indifference curves and look for tangencies.