We constructed in class the exponential function ex, with the property that d(ex)/dx = ex. We also defined ln(x) as the inverse function of ex.
(a) What is ln(ex)? Why?
(b) Using your answer to (a), find the derivative of ln(y) with respect to y, that is, d(ln(y))/dy, where y = ex for some x.
(c) If the relationship between outputQ, labor L, and capital K is given by
ln(Q) = a + b[ln(L)] + c[ln(K)],
Use your answer to (b) to show that holding K constant, b is the elasticity of output with respect to labor. [Hint: If K is constant, treat Q as Q(L), i.e., as a function of L.]
(d) Using the equation in (c) show that if the capital/labor ratio K/L is a constant k, the elasticity of output with respect to labor is b + c.
(e) Why are relationships expressed like those in (c) important for doing econometrics?
(f) Show that the production function specified in (c) is the equivalent of the Cobb-Douglas production function Q(L) = aLbKc