before the price ted by the price increase of Good ) e University wants to minim
ID: 1130737 • Letter: B
Question
before the price ted by the price increase of Good ) e University wants to minimize the cost of creating 100 pamphlets. Ther ction is given by: q = 0.513 K4. The wage rate is $1.00 and the rental ose that State producti 2.00. This production function exhibits rate is a. i. Increasing Returns to Scale ii. Constant Returns to Scale ii. Decreasing Returns to Scale ine the function for the marginal rate of technical substitution of labot MRTS). Determine the quantity of labor the university would hire to minimize the cost of roducing 100 units. C.Explanation / Answer
A.Production function is given as q= 0.5L3K4
In order to determine returns to scale we multiply both the sides of the production function with some constant ‘t’. Hence,
q(tL,tK) = 0.5(tL)3 (tK)4
= t7 0.5L3K4
= t7 q
Here the production function exhibits increasing returns to scale as 7 > 1.
B.The marginal rate of substitution, MRTSLK = MPL/MPK where MPL and MPK are marginal products of labour and capital respectively.
MPL = 1.5L2 K4
MPK = 2L3 K3
Therefore, MRTSLK = 1.5L2 K4/2L3 K3
= 0.75 K/L
C.The university faces the following problem:
Min. C = L + 2K
Subject to: 100 = 0.5L3K4; q=100
Forming a lagrangean expression,
Z = L + 2K – (0.5L3K4 – 100); is the lagrange multiplier
The first order conditions are:
Z/L = 1 - 1.5L2 K4 ..................(1)
Z/K = 2 – 2L3 K3 .....................(2)
Z/ = 0.5L3K4 – 100...............(3)
From 1 we have, = 1/ 1.5L2 K4
From 2 we have, = 1/L3 K3
ó1/ 1.5L2 K4 = 1/L3 K3
ó L3 K3 = 1.5L2 K4
ó1.5K/L =1
=> K = 0.67L
Using in equation 3, 0.5L3K4 – 100 =0
0.5L3(0.67L)4 = 100
0.1 L7 = 100
L7 = 1000
L = (1000)1/7
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.