Show that the two-input CES production function used in this lecture has DRS, CRS or IRS depending on whether ? is smaller than, equal, or larger than one.

12/T1=P= r+ A(r) t' = t + (t ) X f(21, 12) = y 0 Figure 5: The elasticity of substitution o(x) = (dr/r)/(dt/t) = (A(r)/r)/(A(t)/t) In macroeconomic applications, the inputs are typically thought of as labor and capital, owned by households and rented by the firm, and A is interpreted as capturing 'total factor productivity' since increases in A lead to increases in the marginal product of both factors on production in exactly the same proportion. In terms of our Figure 3, an increase in A implies that each isoquant drawn represents a larger level of output. For the Cobb-Douglas production function, we can calculate TRS as TRS(x) = (01/02)(22/71). The TRS for the Cobb-Douglas production function depends only on the ratio r2/a, and not on the actual level of production. That is, all the isoquants have the same slope along a ray like the gray ray going through x in Figure 4. A production function that has this prop- erty is called homothetic. The Cobb-Douglas production function is homogeneous of degree a1 + 02; it is easy to check that a homogenous function is necessarily homothetic. Obtaining r(t) for the Cobb-Douglas production function is easy because of homotheticity. Using TRS(x) = (01/02)(12/x1), and letting r = 12/21 and t = TRS(x), we get r(t) = (02/01)t. Hence the elasticity of substitution is o (x) = r'(t)t/r= 02 / 01 (02/01)t/t = 1. That is, the the elasticity of substitution is constant and equal to one: the percent change in the relative use of inputs is equal to the percent change in the slope. 9Another popular, and useful, production function is the CES (for Constant Elasticity of Substitution) production function: f(21,12) = A(arf + (1 - Q).12)8/p, where A, B > 0, 0 0, and strictly quasiconcave for 21, 22 > 0. For the CES we can find TRS(x) = (a/(1 -Q))(x2/x1)]-p. Thus, the CES production function is homothetic. An exercise in this section asks you to verify that for the CES production function 0 = which is a constant, thus justifying the name of the production function. The CES production function has three interesting limit cases. If we get p -> 0 (hint for those who want to try their hand: take logarithms before taking limits, and then apply I'Hopital rule), we get f(x1, 12) = Arr r, -0)8 which is the Cobb-Douglas function above, with of = 08 and 02 = (1 - 0) 3. If we let p - 1, we get the perfect substitutes case f(21, 12) = A(or + (1-0).12). Finally, if we let p - -oo, we get the perfect complements (also known as Leontief) case f(21, 22) = A(minfox1, (1 -0)22}). Note that neither perfect substitutes nor Leontief production functions are strictly qua- siconcave, and Leontief is not strictly increasing, so they do not satisfy our minimum properties. As we see, for the CES we can treat o as a parameter capturing the ability to substitute one input by the other