18. Assume that a firm produces its product in a system described in the followi
ID: 1176812 • Letter: 1
Question
18. Assume that a firm produces its product in a system described in the following production function and price data:
q = 3X + 5Y + XY
PX = $3
PY = $6
Px and Py are prices per unit of the factors.
here X and Y are the quantities of the two variable in put factors employed in the production of the output, q. All quantities are measured in the number of units per month.
a. What are the optimal input proportions for X and Y in this production system? Is this combination rate constant regardless of the output level? why?
b. (The total cost function associated with the use of X and Y in the production of q is
TC=PxX+PyY or TC=$3X+$6Y) Use the lagrangian technique to deter mine the maximum output that the firm can produce operating under a $1,000 total cost per month constraint, for X and Y. What is the maximum output that can be produced under this cost constrant? Show that the inputs used to produce this level of output meet the optimality condition deived in part A.
c. How much additional output could be obtained from a $1 increase in the budget?
Explanation / Answer
A. Optimal input proportions are found by solving the following relation:
MPX
PX
=
MPY
PY
3 + Y
3
=
5 + X
6
18 + 6Y = 15 + 3X
X = 2Y + 1 or Y = (X - 1)/2
Because Q appears in neither of the above expressions, these optimal unit
proportions are invariant with respect to the output level.
B. The optimization problem faced by the firm can be written
Maximize Q = 3X + 5Y + XY,
Subject to $3X + $6Y = $1,000,
which suggests the following Lagrangian expression:
L
Q = 3X + 5Y + XY + %u03BB($1,000 - $3X - $6Y).
(1) %u039CL
Q/%u039CX = 3 + Y - 3%u03BB = 0
(2) %u039CL
Q/%u039CY = 5 + X - 6%u03BB = 0
(3) %u039CL
Q/%u039C%u03BB = 1,000 - 3X - 6Y = 0
To solve, take 2 times (1) minus (2):
2 %u0397 (1) 6 + 2Y - 6%u03BB = 0
minus (2) - (5 + X - 6%u03BB = 0)
1 + 2Y - X = 0.
Then, multiplying this result by 3 and adding (3) yields the following:
3 %u0397 (above) 3 + 6Y - 3X = 0
plus (3) 1,000 - 6Y - 3X = 0
1,003 - 6X = 0
X = 167.
Then, from (3):
(3) 1,000 - 3(167) - 6Y = 0
6Y = 499
Y = 83.
And, solving for %u03BB using (1):
(1) 3 + 83 - 3%u03BB = 0
3%u03BB = 86
%u03BB = 28.7.
Part A above showed that X and Y should be combined in the ratio
X = ? 2Y + 1
an
Thus, X and Y are combined in optimal proportions.
C. The incremental output obtainable from an additional $1 expenditure on X and Y is
28.7 units as determined by the value of %u03BB in Part B. This implies that the marginal
cost of one output unit is
MC = TC 1
=
Q
%u2202
%u2202 %u03BB
= $0.035 (or 3.5 cents per unit)
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