Here are suggested exercises for the final. Some of them are hard (***). You can skip them if you struggle to
solve. Only some of the questions are provided with answers. The rest you need to work out by yourself.
Question 1
Use the Lagrangian Thm (if possible) to determine the optimal values for each constrained optimization problem’s choice variable and solve for the Lagrange multiplier by doing the following: i) State the maximization
problem showing the choice variables clearly ii) Check the constraint qualification iii) Write down the Lagrangian and find the critical points iv) Find the maximum and minimum points using objective function
evaluation. v) Write the bordered Hessian and verify your results by finding about the definiteness of the
bordered Hessian
(i) max (1/2)a2 + 4b2 − 4a + 8c2 subject to 12 a + 3b + c = 25.
{a,b,c}
(ii) max ln(x + y) subject to xy = 16.
{x,y}
Question 2
Consider the following function f : R2 → R , f (x, y) = x4 − y 4 . Consider the following optimization problem
min f (x, y) subject to (x − 5)3 − y 4 ≥ 0
{x,y}
i) Check the constraint qualification ii) Write down the Lagrangian and find the critical points using KuhnTucker Theorem iv) Find the maximum and minimum points using objective function evaluation.
Question 3
Consider a hungry economics student trying to decide what to each for lunch. Her three-item lunch problem is
to maximize her utility where her utility function is given by U (S, V, J) = 31 ln(S) + 13 ln(V ) + 13 ln(J) and the
constraints are S/4 + V /2 + J/12 ≤ 6 and S + J ≤ 40 where S represents ounces of soup, V represents ounces
of salad, and J represents ounces of juice.
a) Check the constraint qualification for Kuhn-Tucker theorem
b) Find the solution if it exists.
Question 4
Given prices, px , py Income, I and commodities, x, y and a utility function u(x, y) = a ln x + b ln y with parameters a and b. A utility maximization problem can be expressed as:
max a ln x + b ln y subject to px x + py y = I
{x,y}
a) Check the constraint qualification for Lagrange theorem. What is a requirement on parameters for a
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Lagrangian solution to exist.
b)Find a solution to the above problem
Question 5
Fatmagul has a budget of 40YTL to spend on chocolate (C) and hazelnuts (H). Her utility function is given
by U (C, H) = 41 ln C + 12 ln H
Price of a bar of chocolate is Pc = $8 and the price of a package of hazelnuts is Ph = $5.
Suppose the budget of Fatmagul increases by 1 dollar. What’s the increase in the maximized value of her
utility?
Question 6
Consider the following function f : R2 → R , f (x, y) = x − y. Consider the following optimization problem
min f (x, y) subject to (x − 5)4 + (y − 5)4 ≤ 5
x,y
a) Check the rank condition (constraint qualification) for Kuhn-Tucker Method for this constrained optimization problem. Does it hold or not?
b) Can we use Kuhn-Tucker procedure for this problem?
c) Find the solution to the minimization problem.
Question 7
Solve the constrained maximization problem
0.2 0.2
max x0.5
1 x2 x3 s.t 2x1 + 3x2 = 100 , x2 + 4x3 = 20
{x,y}
Question 8
Find the maximum value of the function
f (a, b) = 2a + b
subject to constraints
2a + b ≤ 9
a + b ≥ 16
2
Question 9
As financial advisor to the The Journal of Important Stuff, you need to determine the effect on sales of the
number of pages devoted to important stuff about economics (E), and (un)important stuff about everything
else (U). After careful consideration, you decide that the function describing the relationship between sales (S)
and number of pages on economics and other stuff is
S = 100U + 310E − (1/2)U 2 − 2E 2 − U E
What is your recommendation for the number of pages devoted to economics articles and the number of pages
devoted to articles on other topics in order to maximize sales?
∂S
∂E
= 310 − 4E − U = 0
∂S
∂U
= 100 − U − E = 0
Solving jointly : 210 = 3E, E = 70 and U = 30
Question 10: Difference Equations
i) Solve the following difference equation
xt = 14 xt−1 + 12 xt−2 + 15 given x0 = 0 and x1 = 1
Particular solution:
x̄ = 41 x̄ + 12 x̄ + 15
x̄ = 60
Homogenous solution:
The homogenous equation: xt − 14 xt−1 − 12 xt−2 = 0
Guess:
xt = k t
k t − 41 k t−1 − 12 k t−2 = 0
k t−2 (k 2 − 14 k − 12 ) = 0
k1,2 =
√
−b± b2 −4ac
2a
=
1
8
√
±
33
8
Guess:
xt = Ak1t + Bk2t
Our solution is the summation of the particular and the homogenous solution:
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xt = Ak1t + Bk2t + 60
xt = A( 81 +
q
33 t
8 )
+ B( 18 −
√
33 t
8 )
+ 60
we have to find A and B using initial conditions:
x0 = A( 81 +
q
33 0
8 )
+ B( 18 −
√
33 0
8 )
+ 60 = 0
A = −60 − B (1)
and
x1 = A( 18 +
q
33 1
8 )
+ B( 18 −
√
33 1
8 )
+ 60 = 1(2)
Solving (1) and (2) together
A=
√4
33
− 30
B = −30 −
√4
33
Adding the particular solution we obtain the general solution:
xt = ( √433 − 30)( 18 +
√
33 t
8 )
+ (−30 −
√4 )( 1
33 8
√
−
33 t
8 )
+ 60
You can verify that x0 = 0 and x1 = 1 to check the solution is correct.
