Question

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?

Answer 1

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.