Question 6 (25 marks). Carmen Sandiego receives utility from days spent travelin

Question

Question 6 (25 marks). Carmen Sandiego receives utility from days spent traveling domestically (D) and days spent traveling internationally (F) as given by the utility function 22d U(D;F) = D1 F 1 . The price of a day of traveling domestically is P = $240 and in a foreign country Pf = $320. Carmen Sandiego’s annual budget is $9600. (a) Find Carmen Sandiego’s utility by maximizing the choice of domestic and foreign travel. Find her utility consuming that bundle. (5 marks) (b) Suppose the price of domestic travel increases to PD = $300 per day. Find the new optimal bundle at this price as a function of her budget for traveling I: (5 marks) (c) Find the income needed for Carmen Sandiego to reach the same utility as before the price change. (5 marks) (d) Compute the quantities demanded with the new prices and the new income you found in part c) (3 marks) (e) Compute the quantities demanded with the new prices and the original income. (f) Decompose the changes in the quantity of D due to the income and substitution effects; illustrate with a graph. Give definitions of what the income and substitution effects mean. (5 marks)

Explanation / Answer

finding
D1 and F1 by maximising utility
L=D1F1-u(240D1+320F1-9600)
dL/dD1=F1-240u
F1=240u
and
dL/dF1=D1-320u
D1=320u
dL/du=240D1+320F1-9600
solving
240(320u)+320(240u)=9600
u=4800/(240X320)
u=1/16
F1=240(1/16)=15
D1=320/16=20
the utility is
U(D;F)=D1F1
=15X20=300 unit
b)
increasing the price puting above result in budget constraint
new u= 4800/(300X320).....1
u=1/20
new F1=300/20=15
new D1=320/20=16
new utility bundle=15X16=240
c)
D1F1=300
320uX300u=300
u=0.05590169943
puting this in equation 1
0.05590169943=M/2(300/320)
new M=10733.126292
d)
we can find the quantities
by taking new u value and puting that in the above F1 and D1
F1=300X0.05590169943
=16.7705098312
D1=320X0.05590169943=17.88854382

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