Calculate the marginal rate ofsubstitution for an arbitrary commodity bundle of the form (x, y) >>(0, 0) (that is, where x > 0 and y > 0) for each of the following utilityfunctions.(a) Quasi-Linear Preferences Example 1: U (x, y) = x + py.(b) Quasi-Linear Preferences Example 2: U (x, y) = x + ln (y).(c) Stone-Geary Preferences: U (x, y) = (x x 0 ) ↵ (y y 0 ) 1 ↵ ,where x 0 > 0, y 0 > 0, and ↵ 2 (0, 1) are fixed parameters.(d) Constant-Elasticity-of-Substitution (CES) Preferences:U (x, y) = (↵x ⇢ + y ⇢ ) 1⇢ , where x 0 > 0, y 0 > 0, and ↵ 2 (0, 1)are fixed parameters
Question
Calculate the marginal rate ofsubstitution for an arbitrary commodity bundle of the form (x, y) >>(0, 0) (that is, where x > 0 and y > 0) for each of the following utilityfunctions.(a) Quasi-Linear Preferences Example 1: U (x, y) = x + py.(b) Quasi-Linear Preferences Example 2: U (x, y) = x + ln (y).(c) Stone-Geary Preferences: U (x, y) = (x � x 0 ) ↵ (y � y 0 ) 1�↵ ,where x 0 > 0, y 0 > 0, and ↵ 2 (0, 1) are fixed parameters.(d) Constant-Elasticity-of-Substitution (CES) Preferences:U (x, y) = (↵x ⇢ + �y ⇢ ) 1⇢ , where x 0 > 0, y 0 > 0, and ↵ 2 (0, 1)are fixed parameters
Solution
The marginal rate of substitution (MRS) is the rate at which a consumer is willing to substitute one good for another while maintaining the same level of utility. It is calculated as the negative ratio of the marginal utility of one good to the marginal utility of the other good.
(a) For the utility function U(x, y) = x + py, the marginal utility of x is 1 and the marginal utility of y is p. Therefore, the MRS is -1/p.
(b) For the utility function U(x, y) = x + ln(y), the marginal utility of x is 1 and the marginal utility of y is 1/y. Therefore, the MRS is -y.
(c) For the Stone-Geary utility function U(x, y) = (x - x0)^α * (y - y0)^(1-α), the marginal utility of x is α/(x - x0) and the marginal utility of y is (1-α)/(y - y0). Therefore, the MRS is -α*(y - y0)/[(1-α)*(x - x0)].
(d) For the CES utility function U(x, y) = (αx^ρ + (1-α)y^ρ)^(1/ρ), the marginal utility of x is α*(αx^ρ + (1-α)y^ρ)^((1-ρ)/ρ)x^(ρ-1) and the marginal utility of y is (1-α)(αx^ρ + (1-α)y^ρ)^((1-ρ)/ρ)y^(ρ-1). Therefore, the MRS is -αx^(ρ-1)/[(1-α)*y^(ρ-1)].
Similar Questions
ind Jack’s marginal rate of substitution (MRS) at bundle (x, y).
Consider the following utility functions (in a world with Good X andGood Y ):I. U (x, y) = √xyII. U (x, y) = 4x + 3yIII. U (x, y) = min{x, y} (That is, U (4, 3) = 3, U (1, 1) = 1, U (2, 3) = 2, etc.)IV. U (x, y) = √x + y(a) Fill out the last 3 columns of Table 1. “MRS” stands for Marginal Rate of Substi-tution here. (You would need to copy this table into your answer.)(b) Fill out the the first 4 columns of Table 1 with a Yes/No entry for each cell. Justifyyour answers.(c) Do all utility functions display diminishing MRS? Justify your answers.Note: For the utility function U (x, y) = min{x, y}, calculus cannot be used. How-ever, think about the concept of MRS as how much of y you are willing to give upfor a bit more of x, as represented by the steepness of the indifference curve at apoint. Can you find some value for MRS in this case?(d) Sketch an indifference curve for each of the above utility functions.Table 1. Question 1: Properties of Common Utility FunctionsMonotone Strongly Monotone Convex Strictly Convex MU x MU y MRSI. No 12 x 12 y − 12II. Yes −4/3III. YesIV. Yes Yes1
Question 1Joe’s preferences are described by the following utility functionU (x, y) = xαyβwith α > 0 and β > 0.(a) Let I denote Joe’s income, and px and py denote the prices of good x and y, respectively.Find Joe’s optimal consumption bundle.
Joe’s preferences are described by the following utility functionU (x, y) = xαyβwith α > 0 and β > 0.(a) Let I denote Joe’s income, and px and py denote the prices of good x and y, respectively.Find Joe’s optimal consumption bundle.(b) Now, suppose α = 6, β = 2, px = 2, py = 3 and I = 24. Evaluate Joe’s optimal choice.(c) Suppose px increases by 50%. What is Joe’s new optimal consumption bundle? Calculateboth the Income Effect and the Substitution Effect
Jack consumes only Bread (good x) and Cheese (good y). His utility function is given byU (x, y) = ln(x) + ln(y)(a) Find Jack’s marginal rate of substitution (MRS) at bundle (x, y).(b) The price of good x is $6 and the price of good y is $3. Jack has an income of $120. Findhis utility-maximising consumption bundle.(c) Calculate Jack’s utility from the bundle you have found in part (b). (You can leave youranswer expressed in logarithm.).(d) In a graph with the quantity of x on the x-axis and the quantity of y on the y-axis, drawJack’s budget line, clearly marking the intercepts and the utility maximising bundle youhave solved in part (b) (call it bundle A). Also sketch an indifference curve passing throughbundle A.(e) Now the price of good y increases to $6 while the price of good x remains at $6. Jack’sincome remains at $120. Calculate his new utility-maximising consumption bundle.(f) On the diagram you have drawn for part (d), draw Jack’s new budget line, clearly markingthe intercepts and the new optimal bundle you have solved in part (e) (call it bundle B).Also sketch an indifference curve passing through bundle B.(g) Find the minimal expenditure required for Jack to achieve his original utility level (i.e.,your answer to part (c)) under the new prices. As you are finding this minimal expenditure,solve for the expenditure-minimising bundle that achieves the original utility level underthe new prices.(h) On the diagram you have drawn for parts (d) and (f), draw the budget line associatedwith the minimal expenditure you have solved in part (g), clearly marking the interceptsand the expenditure-minimising bundle you have solved in part (e) (call this bundle H).
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