5. The Solow Model with both Population Growth and Technological Progress in Continuous Time plus Extensions Consider the Solow model in continuous time. The following system of equations fully describe the economy: Y(t) = C(t) + I(t) Y(t) = F[K(t), A(t)L(t)] = K(t)"[A(t)L(t)]¹-a I(t) = S(t) K = -8K (t) + 1(t) S(t) = sy(t) Y defines income, C defines consumption, I investment, S savings, K the capital stock, L labour and A the state of technology; & E (0,1) is the rate of capital depreciation, s E (0,1) the saving rate and a E (0,1) is the capital elasticity of output. The previous equations describe a closed economy with no government. Labour, L, and the state of technology, A, grow at the constant rates n and g, respectively. (a) Derive the fundamental law of motion of the Solow model in per effective labour form and compute capital per effective labour at steady-state equilibrium. For any variable X, let x = X its per effective labour form. AL (25 marks) (b) Solve for capital per labour and output per labour at steady state equilibrium. What determines output per labour in the long run? (25 marks) (c) Derive the growth accounting equation for this production function and determine the Solow residual with respect to growth rates of per worker variables. What does this show? (25 marks) (d) How can the given production function be modified to include also human capital? What does human capital now bring into the analysis? How do the Lucas (1988) and the Mankiw et. al. (1992) models incorporate into their analysis the role of human capital and what are their implications? (25 marks)