please can you help calculate this problem and provide explanation

1. Mr. A derives utility from martinis (m) in proportion to the number he drinks:

U(m) = m (1)

Mr. A is particular about his martinis, however: He only enjoys them made in the exact proportion

of two parts gin (g) to one part vermouth (v). Hence we can rewrite Mr. A's utility function as

U(m) = U(g, v) = min( g 2 , v). (2) (a)

Graph Mr. A's indifference curve in terms of g and v for various levels of utility. Show that, regardless of the prices of the two ingredients, Mr. A will never alter the way he mixes martinis.

(b) Calculate the demand functions for g and v.

(c) Using the results from part (b), what is Mr. A's indirect utility function?

(d) Calculate Mr. A's expenditure function; for each level of utility, show spending as a function of pg and pv. Hint: Because this problem involves a fixed-proportions utility function, you cannot solve for utility-maximizing decisions by using calculus.

2. We looked at the Cobb-Douglas utility function U(x, y) = x y 1, where 0 1. This problem illustrates a few more attributes of that function.

(a) Calculate the indirect utility function for this Cobb- Douglas case.

(b) Calculate the expenditure function for this case. (

c) Show explicitly how the compensation required to offset the effect of an increase in the price of x is related to the size of the exponent .

3. The lump sum principle illustrated in lecture 8 applies to transfer policy and taxation. This problem examines this application of the principle.

(a) Use a graph similar to the graph in lecture 9 to show that an income grant to a person provides more utility than does a subsidy on good x that costs the same amount to the government.

(b) Use the Cobb-Douglas expenditure function presented in Lecture 8 to calculate the extra purchasing power needed to increase this person's utility from U = 2 to U= 3.

Expenditure function is

E(px, py, U) = 2p 0.5 x p 0.5 y U (3) suppose px = 1, py = 4 and income I = 8. 1

(c) Use Expenditure function (Equation (3)) again to estimate the degree to which good x must be subsidized to increase this person's utility from U = 2 to U = 3. How much would this subsidy cost the government? How would this cost compare with the cost calculated in part (b)?

(d) Question 2 asks you to compute an expenditure function for a general Cobb-Douglas utility. Now,use that expenditure function to re-solve parts (b) and (c) here for the case = 0.3. fraction of income that low-income people spend on food.

(e) How would your calculations in this problem have changed if we had used the expenditure function for the Perfect Complements case instead? Say the expenditure function for the Perfect Complements case is

E(px, py, U) = (px + 0.25py)U (