Question 3 thx Shepherd\'s lemma and how can you get it from expenditure functio

Question

Question 3 thx Shepherd's lemma and how can you get it from expenditure function? Explain the concept of indirect utility function. Derive the indirect utility function from the Cobb-Douglas Utility function given by u(x,y) = kx^n y^1-a for a 1 (0,1) and k is positive Consider the case of a quasi-liner utility case with u(x_1, x_2) = x_1^1/2 + x_2, please derive the ordinary demand function. If the utility for a consumer is well-behaved and the government imposes a consumption tax on good 1 and gives the consumer an income subsidy after the tax. The government budget is in equilibrium (which means tax equals subsidy) Is the welfare of the consumer better off? Explain the reason to explain the welfare change of the consumer in graph If a consumer thinks that a cup of coffee is as good as 2 slices of bread. And the endowment for coffee and bread is (2, 2). The prices for coffee and bread are 1 and 1. The income of the consumer is 100. Please help the consumer determine the choice. Determine whether the consumer is a buyer or seller for x Please analyze the slutsky equation for the consumer when price of coffee changes

Explanation / Answer

3. In this case as we have convexity, we can use the tangency condition for deriving the demand function.

tangency condition for utility function u(x1 , x2) = x1a + x2b is

- (p1 / p2) = - (a * x1(a - 1)) / b

So, here the utility function is u(x1 , x2) = x11/2 + x2

So, the tangency condition is

- (p1 / p2) = - [1/2* (x1-1/2) / 1] = 1 / 2x11/2

x1(p1 , p2 , m) = (p2 / 2p1)2

Now to find the demand function for x2, we need to substitute this value of x1 in the budget constraint.

m = p1 * (p2 / 2p1)2 + p2x2

m = (p22 / 4p1) + p2x2

x2(p1 , p2 , m) = (m / p2) - (p2 / 4p1)

So, the demand functions are as follows-

x1(p1 , p2 , m) = (p2 / 2p1)2

x2(p1 , p2 , m) = (m / p2) - (p2 / 4p1)

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