Indeed, if we substitute t = 0 we get x0 = 0 and for t = 1 we get x1 = 1
Question 11
Solve the following difference equation
xt = 98 xt−1 −
7
81 xt−2
+
32
9
and x0 = 3, x1 = 5
DO IT YOURSELF
ANSWER: xt = 53 ( 79 )t − 83 ( 19 )t + 4
Question 12***
In the Coco Republic the population is 5 and everybody is eligible to vote. The government consists of two people;
the prime minister and the finance minister. The remaining three people are unemployed. There are no other jobs.
In fact , the only way to get employed is to form a political party and to win election and then hold office as prime
minister or the finance minister. In order to get elected a party has to win at least 50% of the votes. The number
of votes a party can win, V, depend on the number of unemployed people, U , and the number of people eligible to
P − U2 P ≥ U2
vote, P , .in the following manner. V =
0 otherwise
i) Can the government win the election?
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ii) What’s the maximum rate of unemployment among voters (i.e
still win the election
U
P)
in Coco, below which the government can
i) No, because the number of votes under current conditions is zero (P = 5 < 9 = U 2 )
NOTE: If P ≥ U 2 We had to check the following condition to decide on if the government could win the election.
P − U2 >
P
2
P
2
> U2
P > 2U 2
ii) P − U 2 >
P
2
1
2P
P
2
> U2
2
> (U
P)
√1
2P
√1
> (U
P ) > − 2P
So the maximum unemployment level is
√1
2P
Ex: If the country had 50 people eligible to vote, the max unemployment would be 10 %
Question 13
Solve the constrained maximization problem.
0.2 0.2
max x0.5
1 x2 x3 s.t 2x1 + 3x2 = 100 and x2 + 4x3 = 20
x1 ,x2 ,x3
Question 14
Consider the three-item lunch problem where the utility function is U (S, V, J) = 1/3ln(S) + 1/3ln(V ) + 1/3ln(J)
but the constraints are S/4 + V /2 + J/12 ≤ 6 and S + J ≤ 40 where S represents ounces of soup, V represents ounces
of salad, and J represents ounces of juice.
(a) Set up the Lagrangian function.
(b) Find the optimal solution for this problem.
Question 15
Fatmagul has a budget of 40YTL to spend on chocolate (C) and hazelnuts (H). Her utility function is given by
U (C, H) = 41 ln C + 21 ln H
Price of a bar of chocolate is Pc = $8 and the price of a package of hazelnuts is Ph = $5.
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i) Set up budget constraint.
ii) Set up the Lagrangian function for this maximization problem.
iii) Find the utility maximizing combination of chocolate and hazelnuts
iv) Suppose the budget of Fatmagul increases by 1 dollar. What’s the increase in the maximized value of her utility?
(Hint: Do not resolve the problem from beginning)
Question 16
You are preparing for the midterm exams of EC223 and EC233. You estimate that the grades
√ you will obtain in
each course, as a function of the amount of time spent working on them are gec233 = 20 + 20 tec233
gec223 = −80 + 3tec223 where gi is the grade in course i and ti is the number of hours per week spent in studying
for course i. You wish to maximize your grade average (gec223 + gec233 )/2. You cannot spend in total more than 20
hours studying these courses in the week (perhaps because of the extra burden of EC205 on your poor shoulders?).
Find the optimal values of tec223 andtec233 and discuss the characteristics of the solution. (Since EC233 is a harder
course, can you expect tec233 > tec223 ?)
Question 17
i ) Find the solution to
β
max xα
1 + x2 s.t.I − p1 x1 − p2 x2 ≥ 0, x1 ≥ 0, x2 ≥ 0
x1 ,x2
where α > 0, β > 0, pi > 0 and I > 0 are given.
ii) Let p1 = p2 = 2, and α = 0.8 and β = 0.7 What are the optimal levels of x1 , x2 ?
iii) Let p1 = 2, p2 = 1, and α = 0.8 and β = 0.8 What are the optimal levels of x1 , x2 ?
Question 18***
A firms inventory of a certain commodity I(t) is depleted at a constant rate of dI
dt per unit of time. The firm
reorders an amount of x which is delivered immediately whenever the inventory is zero. The annual demand for the
commodity is A which the firm orders at n installments so that A = nx. The firm incurs two types of costs a unit
holding cost, Ch and the cost of placing an order (no matter what amount), C0 . At any time the average stock is
x/2 so that the cost of holding is Ch x/2 and the ordering cost is C0 n.
i) In a diagram show how the inventory level varies over time. Prove that the average stock of inventory is x/2
ii) Minimize the the total cost of inventory by choice of how much to stock (x) and how many times to order (n)
subject to constraint A = nx Express the solution as a function of parameters C0 , Ch and A
Question 19
A firm produces a single output y using three inputs x1 , x2 and x3 all of which are nonnegative to produce output,
y given the following technology:
y = g(x1 , x2 , x3 ) = x1 (x2 + x3 )
The unit price of y is py > 0 while the price of input i is wi > 0 for i = 1, 2, 3.
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i) Describe the firms profit maximization problem and derive the critical points using the Lagrangian method. ii)
Show that Lagrangean has multiple critical points for any choice of (py , w1 , w2 , w3 , w4 ) ∈ R4++ iii) Show that none
of these critical points identify a solution for the profit maximization problem.
Question 20
Find the maxima and minima of the following functions that are defined on Rn :
i)f (x, y) = xy s.t. x2 + y 2 = 2a2
ii) f (x, y, z) = xyz s.t. x + y + z = 5 and xy + xz + yz = 8
iii) f (x, y) =
(x+y)
y
s.t. xy = 36
Question 21
Review Eigenvalues, eigenvectors, first and second-order linear difference equations, solve the exercises that were
assigned in class.
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