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
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