3. Assume that there are three commodities with amounts $ x_1, x_2, $ and $ x_3 $ and prices $ p_1, p_2, $ and $ p_3 $. Assume that the total value is fixed,\[ p_1 x_1 + p_2 x_2 + p_3 x_3 = w_0 \]where $ w_0 > 0 $ is a fixed constant.For a given utility function $ U(x_1, x_2, x_3) $, the maximum of $ U $ is of interest to economists.(a) Let $ U(x_1, x_2, x_3) = x_1 x_2 x_3 $, i.e., all commodities are valued equally, find the maximum of $ U $ under the constraint on total value.(b) Let $ U(x_1, x_2, x_3) = x_1^{a_1} x_2^{a_2} x_3^{a_3} $ with $ a_i > 0 $, i.e., each commodity may be weighted differently, find the maximum of $ U $ under the constraint on total value.

Utility maximization's idea is that the consumers wish to allocate their money income or endowment…

3. Assume that there are three commodities with amounts ₁, 2 and 3 and prices P₁, P2 and p3. Assume that the total value is fixed, P₁1+P₂æ2+P3x3 = Wo where wo> 0 is fixed constant. For a given utility function U(x₁, x2, x3), the maximum of U is of interest to economists. (a) Let U (x₁, x2, x3) = x₁x2x3, i.e. all commodities are valued equally, find the maximum of U under the constraint on total value. a1a2a3 (b) Let U(T1, T2, T3) = x₁¹x2²x3³ with a > 0, i.e. each commodity may be weighted differently, find the maximum of U under the constraint on total value.

3. Assume that there are three commodities with amounts ₁, 2 and 3 and prices P₁, P2 and p3. Assume that the total value is fixed, P₁1+P₂æ2+P3x3 = Wo where wo> 0 is fixed constant. For a given utility function U(x₁, x2, x3), the maximum of U is of interest to economists. (a) Let U (x₁, x2, x3) = x₁x2x3, i.e. all commodities are valued equally, find the maximum of U under the constraint on total value. a1a2a3 (b) Let U(T1, T2, T3) = x₁¹x2²x3³ with a > 0, i.e. each commodity may be weighted differently, find the maximum of U under the constraint on total value.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Question

3. Assume that there are three commodities with amounts $ x_1, x_2, $ and $ x_3 $ and prices $ p_1, p_2, $ and $ p_3 $. Assume that the total value is fixed,

\[
p_1 x_1 + p_2 x_2 + p_3 x_3 = w_0
\]

where $ w_0 > 0 $ is a fixed constant.

For a given utility function $ U(x_1, x_2, x_3) $, the maximum of $ U $ is of interest to economists.

(a) Let $ U(x_1, x_2, x_3) = x_1 x_2 x_3 $, i.e., all commodities are valued equally, find the maximum of $ U $ under the constraint on total value.

(b) Let $ U(x_1, x_2, x_3) = x_1^{a_1} x_2^{a_2} x_3^{a_3} $ with $ a_i > 0 $, i.e., each commodity may be weighted differently, find the maximum of $ U $ under the constraint on total value.

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1
Suppose that U(f,c) = f + 8c^(1/2)is a utility function that describes Amelia’s preferences over two goods: fish(f)and custard (c). For the following, think of fish as the good graphed on the horizontal axis.a. Derive an expression for her marginal utility (Uf)from a small increase in f holding c fixed. Also find themarginal utility for custard (Uc).b. What is Amelia’s marginal rate of substitution (MRS)? Give a brief (2 sentences maximum) intuitivedescription of what MRS represents. If Amelia has 4 units of custard, holding her utility constant, howmany units of custard would she be willing to give up in order to get one more unit of fish?c. Graph Amelia’s indifference curve for a utility level of 40. Be sure to specify at least 3 bundles of goodson the indifference curve.d. Does the fact that Amelia’s indifference curve intersects with the custard axis violate any of the 5properties of indifference curves? Briefly support your answer.e. Give another utility function that represents…
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Could you just do part c and explain.
2) Which of the following utility functions represent the same preferences? Explain. a) U (x₁, x₂) = X₁ X₂ b) W (x₁, x₂) = 5lnx₁ +5lnx₂ c) V (x₁, x₂) = x₁¹/3x₂ ¹/3 - 0.8 d) Z(x₁, x₂) = 0.5x₁ + 0.5x₂
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5) A consumer of two goods, good 1 and good 2, has a utility function u (X1, X2) = x1 0,6. X2 0,4, where x denotes units of good 1 and X2 units of good 2. The price of good 1 is 5 euros And the price of good 2 is 3 euros per unit and the consumer’s income is 150 euros. Suppose that only the price of good 1 changes and that the new price is p, ‘ = 2. Then the demand function of good 1 that is derived from the utility function is Choose one: A) X1= 0.6m/p1 B) X1= 6m/p1 C) X1= 4m/p1 D) X1= 0.4m/p1 E) Not defined