Consider the following form for the production function of an economy Q = F (K, L,t) where Q, K, L and t denote output, capital, labour and time respectively. If K and L depend only on t, find an expression for dQ/dt in terms of the partial derivatives of the production function and the time derivatives of K and L. Now suppose that K and L have constant proportionate rates of growth m and n respectively. Find the rate of growth of output when the production function takes the following form:

Consider the following form for the production function of an economy Q = F (K, L,t) where Q, K, L and t denote output, capital, labour and time respectively. If K and L depend only on t, find an expression for dQ/dt in terms of the partial derivatives of the production function and the time derivatives of K and L. Now suppose that K and L have constant proportionate rates of growth m and n respectively. Find the rate of growth of output when the production function takes the following form:

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ISBN:9780190931919

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Chapter1: Making Economics Decisions

Section: Chapter Questions

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Question

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3. Consider the following form for the production function of an economy

\[ Q = F(K, L, t) \]

where $ Q, K, L $ and $ t $ denote output, capital, labour and time respectively. If $ K $ and $ L $ depend only on $ t $, find an expression for $ dQ/dt $ in terms of the partial derivatives of the production function and the time derivatives of $ K $ and $ L $.

Now suppose that $ K $ and $ L $ have constant proportionate rates of growth $ m $ and $ n $ respectively. Find the rate of growth of output when the production function takes the following form:

\[ Q = e^{rt} K^{\alpha} L^{\beta} \quad (r, \alpha, \beta > 0). \]

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