Question

In: Economics

Lucy has just been promoted to a managerial position and given a new office. She is...

Lucy has just been promoted to a managerial position and given a new office. She is very fond of small Persian carpets and Native American paintings. Her utility function for carpets(x) and paintings (y) is given by: U = 100 * sqrt(xy)

Her company gave her a budget of $7,200. Persian carpets cost $700 each and Native painting costs $500. Using Excel spreadsheet, develop columns for quantity and of carpets and paintings within her budget. Use 0,1,2,....10 carpets as possible quantities. Develop columns of TU, MU, and MU/$ for carpets and paintings and answer the following questions.

At the maximum total utility level, what is the optimal number of carpets she gets and what is the optimal number of paintings she gets?

Solutions

Expert Solution

Quantity

Persian Carpets

Native Paintings

Total utility

Marginal utility for carpets

Marginal utility for paintngs

Mu of carpets/$

Mu of paintings/$

0

10

0.00

-

0

-

0

1

9

300.00

150

16.6666667

0.214286

0.3

2

8

400.00

100

25

0.285714

0.4

3

7

458.26

76.37626

32.7326835

0.327327

0.458258

4

6

489.90

61.23724

40.824829

0.349927

0.489898

5

5

500.00

50

50

0.357143

0.5

6

4

489.90

40.82483

61.2372436

0.349927

0.489898

7

3

458.26

32.73268

76.3762616

0.327327

0.458258

8

2

400.00

25

100

0.285714

0.4

9

1

300.00

16.66667

150

0.214286

0.3

10

0

0.00

0

-

0

-

The components of table can be described as follows:

  • First two columns represent the quantity of carpets and paintings.
  • Third column shows the total utility derived by substituting the values of carpets (X) and paintings(y) in the utility function i.e.

                                                                    U = 100*sqrt(xy)

  • Fourth column shows the marginal utility of carpets derived by differentiating total utility with respect to carpets. The function for the same is

                

  

And substituting the quantities of x and y.

  • Fifth column shows the marginal utility of paintings derived by differentiating total utility with respect to painting. The function for the same is

                

  

And substituting the quantities of x and y.

  • Sixth and seventh columns are derived using marginal utility of x and y from previous columns and dividing it by dollars spent on acquiring those quantities.

The green color shows the maximum total utility that can be attained by Lucy for given set of possible quantities. At maximum total utility, the number of carpets demanded are 5 and number of paintings are 5.


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