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Unformatted text preview: the combinations of extra good Y and the horizontal segment representing all of the combinations of 26 extra good X. For this reason, Liontief functions are also referred to as "fixedcoefficient" functions. So now let's consider the MRS of a Liontief function. The marginal rate of substitution, as we saw in the last class, is the slope of the tangent line at the point of tangency. We just discussed that the real point of interest to us when dealing with a Liontief function is the kink point. If we want to have the point of tangency at the kink point, do we get a unique solution? NO!!! There are infinitely many tangent lines, with infinitely many different slopes, that are tangent to the kink point. For this reason, Liontief functions are not differentiable at the kink and the marginal utility (or marginal product) concept does not apply to these functions. As such, the concept of the marginal rate of substitution is undefined for the Liontief function (more later). CONSTANT ELASTICITY OF SUBSTITUTION (CES) FUNCTIONS The general form of a CES production function is defined as: Q = f(K,L) = (K- + L-)-1/ or Q = f(K,L) = _____________ [ / K + / L]1/ So what are the marginal products? MPK = Q K = / (Q / K)1 + (holding L constant) MPL = Q L = / (Q / L)1 + (holding K constant) If we take the ratio of these two marginal products, we get the MRTS as follows: MRTS = MPK MPL 27 = / (Q / K)1 + / (Q / L)1 + = (QL)1 + (QK)1 + = [L/K]1 + So let's compute an example using = 1, = = 0.5, and = 1. We get the following CES production function. Q = f(K,L) = _____________ [ / K + / L]1/ Q = f(K,L) = _____1________ [0.5 / K + 0.5 / L] Q = f(K,L) = _____2KL_____ [KL / K + KL / L] Q = f(K,L) = _____2KL_____ L+K with marginal products of: MPK = / (Q / K)1 + = 0.5 (Q / K)2 MPL = / (Q / L)1 + = 0.5 (Q / L)2 and the marginal rate of technical substitution of: MRTS = / (Q / K)1 + / (Q / L)1 + = 0.5 (Q / K)2 0.5 (Q / L)2 = (L / K)2 28 THE IMPORTANCE OF The parameter > 0 measures the degree of flexibility of input (or factor) substitution in the production process. It can also...
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- Spring '10