9780070109100, Chapter 10 7, Problem 11E

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Chapter 10.7, Problem 11E Step-by-step solution Step 1 of 5 It is given that the output 'Q' is a function of capital and labor 'L', both of which are the function of time. That is, the production function is as follows: Step 2 of 5 The following relationship is required to be shown as valid. = + Here, is the rate of growth of output rₖ is the rate of growth of capital (K) is the rate of growth of labor (L) is the elasticity of output with respect to capital is the elasticity of output with respect to labor Step 3 of 5 Differentiate the production function with respect to 't'. a = a (1) of dK of dL = + dt dt Step 4 of 5 Now, write the expression of rate of growth of output as follows: = Q Step 5 of 5 Using equation (1), the above expression is written as follows: of X dK + of X dL = dt Q dt of K + of L dL = K dt Q L dt = 1 of K dK + of L dL Q K dt Q L dt = K K 1 dK dt + of L 1 dt = K + This shows that the expression = + is valid. The above expression shows the rate of growth of output in terms of rate of growth of capital and labor.