CONSTRAINED OPTIMIZATION: Lagrange multipliers
Case problems
1. A monopolistic producer of two goods A and B faces the following demand
functions:
QA= 50 - PA - QB and QB= 25 – 0.25PB – 0.25QA , where QA and QB denote amounts of
production for A and B, and PA and PB - their prices respectively.
The constant cost of production amounts to $70. Production of every unit of A demands
additional cost of $5, and every unit of B – additional cost of $10.
a) What maximum profit can be achieved under these conditions? What amounts of A and
B goods shall be produced then?
Answer: Q*A=15, Q*B=7.5 Π*= $605
b) Say, company intends to spend just $170 on production of A and B. What maximum
profit can be achieved then?
Answer: Q.A=10, Q.B =5 Π** = $530, and λ=3
c) What approximate increase in profit could expect producer in the latter case if
additional $2 were allocated for A and B production?
Answer: Since λ=3 and ΔΠ≈ λ x ΔM, ΔΠ≈ 3 x 2= $6
2. A monopolistic producer of two goods X1 and X2 has the following joint cost
function:
C= 10(Q1+Q2)+Q1Q2, where Q1 and Q2 denote the quantities of the X1 and X2
respectively.
The demand functions for these two goods are identified as
P1 = 50 - Q1 + Q2 and P2 = 30 + 2Q1 - Q2.
The firm has been contracted to produce a total amount of 15 goods of either type. What
would be the optimal production quantities of each good?
Answer: Q*1 =10, Q*2=5, and λ=30
If the production quota is raised by 2 units, what profit increase it would entail?
Answer: Since λ=30 and ΔM=+2, ΔΠ≈ 30x 2= 60
3. A firm which manufactures speciality bicycles has the following profit function
Π = 5X2 + 3Y2 - 10XY +240X, where X denotes the number of frames and Y
denotes the number of wheels.
Find the profit maximizing number of bicycles, assuming that this firm does not want
any spare parts (frames or wheels) left at the end of the production run.
Answer: X =40, Y=80, that is 40 bicycles should be produced.
Tip: Constraint here should be Y=2X (i.e. Y-2X=0), since the number of wheels should
be twice as much as number of frames (because every frame has to be equipped with
two wheels – in case you never saw a bicycle 😊).
