Assume a production function with constant returns to scale. Labour's share of income is 4/5 and capital's share is V5. If labour grows at 3%, capital at 2%, and the rate of technological advance is 1.2%, roughly how many years would it take to double the current level of output? Multiple Choice 23

In class we argued that if people could accumulate human as well as physical capital, the production function would look like the “AK” production function. • (a) If the production function is AK and the savings rate is constant at rate “s”, and the rates of depreciation and populati on growth are δ and n respectively, what would the growth rate of the economy be? • (b) What would be the macroeconomic consequences of decreasing the savings rate in this economy? • (c) What would be the consequences of an increase in fertility in this economy? • (d) Would the consequences of decreasing fertility be UNAMBIGUOUSLY GOOD? • (e) Can human capital grow without bounds? Explain why or why not (make sure you discuss the physical nature of human capital). • (f) What is the growth rate of the economy (in the absence of technological progress) if human capital cannot grow without bounds?

QUESTION 1 The first observation to be made about technical progress is that historically, the rate of growth of outputs over time has exceeded the growth rate that can be attributed to the growth in conventionally defined inputs. Letting Q = A(t) f(K, L) (1) be the production function for society's output as a whole. The term A(t) in the function represents all the influences that go into determining Q other than K (machine-hours) and L (labor-hours) so that changes in A over time represent technical progress. For this reason, A is shown as a function of dA >0 time and presumably dt ; that is, particular levels of labor and capital become more productive over time. Using equation (1) above, show: (e,«.) a) The elasticity of output with respect to capital input b) The elasticity of output with respect to labor input (e,.) c) Show that the growth rate in output can be broken into the sum of two components, that is, growth attributed to changes in inputs (K and L) and other "residual"…

A CES production function with physical and human capital Consider the CES production function in terms of physical capital, K, and human capital, H: where 0 a. Set up the Hamiltonian and find the first-order conditions. b. What is the optimal relation between K and H? Substitute this relation into the given production function to get a relation between Y and K. What does this “reduced-form” production function look like? c. What is the steady-state value of the ratio of physical to human capital, (K/H)∗? d. Describe the behavior of the economy over time if the initial condition is such that K(0)/H(0)? e. Suppose that the inequality restrictions IK ≥ 0 and IH ≥ 0 apply. How do these constraints affect the dynamics if the economy begins with K(0)/H(0)∗?