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.
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. 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.
A. To draw the indifference curve and budget line, we need to determine the different combinations of goods A and B that give a total utility level of 12 and the combinations that the consumer can afford given the price of A and the consumer income.
First, we need to find the combinations that give a total utility level of 12. Since the utility function is U(QA, QB) = QA * QB, we can set up the equation QA * QB = 12 and solve for QB:
QB = 12 / QA
Now, we can plot a few points on the graph with different combinations of QA and QB that satisfy the equation:
QA | QB
---------------
1 | 12
2 | 6
3 | 4
4 | 3
Next, we need to determine the budget line. The budget line represents the different combinations of goods A and B that the consumer can afford given their income and the price of A.
The equation for the budget line is: PA * QA + PB * QB = M, where PA is the price of A, PB is the price of B, QA is the quantity of A consumed, QB is the quantity of B consumed, and M is the consumer's income.
In this case, PA = 5, QA = x (variable), PB = 1 (based on the utility function since prices are not specified), and M = 60. Plugging these values into the equation, we get:
5x + QB = 60
Solving for QB, we get:
QB = 60 - 5x
Now, we can plot a few points on the graph with different combinations of QA and QB that satisfy the budget constraint:
QA | QB
---------------
0 | 60
12 | 0
10 | 10
6 | 30
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 an additional unit of B while keeping the same level of utility.
The MRS can be calculated using the formula: MRS = MUa / MUb, where MUa and MUb stand for marginal utilities of A and B, respectively.
In this case, the utility function is U(QA, QB) = QA * QB. To find the marginal utilities, we take partial derivatives of the utility function with respect to each good.
MUa = ∂U/∂QA = QB
MUb = ∂U/∂QB = QA
Now we can calculate the MRS:
MRS = MUa / MUb = (QB) / (QA)
Substituting QB = 12 / QA into the MRS formula, we get:
MRS = (12 / QA) / QA = 12 / (QA^2)
C. To determine the optimum level of consumption, we need to find the point where the consumer's indifference curve is tangent to the budget line. At this point, the consumer maximizes their utility while fully exhausting their income.
Based on the graph, we can see that the indifference curve is convex, and the budget line is straight. Therefore, the optimum level of consumption is the point where the tangency occurs.
By comparing the equations of the indifference curve and the budget line, we have:
QB = 12 / QA
QB = 60 - 5QA
Setting these two equations equal, we have:
12 / QA = 60 - 5QA
Rearranging this equation, we get:
5QA^2 - 60QA + 12 = 0
Solving for QA using quadratic formula, we find two solutions: QA = 2.33 and QA = 2.14.
Substituting these QA values back into either of the equations, we can find the corresponding QB values.
Therefore, the optimal level of consumption is:
QA = 2.14 and QB = 12 / 2.14
A. To draw the indifference curve and the budget line, we need to plot the combinations of goods A and B that yield the same level of utility and the combinations that the consumer can afford with their given income and prices.
1. Indifference Curve:
The utility function U(QA,QB) = QA * QB represents a Cobb-Douglas utility function, which implies that the consumer's preferences exhibit constant elasticity of substitution between goods A and B. This means that the consumer is willing to trade off one unit of good A for a fixed proportion of good B.
To draw the indifference curve, we need to find different combinations of QA and QB that lead to a total utility level of 12. For example, when QA = 1, QB = 12, or when QA = 2, QB = 6.
Plotting these combinations on a graph with QA on the x-axis and QB on the y-axis, and connecting these points, we can draw the indifference curve. The shape of the indifference curve will depend on the specific values chosen for QA and QB.
2. Budget Line:
The budget line represents all the combinations of goods A and B that the consumer can afford given their income and prices of the goods. The equation for the budget line is given by:
Price of A * QA + Price of B * QB = Income
In this case, the price of A is 5 birr, the price of B is unknown, and the consumer's income is 60 birr. Let's assume the price of B is 1 birr for simplicity.
The equation for the budget line will be:
5 * QA + 1 * QB = 60
Plotting this equation on the same graph as the indifference curve, with QA on the x-axis and QB on the y-axis, we can draw the budget line.
B. The marginal rate of substitution (MRS) of good B for good A is defined as the rate at which the consumer is willing to give up good A for an additional unit of good B while keeping the utility constant.
In this case, the utility function U(QA,QB) = QA * QB implies that the marginal rate of substitution is given by the ratio of the marginal utilities of the two goods:
MRS = MU(QA) / MU(QB)
The marginal utility of a good is the derivative of the utility function with respect to that good. In this case, the marginal utility of good A (MU(QA)) is equal to QB, and the marginal utility of good B (MU(QB)) is equal to QA.
Therefore, the MRS of B for A is equal to QB / QA.
C. To determine the optimum level of consumption, we need to find the point where the indifference curve is tangent to the budget line. At this point, the consumer is maximizing their utility given their income and prices.
Visually, the optimum level of consumption is where the indifference curve and the budget line touch each other.
To find the exact coordinates of this point, we can solve the equations of the indifference curve and the budget line simultaneously. This will give us the specific quantities of goods A and B that the consumer should consume to maximize their utility.
Once we have the quantities, we can substitute them back into the utility function to calculate the maximum utility level.