We constructed in class the exponential function e x , with the property that d

Question

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? [Hint: This is a giveaway, not a trick question.]

(b) Using your answer to (a), find the derivative of ln(y) with respect to y, that is, d(ln(y))/dy, where y = exfor some x. [Hint: The Chain Rule may help.]

(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.  

Explanation / Answer

d) K/L = k

=> K = k*L

substitute this in ln(Q) = a + b[ln(L)] + c[ln(K)]

we get,

ln(Q) = a + b*ln(L) + cl*ln(k*L)

=> ln(Q) = a + b*ln(L) + c*ln(L) + c*ln(k)

=> ln(Q) = a + (b+c)*ln(L)

now differenting like in part (c)

we get (b+c) is the elasticity of output with respect to labor.

e) Such equations are important because they are used in estimation of production functions

f) given ln(Q) = a + b[ln(L)] + c[ln(K)],

now, b[ln(L)] = ln (L^b) and c[ln(K)] = ln(K^c)

=> ln(Q) = a + ln(L^b)+ ln(K^c)

=> ln(Q) = a + ln( (L^b)*(K^c))

=> Q(L) = a*(L^b)*(K^c) (taking anti-logarithm and treating 'a' as a constant)

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