You are choosing between two goods. X and Y. and your marginal utility from each

Question

You are choosing between two goods. X and Y. and your marginal utility from each is as shown below. If your income is $18.00 and the prices of X and Y are $4. 00 and $2.00, respectively, what quantities of each will you purchase to maximize utility? X = units. Y = units. What total utility will you realize? Assume that, other things remaining unchanged the price of X falls to $2.00. What quantities of X and Y will you now purchase? X = units. Y = units. Using the two prices and quantities for X, derive a demand schedule (a table showing prices and quantities demanded) for X.

Explanation / Answer

(a) Utility is maximized when (MUx / Price of X) = (MUy / Price of Y)

This condition holds true for following bundles:

(i) X = 2 (MUx / Px = 16 / 4 = 4) and Y = 5 (MUy / Px = 8 / 2 = 4)

(ii) X = 4 (MUx / Px = 8 / 4 = 2) and Y = 6 (MUy / Px = 4 / 2 = 2)

For bundle (i), Total cost = 2 x $4 + 5 x $2 = $(8 + 10) = $18 (budget is exhausted).

For bundle (ii), Total cost = 4 x $4 + 6 x $2 = $(16 + 12) = $28 (Budget is exceeded).

So, bundle (i) is optimal with X = 2 units, Y = 5 units.

(b) MU = Change in Total utility / Change in quantity

For X, when X = 2, Total utility = (MUx when X = 1 + MUx when X = 2) = 23 + 16 = 39

For Y, when Y = 5, Total utility = (MUy when Y = 1 + MUy when Y = 2 + MUy when Y = 3 + MUy when Y = 4 + MUy when Y = 5) = 18 + 16 + 14 + 10 + 8 = 66

Total utility = 39 + 66 = 105

(c) The bundles are:

(i) X = 2 (MUx / Px = 16 / 2 = 8) and Y = 2 (MUy / Py = 16 / 2 = 8).

Total cost = 2 x $2 + 2 x $2 = $(4 + 4) = $8 (underutilized budget, so sub-optimal)

(ii) X = 4 (MUx / Px = 8 / 2 = 4) and Y = 5 (MUy / Py = 8 / 2 = 4)

Total cost = 4 x $2 + 5 x $2 = $(8 + 10) = $18 (exhausted budget, so optimal)

(iii) X = 5 (MUx / Px = 4 / 2 = 2) and Y = 6 (MUy / Py = 4 / 2 = 2)

Total cost = 5 x $2 + 6 x $2 = $(10 + 12) = $22 (over the budget, so sub-optimal)

(d) Demand schedule

Price ($) Quantity demanded 4 2 2 4

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