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If the technologies of the two producers (of X and Y) are given, initially endowment with...

If the technologies of the two producers (of X and Y) are given, initially endowment with L and K are given for both consumers. How can we find the equilibrium production inputs (Kx, Ky etc.)?

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Answer #1

c) MRSX = MULX / MUKX = (1/2 LX-1/2KX1/2) / (1/2 LX1/2KX-1/2) = KX/LX

MRSY = MULY / MUKY = (1/3 LY-2/3KY2/3) / (1/2 LY1/3KY-1/3) = KY/2LY

KX + KY = 100

LX + LY = 100

In eq, MRSX = MRSY = PX / PY

Let PX =1 and PY = p

This implies, KX/ LX = (100 - KX )/ 2 (100 - LX) = 1/p - pareto optimal locus of production.

Now solving this equation we find that

LX/ 100- LX = 1/p or LX = 100/ (1+p)

LY = 100 - LX = 100p/ (1+p)

KX / 2(100 -KX) = 1/p or KX = 200/ (p+2)

KY= 100 - KX = 100p/ (p+2)

d) Slope of the PPF is called the marginal rate of technical substitution and is also equal to 1/2.

e) output of x would increase relatively and that of y would fall, as x becomes relatively more profitable. As y will fall less of relatively capital would now be used, thus wage rate would increase while rental rate would fall.

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