# 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 = ¦_{K}K + ¦ _{L}L, thus our expression becomes immediately:

s ={¦ _{L}¦ _{K}Y}/{ KL(2¦ _{KL}¦ _{L}¦ _{K} – ¦ _{LL}¦ _{K}^{2} – ¦ _{KK}¦ _{L}^{2})}

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:

¦ _{KK}K + ¦ _{KL}L = 0

¦ _{LK}K + ¦ _{LL}L = 0

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

s ={¦ _{L}¦ _{K}Y}/{ KL(2¦ _{KL}¦ _{L}¦ _{K} + ¦ _{LK}(K/L)¦ _{K}^{2} + ¦ _{KL}(L/K)¦ _{L}^{2})}

={¦ _{L}¦ _{K}Y}/{¦ _{KL}(2¦ _{L}¦ _{K}KL + K^{2}¦ _{K}^{2} + L^{2}¦ _{L}^{2})}

={¦ _{L}¦ _{K}Y}/{¦ _{KL}(¦ _{K}K + ¦ _{L}L)^{2}}

so, by Euler’s Theorem again:

s =¦ _{L}¦ _{K}Y/¦ _{KL}Y^{2}

or simply:

s = ¦ _{L}¦ _{K}/¦ _{KL}Y

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