1) Imagine that the utility a consumer derives from consuming goods A and B is given by the utility function U (QA,QB)= QAQB with total utility levels of 12. Furthermore, price of A equals birr 5 and consumer income equals birr 60.

Question

A. Draw the corresponding indifference curve and the budget line.
B. Calculate the marginal rate of substitution of B for A.
C. Determine the optimum level of consumption.

Answer 1

A. In order to draw the indifference curve and the budget line, we need to determine the combinations of goods A and B that give the consumer a total utility level of 12.

To find these combinations, we can set up the utility function equation as follows:
U(QA, QB) = QA * QB = 12

We can rearrange this equation to solve for QB:
QB = 12 / QA

Now, let's plot the indifference curve on a graph where the x-axis represents the quantity of good A (QA) and the y-axis represents the quantity of good B (QB).

- Start by assigning different values to QA (such as 1, 2, 3, etc.).
- Use the equation QB = 12 / QA to determine the corresponding values of QB.
- Plot the points (QA, QB) on the graph.
- Connect the points to create the indifference curve.

Next, let's draw the budget line. The budget line represents the combinations of goods A and B that the consumer can afford given their income and the prices of the goods.

The equation for the budget line is:
P_A * QA + P_B * QB = Income

In this case, the price of A (P_A) is birr 5 and the consumer income is birr 60.

- Let's assume that the price of B (P_B) is also birr 5 (for simplicity).
- Substitute the values into the budget line equation and solve for QB:
5 * QA + 5 * QB = 60
QA + QB = 12 - divide by 5
QB = 12/5 - QA/5

Again, assign different values to QA and use the budget line equation to determine the corresponding values of QB.

Plot the points (QA, QB) on the graph and connect them to create the budget line.

B. The marginal rate of substitution (MRS) of B for A measures how much of good A a consumer is willing to give up in order to consume one additional unit of good B while keeping their utility level constant.

In this case, the utility function U(QA, QB) = QA * QB implies that the MRS is the ratio of the marginal utility of good A to the marginal utility of good B.

To find the marginal utility of each good, we take the partial derivatives of the utility function with respect to each good:

MU_A = ∂U/∂QA = QB
MU_B = ∂U/∂QB = QA

The MRS can then be calculated as:
MRS = MU_A / MU_B = QB / QA

So, the MRS of B for A in this case is QB / QA.

C. The optimum level of consumption can be found at the point where the indifference curve is tangent to the budget line.

This occurs when the slope of the indifference curve (MRS) is equal to the slope of the budget line (the negative of the price ratio).

In this case, the MRS is QB / QA, and the price ratio of A to B is 5:5, or 1.

So, the optimum level of consumption is where QB / QA = 1, or QB = QA.