Answer :
The teacher's Marginal Rate of Substitution (MRS) is -2, meaning they are willing to substitute 2 Workbook As for 1 Workbook B. The budget constraint is $100, and the demand functions for Workbook A and B follow from maintaining this budget while substituting according to the MRS.
Given:
Budget: $100
Cost of Workbook A: $4
Cost of Workbook B: $6
1. Understanding MRS:
MRS is the rate at which the teacher is willing to substitute workbook B for workbook A while maintaining the same level of utility.
Given that 2 Workbook As replace 1 Workbook B, we have:
MRS = -2
This is because giving up 1 Workbook B requires gaining 2 Workbook As to maintain utility.
2. Demand Function for Workbooks:
Let x1 be the quantity of Workbook A.
Let x2 be the quantity of Workbook B.
Our budget equation is: [tex]4x_1 + 6x_2 = 100[/tex]
Using the MRS of -2, we derive the demand functions:
From the budget constraint:
[tex]4x_1 + 6x_2 = 100[/tex]
Isolate x2:
[tex]x_2 = (100 - 4x_1) / 6[/tex]
The teacher's demand for Workbook A and B will adjust according to their prices maintaining the budget constraint of $100.
In conclusion, the Marginal Rate of Substitution (MRS) indicates that the teacher needs 2 workbooks of A to substitute 1 workbook of B. The demand functions derived are constrained by the given total budget of $100.
The teacher's MRS (Marginal Rate of Substitution) is 2, and the demand functions for workbook A and workbook B are [tex]\( x_A = \frac{100 - 6p_B}{4} \)[/tex] and [tex]\( x_B = \frac{100 - 4p_A}{6} \)[/tex] respectively.
Given that the teacher views workbook A and workbook B as substitutes and that two workbook As can replace one workbook B, we can infer that the teacher's utility is constant along a line where one workbook B is equivalent to two workbook As. This implies that the MRS, which is the rate at which the teacher is willing to substitute workbook A for workbook B and remain indifferent, is 2. Mathematically, the MRS is defined as the negative of the slope of the indifference curve, which in this case is the ratio of the marginal utilities of workbook A and workbook B:
[tex]\[ MRS = -\frac{MU_A}{MU_B} \][/tex]
Using the change definition of marginal utilities provided in the question, we have:
[tex]\[ MU_A = \frac{u(x_A + \Delta x_A, x_B) - u(x_A, x_B)}{\Delta x_A} \][/tex]
[tex]\[ MU_B = \frac{u(x_A, x_B + \Delta x_B) - u(x_A, x_B)}{\Delta x_B} \][/tex]
Since the teacher is indifferent between one workbook B and two workbook As, we can say that the utility gained from one additional workbook B is equal to the utility gained from two additional workbook As:
[tex]\[ u(x_A, x_B + \Delta x_B) = u(x_A + 2\Delta x_A, x_B) \][/tex]
Therefore, the MRS simplifies to:
[tex]\[ MRS = -\frac{MU_A}{MU_B} = -\frac{\Delta x_B}{\Delta x_A} = -\frac{1}{2} \][/tex]
The negative sign indicates that the teacher must give up workbook A to gain more of workbook B, but since we are considering the absolute value of the rate of substitution, we take the MRS as 2.
Now, to derive the demand functions for each workbook, we use the budget constraint:
[tex]\[ 4x_A + 6x_B = 100 \][/tex]
Solving for [tex]x_A[/tex] in terms of [tex]p_B[/tex] and the budget:
[tex]\[ x_A = \frac{100 - 6p_B}{4} \][/tex]
Similarly, solving for [tex]x_B[/tex] in terms of [tex]p_A[/tex] and the budget:
[tex]\[ x_B = \frac{100 - 4p_A}{6} \][/tex]
These demand functions show the quantity of each workbook that the teacher will demand at different prices, given the budget constraint of $100.
