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. To draw the indifference curve and the budget line, we need to find the combinations of goods A and B that yield a total utility of 12, and plot them on a graph.

We can start by setting up the equation for total utility:
U(Qa, Qb) = Qa * Qb = 12

Next, let's plot a few points on the graph using different combinations of Qa and Qb that satisfy the total utility equation. For example, we can use the following combinations:
(Qa, Qb) = (1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1)

Now, draw a curve passing through these points. This curve represents the indifference curve, which shows different combinations of Qa and Qb that yield the same level of utility (in this case, 12).

Next, let's plot the budget line on the same graph. The budget line represents the combinations of Qa and Qb that the consumer can afford given their income and the prices of the goods.

The price of A is birr 5, so the equation for the budget line can be written as:
Pa * Qa + Pb * Qb = Income
(5 * Qa) + (Pb * Qb) = 60

Let's assume the price of B is also birr 5 (Pb = 5) for simplicity. Then, the equation becomes:
(5 * Qa) + (5 * Qb) = 60
Qa + Qb = 12

Now, let's plot this equation on the graph. Connect the points (0,12) and (12,0) to form the budget line.

B. The marginal rate of substitution (MRS) of B for A can be calculated by the slope of the indifference curve at any given point.

Let's choose the point (Qa, Qb) = (3, 4) on the indifference curve. The slope of the indifference curve at this point tells us how many units of B the consumer is willing to give up to obtain an additional unit of A.

The slope can be calculated as:
MRS = -(∂U/∂Qa) / (∂U/∂Qb)

∂U/∂Qa = Qb
∂U/∂Qb = Qa

MRS = -(Qb/Qa) = -4/3

Therefore, the marginal rate of substitution of B for A is -4/3, or -1.33.

C. To determine the optimum level of consumption, we need to find the point on the budget line that is tangent to the highest indifference curve.

In this case, it is the point where the indifference curve with a total utility of 12 is tangent to the budget line. This point represents the highest level of utility that the consumer can achieve given their income and the prices of the goods.

From the graph, it appears that the tangent point is around (Qa, Qb) = (6, 2).

Therefore, the optimum level of consumption is Qa = 6 units of A and Qb = 2 units of B.