Question

Y = X₁ X₂    P_X1 = 20,   and P_X2 = 10

Which combination of X₁ and X₂ will produce maximum output when the manager has only $100 to spend on  X₁ and X₂

Answer 1

Ans. Production function, Y = X1X2

Marginal product of X1, MP1 = dY/dX1 = X2

and

Marginal product of X2, MP2 = dY/dX2 = X1

=> Marginal rate of technical substitution,

MRTS = MP1/MP2 = X2/X1

Cost constraint,

20X1 + 10X2 = 100

At equilibrium,

MRTS = Px1/Px2

=> X2/X1 = 20/10 = 2

=> X2 = 2X1

Substituting this in cost constraint, we get

20X1 + 10 * 2X1 = 100

=> X1 = 2.5 units and X2 = 5 units

Therefore, 2.5 units of X1 and 5 units of X2 will produce maximum output of 12.5 units in the cost of $100.

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