Question

This question is very important and I need solution for this issue with all the details a.b.c, and help me with all the details? BR/Ha

Question

Assume that the production function is given by f(x₁, x₂) = x₁^1/2 x₂^1/2 .

a) Calculate the return to scale of this technology.

b) Calculate the technical rate of substitution by setting up the Lagrangian and draw a graph illustrating the solution. Calculate the conditional factor demands.

Answer 1

Q = f(x1, x2) = x1^1/2x2^1/2

(a)

When both inputs are doubled, new production function becomes

Q1 = (2x1)^1/2(2x2)^1/2 = 2^1/2.2^1/2. x1^1/2x2^1/2 = 2 x Q

Q1/Q = 2

Since doubling both inputs exactly doubles the output, there is constant returns to scale.

(b)

Isocost line: C = p1.x1 + p2.x2

Lagrangian function is:

L = Q + (C - p1.x1 - p2.x2)

L = x1^1/2x2^1/2 + (C - p1.x1 - p2.x2)

Utility is maximized when L/x1 = 0, L/x2 = 0 and ​​​​​​​L/ = 0.

L/x1 = (1/2).(x2/x1)^1/2 - p1 = 0, therefore,

(1/2).(x2/x1)^1/2 = p1

(x2/x1)^1/2 = 2p1...........(1)

L/x2 = (1/2).(x1/x2)^1/2 - p2 = 0, therefore,

(1/2).(x1/x2)^1/2 = p2

(x1/x2)^1/2 = 2p2...........(2)

​​​​​​​L/ = C - p1.x1 - p2.x2 = 0, therefore,

C = p1.x1 + p2.x2.........(3)

Dividing (1) by (2),

x2/x1 = p1/p2

Technical rate of substitution = x2/x1

From above relationship,

p1.x1 = p2.x2

x2 = (p1.x1)/p2 = x1.(p1/p2)

Substituting in production function,

Q = x1^1/2[(p1.x1)/p2]^1/2

Q = x1^1/2.x1^1/2.(p1/p2)^1/2

Q = x1.(p1/p2)^1/2

x1 = Q x (p2/p1)^1/2

x2 = x1/(p1/p2) = [Q x (p2/p1)^1/2].(p1/p2) = Q x (p1/p2)^1/2

In following graph, Q0 is the isoquant and AB is the isocost line. Conditional factor input combination is at point E where Q0 is tangent to AB with optimal (x1, x2) bundle being (x1*, x2*).

​​​​​​​