A consumer consuming two commodities X and Y has the following utility function .If the price of the two commodities are Birr 1 and 4 respectively and his/her budget is birr 10.
Find the quantities of good X and Y which will maximize utility.
Total utility at optimum point
Find the at optimum point
Show optimum point graphically
To find the quantities of goods X and Y that will maximize utility, we need to determine the optimal combination of X and Y that maximizes total utility, given the budget constraint.
The utility function of the consumer is not provided in the question, so we cannot calculate the specific quantities or total utility at the optimum point without this information.
However, we can still describe the general steps to find the optimum point and draw a graphical representation:
1. Set up the budget constraint equation:
The budget constraint equation is given by: Px*X + Py*Y = Income
In this case, the price of X is Birr 1 and the price of Y is Birr 4. The total income is Birr 10.
Therefore, the equation becomes: 1*X + 4*Y = 10.
2. Determine the utility function:
The utility function specifies the relationship between the consumption of X and Y and the consumer's level of satisfaction. Without the utility function, we cannot proceed with the calculation.
3. Transform the budget constraint into a simplified equation:
In this case, we can rewrite the budget constraint as: X = 10 - 4Y.
4. Substitute the simplified equation into the utility function:
Using the utility function, we can replace X with (10 - 4Y). This will give us a utility function in terms of Y only.
5. Maximize utility by finding the first-order condition:
Taking the derivative of the utility function with respect to Y and setting it equal to zero will give us the first-order condition. Solving this equation will yield the value of Y that maximizes utility.
Without the specific utility function, we cannot proceed with the steps mentioned above or calculate the quantities and total utility at the optimum point.
To show the optimum point graphically, we can plot the budget constraint line (1*X + 4*Y = 10) on a graph with X and Y as the two axes. Then, we can plot indifference curves representing different levels of utility and identify the point where the budget constraint line is tangent to the highest indifference curve. This point represents the optimum point. However, without the specific utility function, we cannot plot the indifference curves or identify the optimum point on the graph.
To find the quantities of goods X and Y that maximize utility, we need to use the concept of marginal utility and the budget constraint.
The utility function is not provided in the question, so we cannot determine the exact quantities that will maximize utility or the total utility at the optimum point. However, we can explain the general steps to find the optimal quantities and total utility.
Step 1: Determine the marginal utilities for each commodity.
The marginal utility represents the additional utility gained from consuming an additional unit of a commodity. It helps us understand how the consumer values each unit.
Step 2: Calculate the price ratio.
The price ratio is the ratio of the prices of the two commodities (P_X / P_Y). In this case, the price ratio is (1 / 4) = 0.25.
Step 3: Compare the marginal utility per birr for each commodity.
Calculate the marginal utility per birr for each commodity by dividing the marginal utility of X by its price (MU_X / P_X) and the marginal utility of Y by its price (MU_Y / P_Y). Compare these ratios.
Step 4: Allocate the budget such that the marginal utility per birr is equal for both commodities.
To maximize utility, the consumer should allocate the budget such that the marginal utility per birr is equal for both commodities. This is known as the rule of equal marginal utility per birr.
Step 5: Calculate the quantities of X and Y at the optimum point.
Once the marginal utility per birr is equal, determine the quantities of X and Y that can be purchased within the budget. The consumer should spend the entire budget.
Step 6: Calculate the total utility at the optimum point.
The total utility at the optimum point can be calculated by plugging in the quantities obtained in step 5 into the utility function. However, since the utility function is not provided, we cannot give an exact value.
To show the optimum point graphically, we would need to plot the indifference curves (representing different levels of utility) and the budget constraint line (representing the combinations affordable with the given budget). However, without the specific utility function, we cannot accurately depict the graphical representation.
To find the quantities of goods X and Y that maximize utility, we need to maximize the consumer's utility function subject to the budget constraint.
The utility function is not provided in the question, so we will work with a general utility function to explain the process. Let's assume the utility function is given by U(X, Y) = X^α * Y^β, where X represents the quantity of good X consumed, Y represents the quantity of good Y consumed, and α and β are positive constants.
Step 1: Set up the Problem
The objective is to maximize the utility function U(X, Y) subject to the budget constraint.
Maximize: U(X, Y) = X^α * Y^β
Subject to: PₓX + PᵧY = M, where Pₓ and Pᵧ are the prices of goods X and Y, respectively, and M is the consumer's budget.
In the given question, Pₓ = 1, Pᵧ = 4, and the budget M = 10.
Step 2: Rewrite the Constraint in terms of one variable
We can rewrite the budget constraint as Y = (M - PₓX) / Pᵧ. Substituting the given values, we have Y = (10 - X) / 4.
Step 3: Substitute the rewritten constraint into the utility function
Now, substitute the expression for Y from Step 2 into the utility function from Step 1, and rewrite it in terms of one variable.
U(X) = X^α * [(10 - X) / 4]^β
Step 4: Maximize the Utility Function
To maximize the utility function, we can take the derivative of U(X) with respect to X, set it equal to zero, and solve for X.
dU(X) / dX = 0
Solve for X using the first-order condition.
Step 5: Find the optimal quantities and total utility
Once we find the optimal value of X, we can substitute it back into the rewritten constraint equation from Step 2 to find the corresponding value of Y. These quantities of goods X and Y will maximize utility.
To find the total utility at the optimum point, substitute the values of X and Y into the utility function.
Step 6: Plotting the Optimum Point Graphically
To show the optimum point graphically, plot the budget constraint line and the level curves of the utility function. The point where the budget constraint is tangent to a level curve represents the optimum point of consumption.
Without knowing the specific utility function or the values of α and β, we cannot provide numerical results. However, by following the steps outlined above, you can find the optimal quantities of good X and Y, calculate the total utility at the optimum point, and graphically represent the optimum point in the X-Y plane.