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.
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.
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.
A. To draw the indifference curve and the budget line, we need to determine the quantities of goods A and B that the consumer can afford given their income and the price of A.
First, let's find the maximum quantity of A that the consumer can afford:
Maximum quantity of A = Income / Price of A
= 60 / 5
= 12
Next, let's plot the points on the graph. Since the utility function is U(Qa, Qb) = Qa * Qb, we can assign different values to Qa and Qb that will result in a total utility of 12. For example, if Qa = 1 and Qb = 12, or if Qa = 2 and Qb = 6, the total utility would still be 12.
Indifference curve: An indifference curve represents all the combinations of goods A and B that give the consumer the same level of utility. In this case, the utility is fixed at 12, so we can plot multiple points on the graph where the total utility is 12.
Budget line: The budget line represents all the combinations of goods A and B that the consumer can afford given their income and the prices of the goods. The budget line equation is given by:
Price of A * Quantity of A + Price of B * Quantity of B = Income
5 * Qa + Price of B * Quantity of B = 60
Plot the points where the budget line intersects with the indifference curve to see the optimum level of consumption.
B. The marginal rate of substitution (MRS) of B for A represents the rate at which the consumer is willing to give up units of A to obtain one more unit of B while remaining on the same indifference curve. Mathematically, it is given by:
MRS = - (MUa / MUb)
Here, MUa is the marginal utility of A and MUb is the marginal utility of B. In this case, the utility function is U(Qa, Qb) = Qa * Qb. Taking partial derivatives with respect to Qa and Qb, we find:
MUa = Qb
MUb = Qa
Therefore, the MRS of B for A is:
MRS = - (MUa / MUb) = - (Qb / Qa)
C. To determine the optimum level of consumption, we need to find the combination of goods A and B that maximizes the consumer's utility while staying within their budget constraint. This occurs at the point where the indifference curve is tangent to the budget line.
At this point, the slope of the indifference curve equals the slope of the budget line. The slope of the indifference curve is given by the MRS of B for A, which we calculated in part B. The slope of the budget line is equal to the price ratio:
Price of A / Price of B = 5 / Price of B
Setting the MRS equal to the price ratio and solving for quantities Qa and Qb will give us the optimum level of consumption.
(Qb / Qa) = 5 / Price of B
Using this equation, we can substitute the value of Qa from the budget line equation (5Qa + Price of B * Quantity of B = 60) and solve for Qb. This will give us the optimum level of consumption for goods A and B.
A. To draw the indifference curve and the budget line, we need to plot the utility function U(Qa, Qb) = Qa * Qb with a total utility level of 12.
Indifference curve: An indifference curve represents combinations of goods A and B that provide the same level of utility (in this case, 12). To draw the indifference curve, we can choose different combinations of Qa and Qb that result in a utility level of 12. For example:
Qa = 1, Qb = 12 (12 * 1 = 12)
Qa = 2, Qb = 6 (2 * 6 = 12)
Qa = 3, Qb = 4 (3 * 4 = 12)
Plotting these points on a graph will give us the indifference curve.
Budget line: The budget line represents the combinations of goods A and B that a consumer can afford given their income and the prices of the goods. In this case, the price of A is 5 birr and the consumer's income is 60 birr.
To find the maximum quantity of each good that the consumer can afford, we divide the income by the price of each good:
Maximum quantity of A = expenditure on A / price of A = (60 birr) / (5 birr) = 12
Maximum quantity of B = expenditure on B / price of B = (60 birr) / (Pb)
Now, we can plot the budget line by putting A on the horizontal axis and B on the vertical axis and connecting the points (0, (60/Pb)) and ((12/(Pb)), 0).
B. The marginal rate of substitution (MRS) of B for A represents the rate at which the consumer is willing to trade one unit of good A for good B while keeping the utility level constant. It can be calculated using the partial derivatives of the utility function:
MRS = - (∂U/∂Qa) / (∂U/∂Qb)
For this utility function, MRS = - (Qb/Qa) = - (Qa/Qb)
So, the marginal rate of substitution of B for A is constant and equals -1.
C. To determine the optimum level of consumption, we need to find the combination of goods A and B that maximizes utility while considering the budget constraint.
The optimal consumption point occurs where the indifference curve is tangent to the budget line. At this point, the marginal rate of substitution (MRS) is equal to the price ratio.
Since the MRS is constant and equal to -1, it should be equal to the price ratio:
MRS = - (Qa/Qb) = (price of A)/(price of B)
- (Qa/Qb) = 5/Pb
Solving for Qa:
Qa = -5(Qb/Pb)
Substituting Qa into the budget constraint:
5Qa + PbQb = 60
5(-5(Qb/Pb)) + PbQb = 60
Simplifying the equation gives us:
-25(Qb/Pb) + PbQb = 60
Now, we can solve this equation to find the optimal level of consumption.