A consumer consuming two commodities X and Y has the following utility function U=X1.5Y.If the price of the two commodities are 3 and 4 respectively and his/her budget is birr 100.
Find the quantities of good X and Y which will maximize utility.
Total utility at equilibrium.
Find marginal utility of income and provide its interpretation
Find the at optimum point
Show optimum point graphically
To find the quantities of goods X and Y that will maximize utility, we can use the concept of consumer equilibrium, which states that the consumer's optimal choice occurs when the marginal utility per birr spent on each good is equal.
First, let's set up the problem. The consumer's utility function is given as U = X^1.5 * Y, and the prices of the two commodities are given as P_X = 3 and P_Y = 4. The consumer's budget is 100 birr.
To maximize utility, we need to find the combination of X and Y that satisfies the condition of marginal utility per birr being equal for both goods. Mathematically, this can be expressed as:
MU_X / P_X = MU_Y / P_Y,
where MU_X is the marginal utility of X, MU_Y is the marginal utility of Y, P_X is the price of X, and P_Y is the price of Y.
Differentiating the utility function with respect to X and Y, we can find the marginal utilities:
MU_X = 1.5X^0.5 * Y,
MU_Y = X^1.5.
Substituting these into the equilibrium condition, we have:
(1.5X^0.5 * Y) / 3 = (X^1.5) / 4.
This equation can be simplified to:
2X^0.5 * Y = X^1.5.
Now, let's solve for X and Y. Squaring both sides of the equation, we get:
4X * Y^2 = X^3.
Rearranging and simplifying, we have:
4Y^2 = X^2.
Taking the square root of both sides, we get:
2Y = X.
Next, we need to determine the budget allocation between X and Y. Let's assume the consumer spends an amount 'k' on X and '100 - k' on Y.
The price of X is 3, so the quantity of X purchased is given by 'k / 3'.
The price of Y is 4, so the quantity of Y purchased is given by '(100 - k) / 4'.
Substituting the earlier relation between X and Y, we have:
2Y = X,
2(100 - k) / 4 = k / 3.
Simplifying this equation, we get:
200 - 2k = 4k / 3,
3(200 - 2k) = 4k,
600 - 6k = 4k,
600 = 10k,
k = 60.
Therefore, the consumer should spend 60 birr on X and 40 birr on Y.
To find the total utility at equilibrium, we substitute the quantities of X and Y into the utility function:
U = X^1.5 * Y,
U = (60)^1.5 * 40,
U = 138240.
So, the total utility at equilibrium is 138240.
Now, let's find the marginal utility of income and interpret it. The marginal utility of income (MU_I) represents the additional utility gained from spending an additional unit of money. In this case, it can be calculated as the derivative of the utility function with respect to income (I):
MU_I = dU / dI.
Since the budget is fixed at 100 birr, the marginal utility of income becomes:
MU_I = dU / dI = 0.
This means that any additional income does not contribute to the consumer's utility in this scenario, as the consumer has already maximized their utility with the given budget.
Finally, let's graphically show the optimum point. We can plot a graph with X on the x-axis and U on the y-axis. We know that the relationship between X and Y is 2Y = X, which means that as X increases, Y also increases proportionally. However, we should also consider the budget constraint. Since the budget is limited to 100 birr, we can draw a line representing this constraint on the graph.
The optimum point will be the intersection of the utility function curve and the budget constraint line. The tangent line from the origin to this point represents the marginal rate of substitution (MRS), indicating the rate at which the consumer is willing to exchange X for Y to maintain the same level of utility.
Unfortunately, as a text-based AI, I am unable to provide a graphical representation. However, you can use the information provided to plot the graph yourself and locate the optimum point.
To maximize utility, we need to allocate the consumer's budget in a way that maximizes the utility function U(X, Y) = X^1.5 * Y.
Let's assume the consumer spends a certain amount of money on X, denoted by X* and the remaining budget on Y, denoted by Y*.
