The table below shows the total utility associated with various units of goods X and Y. Table 6.5
Refer to Table 6.5.Given that the price of good X is $2 per unit, the price of good Y is $5 per unit, and that the consumer spends a total of $14 on both goods, what is the total utility associated with the utility-maximizing combination?

Question

The table below shows the total utility associated with various units of goods X and Y. Table 6.5
 Refer to Table 6.5.Given that the price of good X is $2 per unit, the price of good Y is $5 per unit, and that the consumer spends a total of $14 on both goods, what is the total utility associated with the utility-maximizing combination?

Quizplus · Accepted Answer

We can use the formula for calculating the total utility-maximizing combination:

MUx/Px = MUy/Py

where MUx and MUy are the marginal utilities of X and Y respectively, and Px and Py are the prices of X and Y respectively.

To find the marginal utilities, we need to look at the change in total utility when the consumption of each good increases by one unit.

For X: 
- The TU increases from 0 to 18 when consumption increases from 0 to 1. So, the marginal utility of X when one unit is consumed is 18.
- The TU increases from 18 to 29 when consumption increases from 1 to 2. So, the marginal utility of X when the second unit is consumed is 11.
- The TU increases from 29 to 35 when consumption increases from 2 to 3. So, the marginal utility of X when the third unit is consumed is 6. 
- The TU increases from 35 to 38 when consumption increases from 3 to 4. So, the marginal utility of X when the fourth unit is consumed is 3. 
- The TU increases from 38 to 40 when consumption increases from 4 to 5. So, the marginal utility of X when the fifth unit is consumed is 2.

For Y: 
- The TU increases from 0 to 23 when consumption increases from 0 to 1. So, the marginal utility of Y when one unit is consumed is 23.
- The TU increases from 23 to 41 when consumption increases from 1 to 2. So, the marginal utility of Y when the second unit is consumed is 18.
- The TU increases from 41 to 54 when consumption increases from 2 to 3. So, the marginal utility of Y when the third unit is consumed is 13. 
- The TU increases from 54 to 64 when consumption increases from 3 to 4. So, the marginal utility of Y when the fourth unit is consumed is 10. 
- The TU increases from 64 to 71 when consumption increases from 4 to 5. So, the marginal utility of Y when the fifth unit is consumed is 7.

Now, we can use the formula:

18/2 = MUy/5 
MUy = 90

So, when the consumer spends $2 on X and $12 on Y (i.e. buying 1 unit of X and 2 units of Y), the total utility is:

TUx + TUy = 18 + 41 = 59

If the consumer spends $4 on X and $10 on Y (i.e. buying 2 units of X and 2 units of Y), the total utility is:

TUx + TUy = 29 + 41 = 70

If the consumer spends $6 on X and $8 on Y (i.e. buying 3 units of X and 1 unit of Y), the total utility is:

TUx + TUy = 35 + 23 = 58

If the consumer spends $8 on X and $6 on Y (i.e. buying 4 units of X and 1 unit of Y), the total utility is:

TUx + TUy = 38 + 23 = 61

If the consumer spends $10 on X and $4 on Y (i.e. buying 5 units of X and 1 unit of Y), the total utility is:

TUx + TUy = 40 + 23 = 63

If the consumer spends $12 on X and $2 on Y (i.e. buying 6 units of X and 0 units of Y), the total utility is:

TUx + TUy = 40 + 0 = 40

Therefore, the total utility is maximized when the consumer spends $6 on X and $8 on Y (i.e. buying 3 units of X and 1 unit of Y), and the total utility at this point is 58 + 23 = 81.

The closest answer choice is C) 172, which is not the exact answer but gives us an indication that the total utility is likely above 81.

The Table Below Shows the Total Utility Associated with Various