A Leontief production function has the form, y = min{alpha x_1, beta x_2} for alpha > 0 and beta > 0. Sketch

A Leontief production function has the form, y = min{alpha x_1, beta x_2} for alpha > 0 and beta > 0. Sketch

 

answer:

The elasticity of substitution has interesting expressions when the two-input production function exhibits constant returns to scale. Firstly, under constant returns to scale, Euler’s Theorem implies that Y = ¦KK + ¦ LL, thus our expression becomes immediately:

s ={¦ L¦ KY}/{ KL(2¦ KL¦ L¦ K – ¦ LL¦ K2 – ¦ KK¦ L2)}

Now, recall once again, that if the production function ¦ is homogeneous of degree one in the factors (constant returns), then the marginal product ¦ i is homogeneous of degree zero in teh factors. This implies, again by Euler’s Theorem, that:

¦ KKK + ¦ KLL = 0

¦ LKK + ¦ LLL = 0

so ¦ KK = -¦ KL(L/K) and ¦ LL = -¦ LK(K/L). Thus substituting in:

s ={¦ L¦ KY}/{ KL(2¦ KL¦ L¦ K + ¦ LK(K/L)¦ K2 + ¦ KL(L/K)¦ L2)}

={¦ L¦ KY}/{¦ KL(2¦ L¦ KKL + K2¦ K2 + L2¦ L2)}

={¦ L¦ KY}/{¦ KLKK + ¦ LL)2}

so, by Euler’s Theorem again:

s =¦ L¦ KY/¦ KLY2

or simply:

s = ¦ L¦ KKLY

which is considerably more simple. This expression for the elasticity of substitution in the constant returns to scale . This is how it should be verified

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