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.

Question

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

Answer 1

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.