Indicate the tutorial in which you are enrolled:
TUT0101 TUT0102 TUT0103 TUT0104 TUT0105 TUT0108
T1200 T1600 T1500 W0900 W1000 W1100
TUT0109 TUT0111 TUT0112 TUT0114 TUT0116 TUT0117
W1600 W1700 R1100 R1300 R1400 R1500
TUT0118 TUT0119 TUT0120 TUT0121 TUT0122 TUT0123
R1800 R1600 R1700 R1900 F1000 F1200
Answer the questions in the space provided. If you run out of room for an answer,
continue on the back side of the sheet or attach a sheet of paper detailing your
work, clearly indicating when you’ve done so. You are expected to show all work
unless otherwise indicated.
Due at the beginning of tutorial in the week of March 19
Student Name: Student Number:
You’re assembling the perfect trail mix and need to determine the optimal number of nuts to
add. You have the option between Almonds, Brazil Nuts, and Cashews, which you buy in quantities
x;y, z, at prices px;py; and pz, respectively. Your nut budget is M dollars.
1. [1 Mark] Determine an equation which describes your budget constraint.1
2. [1 Mark] De ne an appropriate constraint function g(x;y;z) so that your solution to part (1)
is written as g(x;y;z) = M. What is rg?
3. [1 Mark] You really like Brazil nuts, so your happiness (utility) with the mix is dictated by
f(x;y;z) = xy2z. Write down rf.
4. [1 Mark] Our goal is to maximize the utility function f subject to our budget constraint,
write down the three equations one derives from using the method of Lagrange multipliers.
1Note that we are assuming that x;y;z are continuous, so this should be an equality rather than an inequality.
Student Name: Student Number:
5. [7 Marks] Solve the optimization problem. State both the optimal mix of nuts and the
maximum utility.
Student Name: Student Number:
6. [4 Marks] Suppose that your budget is allowed to change, so your solution to the above
problem depends upon M; namely, you have three functions (x(M);y(M);z(M)). Recall
that the marginal utility of income is the rate of change of your utility with respect to your
budget. Show that the marginal utility of income is . Hint: This is always true, so you can
either show this is true for the example above, or you can do this abstractly.