A
State the reason behind the utility function.
A
Answer to Problem 2.10P
The decimal exponents resulting in
Explanation of Solution
The utility function used is under the Cobb Douglas form:
From the above function, it can be derived that
Similarly, in the other given problem also;
This result makes this function a special case.
B
Prove that the individual will spend a fraction of the income on commodity X and a fraction of income on commodity Y.
B
Answer to Problem 2.10P
While the fraction of income spent on commodity X is
Explanation of Solution
The Marginal Rate of Substitution for the utility function will be as follows:
Further,
The condition according to the utility maximization states that MRS equals to the ratio between the price of X and price of Y.
Now, rearrange it in terms of PXX,
Now, the budget constraint will be as follows:
Now, substituting the value of PXX in the above equation;
Now, substituting the value of PYY in the equation, derive
Thus, now the fraction of income spent on commodity X is
Introduction: The consumer waiving off a commodity in exchange for another good through maintaining the same level of utility is called the marginal rate of substitution.
C
Prove that the determined expenditure on commodity X does not alter.
C
Answer to Problem 2.10P
After careful observations, the fraction of income spent on commodity X does not alter and remains a.
Explanation of Solution
First, let us assume that the price of commodity X has changed to P`x.
With this value, the utility maximizing condition will be as follows:
Now, rearrange this condition in terms of P`xX;
The budget constraint function of the individual consuming commodities X and Y will be as follows:
Now, substitute this value of P`xX in the above equation;
Now, substitute this value of P`YY in the equation;
Now, the fraction of income spent on commodity X remains a, this means,
Introduction: Utility in Economics refers to the total satisfaction received by the individual from consuming goods and services in consideration. And hence, the utility level implies a direct influence on the demand and price of those goods and services in consideration.
D
Prove that the altering of price of Y does not have any impact on the quantity purchased of X.
D
Answer to Problem 2.10P
Since the fraction of income spent on commodity X remains a, the quantity purchased of X also does not change.
Explanation of Solution
Now, supposing that the price of commodity Y changes to PY1, the condition of utility maximization will be as follows:
As usual the budget constraint function;
Now, substitute the value of
Now, substitute this value of
Thus, since the fraction of income spent on commodity X remains a, the quantity purchased of commodity X does not change.
Introduction: Utility in Economics refers to the total satisfaction received by the individual from consuming goods and services in consideration. And hence, the utility level implies a direct influence on the demand and price of those goods and services in consideration.
E
Using this utility function prove that the income when doubled with no price changes of the commodities will lead to the purchases of these commodities being doubled.
E
Answer to Problem 2.10P
By observing the doubled fraction of income spent on commodities, the purchases of both X and Y being doubled.
Explanation of Solution
As usual the budget constraint function, with income being doubled;
Now, substitute the value of
Now, substitute this value of
Thus, since the fraction of income spent on commodities X and Y are doubled, the quantity purchased of commodities X and Y are doubled.
Introduction: Utility in Economics refers to the total satisfaction received by the individual from consuming goods and services in consideration. And hence, the utility level implies a direct influence on the demand and price of those goods and services in consideration.
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Chapter 2 Solutions
EBK INTERMEDIATE MICROECONOMICS AND ITS
- You have k20 per week to spend and two possible uses for the money: telephoning friends back home and drinking coffee. Each Hour of phoning costs k2 and each cup of coffee costs k1. Your utility function is U(X,Y)=XY,where X is the hours of phoning you do and Y the number of cups of coffee you drink. What are your optimal choices? What is the resulting utility levels? You can use the standard result on the constrained maximization of such a function, but must state in clearlyarrow_forwardConsider a hypothetical consumer named Hayden who is shopping for bread and brie. The graph with bread and brie on the axes presents the utility‑maximizing combinations of bread and brie that Hayden chooses when the price of bread is $1.00$1.00 per loaf and the price of brie is $4.00$4.00 and $6.00$6.00 per wheel, respectively. The other graph shows Hayden's demand curve for brie. The two points and associated values in the graph for bread and brie combinations correspond to points A and B in the graph of the demand curve for brie. What are the specific prices and quantities of brie associated with points A and B on Hayden's demand curve? price of brie at point A: $$ quantity demanded at point A: price of brie at point B: $$ quantity demanded at point B:arrow_forwardWhich of the following statements is true? Group of answer choices If the more is better assumption is satisfied, the marginal utility from consuming the second unit must be greater than the marginal utility from consuming the first unit. Marginal utility is the slope of total utility. If the more is better assumption is satisfied, total utility will increase as consumption increases. Marginal utility may be negative.arrow_forward
- Marina decides to purchase a ring made from an alloy composed exclusively of gold (G) and titanium (T). The price of gold is $60 per gram, and the price of titanium is $30 per gram. Her total budget for the ring is $600. Her utility function is given by U(G,T) =GT. Suppose the price of titanium falls to $20 per gram. At the final basket, the optimal amount of titanium is()grams.arrow_forwardJane receives utility from days spent traveling on vacation domestically (D) and days spent traveling on vacation in a foreign country (F), as given by the utility function U(D,F) = 10DF. In addition, the price of a day spent traveling domestically is $100, the price of a day spent traveling in a foreign country is $400, and Jane’s annual travel budget is $4000. Suppose F is on the horizontal axis and D is on the vertical axis. Jane's marginal rate of substitution between F and D is equal to 10 1 F/D D/Farrow_forwardI am unsure the direction the utility functions would go in , with this specific scenarioarrow_forward
- Suppose the function for the utility from good c is denoted as U(c)=2c1/2 . Which of the following expressions indicates the marginal utility of c?arrow_forwardExplain the technique of constrained optimization in relation to utility functions and how it relates to consumer equilibrium.arrow_forwardYou have £20 per week to spend, and two possible uses for this money: telephoning friends back home, and drinking coffee. Each hour of phoning costs £2, and each cup of coffee costs £1. Your utility function is U(X,Y) = XY, where X is the hours of phoning you do, and Y the number of cups of coffee you drink. What are your optimal choices? What is the resulting utility level? You can use the standard result on the constrained maximization of such a function, but must state it clearly. Now suppose the price of telephone calls drops to £1 per hour. What are your optimal choices? What is the resulting utility level? How much income per week will enable you to achieve the same quantities at the new prices as the ones you chose before? What income will enable you to attain the same utility as you did before? Comment on your answer in the context of equivalent variation and compensating variation.arrow_forward
- for which utility function recardian equivalence does not hold?arrow_forwardA consumer’s indirect utility function is given byarrow_forwardFor the utility function (x^3)(z^2)Provide a positive linear transformation of the utility function, then, provide a nonlinear monotone transformation of the utility function.arrow_forward