1)Suppose the production function for x is: x = LaKbwith 0 < a <1 and 0 < b < 1 and a + b = 1.a)Show that each input, L and K, has a positive marginal product.ANS:Calculate the marginal products, so take the partial derivatives with respect to each input and show that this expression is positive. So: mpLx= aL(a-1)Kbwhich has to be a positive number and likewise, mpKx= bLaK(b-1)which also has to be a positive number.b)Show that each input, L and K, has diminishing marginal productivity.ANS:Now take the partial of each marginal product you just got above with respect to that input and show that what you get it negative, i.e., that the value of marginal product is getting smaller and smaller. So: ∂mpLx/∂L = (a-1)aL(a-2)Kbwhich is negative since (a-1)<1. Do the same thing again for K and get ∂mpKx/∂K = (b-1)bLaK(b-2)which will be negative since (b-1)<1.c)Show that the marginal rate of technical substitution, the MRTS, depends only on the kapital to labor ratio, that is, on K/L, but not on the scale of production.
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