Max Imization s preferences are represented by a utility function: U = Max has g

Question

Max Imization s preferences are represented by a utility function: U = Max has got an income Y and the prices of x_1 and x_2 are p_1 and p_2, respectively. Derive Max's demand functions for x_1 and x_2, respectively. Suppose initially p_1 = 1 and p_2 = 1. The government imposes a per unit tax t = 3 on the first commodity, so that p_1 + t = 4. How much increased income does Max need to not be worse-off? Jenny's company produces biscuits at a constant cost of $6 each. The company has groups of buyers-rich with demand Qrich = 2000 - P, and poor withdemand poor=2000-2P. Calculate profit-maximizing price, quantity, and profits, assuming that it is

Explanation / Answer

(16)

U = x10.5x20.5

Budget line is: Y = p1x1 + p2x2

(a) Under optimal condition, MU(x1) / MU(x2) = p1 / p2

MU(x1) = dU / dx1 = 0.5 (x2 / x1)0.5

MU(x2) = dU / dx2 = 0.5 (x1 / x2)0.5

So, the optimality condition requires that (x2 / x1) = p1 / p2

Or,

p1x1 = p2x2

Substituting in budget line:

Y = p1x1 + p2x2 = p1x1 + p1x1 = 2p1x1

X = Y / 2p1 [Demand function for x1]

Similarly,

Y = 2p2x2

x2 = Y / 2p2 [Demand function for x2]

(b)

Initially, p1 = p2 = 1

Y = x1 + x2

Next, p1 = 1 + 3 = 4 & p2 = 1

Y1 = p1x1 + p2x2 = 4x1 + x2

Y1 - Y = Change in income (compensation in income to maintain same utility) = 3x1

Exact quantitative value for (Y1 - Y) can be derived only if value for Y is specified, but this information is not provided.

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