Step 1: Set up the problem
We have the following constraints:
1. The price of X is 3, so the budget spent on X, denoted by Px, is 3 * X*.
2. The price of Y is 4, so the budget spent on Y, denoted by Py, is 4 * Y*.
3. The total budget is 100, so we have the equation: Px + Py = 100.
Step 2: Substitute the constraints into the utility function
Replace X* and Y* in the utility function with Px/3 and Py/4, respectively:
U = (Px/3)^1.5 * (Py/4)
Step 3: Optimize the utility function
Take the derivative of the utility function with respect to Px and Py, and set them equal to zero to find the maximum point.
dU/dPx = 0 and dU/dPy = 0
dU/dPx = (1.5/3) * (Px/3)^0.5 * (Py/4) = 0
dU/dPy = (1/4) * (Px/3)^1.5 * (Py/4)^0.5 = 0
Solving these two equations will give us the values of Px and Py at the point of maximum utility.
Step 4: Solve the equations
dU/dPx = (1.5/3) * (Px/3)^0.5 * (Py/4) = 0
(1/2) * (Px/3)^0.5 * (Py/4) = 0
Since the second term is always positive, we conclude that Px = 0.
dU/dPy = (1/4) * (Px/3)^1.5 * (Py/4)^0.5 = 0
(1/4) * (0/3)^1.5 * (Py/4)^0.5 = 0
Since the first term is always zero, we can't determine the value of Py from this equation.
Step 5: Total utility at equilibrium
To find the total utility at equilibrium, substitute the values of Px and Py into the utility function:
U = (0/3)^1.5 * (Py/4)
U = 0
Step 6: Marginal utility of income and interpretation
The marginal utility of income measures the increase in utility when income increases by one unit. It can be calculated as the derivative of the total utility with respect to the total budget.
To find the marginal utility of income, take the derivative of the utility function with respect to the budget:
dU/d(Py/4) = 0/4 = 0
The interpretation of a marginal utility of income of zero is that the consumer's utility does not change when the budget changes. In other words, increasing the budget does not increase the consumer's satisfaction.
Step 7: Optimum point graphically
Since Px is zero, the optimum point graphically will be at the y-axis, where the consumer spends the entire budget on Y.
To find the quantities of goods X and Y that will maximize utility, we need to maximize the utility function U(X, Y) = X^1.5 * Y subject to the budget constraint.
Step 1: Set up the problem:
We have the utility function U = X^1.5 * Y and a budget constraint given by the equation P_X*X + P_Y*Y = I, where P_X and P_Y are the prices of X and Y respectively, and I is the total budget.
In this case, P_X = 3, P_Y = 4, and the budget I = 100.
Step 2: Substitute the budget constraint into the utility function:
Substituting the budget constraint, we have U = X^1.5 * Y = (100 - 3X - 4Y)^1.5 * Y.
Step 3: Maximize the utility function:
To maximize the utility function, we need to find the critical point by taking the partial derivatives of U with respect to X and Y and equating them to zero.
∂U/∂X = 1.5 * (100 - 3X - 4Y)^0.5 * (-3)Y + (100 - 3X - 4Y)^1.5 * (-4) = 0
∂U/∂Y = (100 - 3X - 4Y)^1.5 + 1.5 * (100 - 3X - 4Y)^0.5 * (-4)Y = 0
Solving these two equations simultaneously will give us the values of X and Y that maximize utility.
Step 4: Find the total utility at equilibrium:
To find the total utility at equilibrium, substitute the values of X and Y obtained from the previous step into the utility function U = X^1.5 * Y.
Step 5: Find the marginal utility of income:
The marginal utility of income measures how the utility of the consumer changes with a one-unit increase in income. It is the derivative of the utility function with respect to income (MUI = dU/dI).
Step 6: Interpret the marginal utility of income:
The interpretation of the marginal utility of income depends on its value. If MUI > 0, then an increase in income leads to an increase in utility, indicating that the consumer is in a state of diminishing marginal utility. If MUI = 0, then the consumer has maximized their utility given their current preferences and budget. If MUI < 0, then an increase in income leads to a decrease in utility, indicating that the consumer is already consuming beyond their optimal level.
Step 7: Find the equilibrium point graphically:
To show the optimum point graphically, plot the budget constraint on a graph with X on the horizontal axis and Y on the vertical axis. The optimal point is where the budget constraint is tangent to a utility curve. This tangent point represents the maximum level of utility achievable with the given budget.
By following these steps, you should be able to find the quantities of goods X and Y that maximize utility, the total utility at equilibrium, interpret the marginal utility of income, and illustrate the optimum point graphically.