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(b) Now use your calculations in (a) above to approximate the change in the level of utility of the consumer from the original location (x, y) = (1,2) to the point (x, y) = (1,2½/) (without using direct substitution). (c) What are the condition(s) for a point to be a regular point on a function? Show that these condition(s) hold in the case point (1, 2) and function U₂(x, y). Now show that the gradient vector VU₁(x, y) will be perpendicular to the indifference curve of U¿ at (1,2). Show all calculations.

(b) Now use your calculations in (a) above to approximate the change in the level of utility of the consumer from the original location (x, y) = (1,2) to the point (x, y) = (1,2½/) (without using direct substitution). (c) What are the condition(s) for a point to be a regular point on a function? Show that these condition(s) hold in the case point (1, 2) and function U₂(x, y). Now show that the gradient vector VU₁(x, y) will be perpendicular to the indifference curve of U¿ at (1,2). Show all calculations.

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Could you please help with b and c. 

I've attached the formula for Utility. 

1. Consider the following utility function for consumer i, who consumes two goods, namely x and y : U;(x, y) = 0.2xy³ + 0.8y2 (a) Make use of the Implicit Function Theorem to calculate the marginal rate of substitu- tion (MRS) for consumer i at the point (x, y) = (1,2) in order to illustrate how much less of good y the consumer would need to consume to compensate for gaining 1 unit of good x while remaining on the same indifference curve. (b) Now use your calculations in (a) above to approximate the change in the level of utility of the consumer from the original location (x, y) = (1,2) to the point (x, y) = (1,2;) (without using direct substitution).. (c) What are the condition(s) for a point to be a regular point on a function? Show that these condition(s) hold in the case of point (1, 2) and function U;(x, y). Now show that the gradient vector VU;(x,y) will be perpendicular to the indifference curve of U; at (1,2). Show all calculations.
Transcribed Image Text:1. Consider the following utility function for consumer i, who consumes two goods, namely x and y : U;(x, y) = 0.2xy³ + 0.8y2 (a) Make use of the Implicit Function Theorem to calculate the marginal rate of substitu- tion (MRS) for consumer i at the point (x, y) = (1,2) in order to illustrate how much less of good y the consumer would need to consume to compensate for gaining 1 unit of good x while remaining on the same indifference curve. (b) Now use your calculations in (a) above to approximate the change in the level of utility of the consumer from the original location (x, y) = (1,2) to the point (x, y) = (1,2;) (without using direct substitution).. (c) What are the condition(s) for a point to be a regular point on a function? Show that these condition(s) hold in the case of point (1, 2) and function U;(x, y). Now show that the gradient vector VU;(x,y) will be perpendicular to the indifference curve of U; at (1,2). Show all calculations.
This use of the Implicit Function Theorem is the natural approach when studying the slope of an indifference curve of a utility function and the slope of an isoquant of a production function, since in these situations we really are interested in which directions to move to keep the function constant. Recall that the level curve of a utility function U(x, y) is called an indifference curve of U. Its slope at (xo, yo) is called the marginal rate of substitution (MRS) of U at (*o, yo) since it measures, in a marginal sense, how much more of good y the consumer would require to compensate for the loss of one unit of good x to keep the same level of satisfaction. By the Implicit Function Theorem, the MRS at (xo, yo) is: (xx, yo) dx ду
Transcribed Image Text:This use of the Implicit Function Theorem is the natural approach when studying the slope of an indifference curve of a utility function and the slope of an isoquant of a production function, since in these situations we really are interested in which directions to move to keep the function constant. Recall that the level curve of a utility function U(x, y) is called an indifference curve of U. Its slope at (xo, yo) is called the marginal rate of substitution (MRS) of U at (*o, yo) since it measures, in a marginal sense, how much more of good y the consumer would require to compensate for the loss of one unit of good x to keep the same level of satisfaction. By the Implicit Function Theorem, the MRS at (xo, yo) is: (xx, yo) dx ду
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