(a) Show that under some conditions on the value of the technology parameters α > 0 and β > 0 the production function F˜ defined by (1), satisfies the following properties: • Strictly positive and diminishing marginal returns to capital and labour; • Constant returns to scale; • The Inada conditions. State the conditions related to α and β under which the above assumptions are satisfied. [5 points] (b) Demonstrate that under the appropriate conditions highlighted in question (a), we have the equality: Y(t) = F˜ K(K(t), L(t)) × K(t) + F˜ L(K(t), L(t)) × L(t), (3) where F˜ K and F˜ L denotes partial derivatives of F˜ with respect to K and L. [5 points] From now on, assume the conditions highlighted in question (a) is satisfied.
(a) Show that the production function F˜ can represented by an equivalent function F, where the technology is purely labour-augmenting, i.e., by a function F such that Y(t) = F˜(K(t), L(t)) = F(K(t), A(t)L(t)), (4) where A(t) is a function of AK(t) and AL(t). Determine the expression for A in terms of AK and AL and the parameters. [5 points] (b) Compute the growth rate over time of the labour-augmenting technology variable A(t). [4 points] (c) Write the production function F in its intensive form given by y(t) = f (k(t)), where y(t) ≡ Y(t)/(A(t)L(t)) and k(t) ≡ K(t)/(A(t)L(t)) denote respectively total output and capital per effective units of labour A(t)L(t). [1 point]
(a) Using the capital accumulation equation (2) and the appropriate equilibrium conditions, derive the fundamental law of motion of the Solow model, which expresses the rate of change of k with respect to time as a function of its level and the parameters. [5 points] 2 (b) Deduce an expression for the steady-state level of k characterized by a constant level of k over time, and that is eventually reached at some time t > 0. This expression should express the steady-state ratio k ∗ as an explicit function of the parameters. [5 points]
Demonstrate that in steady state, output, consumption, and the capital stock per capita grow at the same rate, and determine this growth rate. [15 points]
Determine the golden-rule saving rate sGR and the associated level of consumption per capita and output per capita. [10 points]
Consider now the problem of a representative producer that uses capital and labour to maximize its profits at all times. Recall that it is assumed that producers are competitive, i.e., they take prices as given. (a) Write down the maximization problem of the producer and solve for the equilibrium values of prices, w(t) and R(t) in terms of the capital-effective labour ratio k(t) and the parameters. [5 points] (b) Show that in equilibrium, producers’ profits are zero at all times. Moreover, show that with the assumed Cobb-Douglas functional form, the share of aggregate income devoted to the remuneration of capital is α, whereas the share going to the remuneration of labour is 1 − α. [5 points]
Assume that the economy is in steady state. Analyze the effect of a permanent increase in the growth rate of the capital-augmenting technology index, gK, by representing graphically the path over time followed by the economy. Specifically, assume that the growth rate of AK is gK ≥ 0 for t ∈ [0, t0] and g ′ K > gK for t > t0, and assume that the economy is in steady-state at t = t0. Then, represent, in a graph with t on the horizontal axis, the trajectory followed by the following equilibrium variables: [20 points] (a) output per capita, Y(t)/L(t); (b) consumption per capita, C(t)/L(t); (c) the growth rate of output per capita; (d) the wage rate, w(t); (e) the rental rate of capital R(t). 3
Now, assume that α = 1 and β = 0. (a) Going back to question 1., which of the stated assumptions does the production function satisfies and does not satisfy? [2 points] (b) Determine the fundamental law of motion describing the dynamics of k(t), and determine the value of k(t) in terms of the exogenous parameters and in terms of the initial value k(0). [3 points] (c) Contrast the behaviour of this economy with the one where 0 < α < 1 and where the production function has constant return to scale. Specifically, represent on the same graph the path followed by income per capita and by the capital-to-aggregate income ratio in this economy and in an economy with α ∈ (0, 1) for a given initial capital stock K(0) > 0. [10 points]
In the general case where α ∈ [0, 1], solve for the capital-effective labour ratio k(t) in terms of t and in terms the models’ exogenous variables only. [Bonus question] [10 points]