Problem: the Solow growth model with a Cobb-Douglas technology of production. Consider the following continuous-time economy...

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Problem: the Solow growth model with a Cobb-Douglas technology of production. Consider the following continuous-time economy with time indexed by t 0. The economy is populated by identical households. The population of households at time t > 0 is L(t) > 0 and is exogenous. The economy is also composed of a large number of identical competitive producers which have access to a technology that uses capital and labour to produce a final homogeneous good Y(t) 0 that can be used for consumption C(t) 0 and investment I(t). Capital is held by households and has stock denoted by K(t) 0; each household supplies one unit of labour at each time so total employment is equal to population L(t). Using the fact that all producers all identical, we will consider a representative producer for this economy. We assume that the aggregate production function takes the following Cobb- Douglas form: Y(t) = (K(t), L(t), t) = (Ak(t)K(t)) (Az(t)L(t)), where a > 0 and > 0 are parameters and where Ak(t) > 0 and A(t) > 0 are, respectively, capital and labour-augmenting technological factors governing the efficiency of these inputs. Households and producers exchange labour and capital in the markets for inputs; the wage rate is w(t) and the rental rate of capital is R(t). Markets are competitive and households and producers are price-takers, i.e., they treat the prices as given. The growth rate of the population is exogenous and constant and given by n > 0, and the initial population level is L(0) = Lo. The growth rate of the technology indexes AK and A are denoted by 8K 0 and g 0, respectively (assumed to be exogenous and constant as well). The capital stock accumulates following (1) K(t) = I(t) 8K(t), (2) where X(t) designates the time derivative of a given variable X(t), and 8 > 0 represents the depreciation rate of capital. The initial aggregate capital stock K(0) = Ko > 0 is given. Finally, the household's savings rate is exogenous and given by s (0, 1). The flow of aggregate savings at time t is denoted by S(t). 1. (a) Show that under some conditions on the value of the technology parameters a > 0 and > 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 a and under which the above assumptions are satis- fied. [2.5 points] (b) Demonstrate that under the appropriate conditions highlighted in question (a), we have the equality: Y(t) = k(K(t), L(t), t) K(t) + F (K(t), L(t), t) L(t), (3) where FK and F denotes partial derivatives of F with respect to K and L. [2.5 points] From now on, assume the conditions highlighted in question (a) are satisfied. Problem: the Solow growth model with a Cobb-Douglas technology of production. Consider the following continuous-time economy with time indexed by t 0. The economy is populated by identical households. The population of households at time t > 0 is L(t) > 0 and is exogenous. The economy is also composed of a large number of identical competitive producers which have access to a technology that uses capital and labour to produce a final homogeneous good Y(t) 0 that can be used for consumption C(t) 0 and investment I(t). Capital is held by households and has stock denoted by K(t) 0; each household supplies one unit of labour at each time so total employment is equal to population L(t). Using the fact that all producers all identical, we will consider a representative producer for this economy. We assume that the aggregate production function takes the following Cobb- Douglas form: Y(t) = (K(t), L(t), t) = (Ak(t)K(t)) (Az(t)L(t)), where a > 0 and > 0 are parameters and where Ak(t) > 0 and A(t) > 0 are, respectively, capital and labour-augmenting technological factors governing the efficiency of these inputs. Households and producers exchange labour and capital in the markets for inputs; the wage rate is w(t) and the rental rate of capital is R(t). Markets are competitive and households and producers are price-takers, i.e., they treat the prices as given. The growth rate of the population is exogenous and constant and given by n > 0, and the initial population level is L(0) = Lo. The growth rate of the technology indexes AK and A are denoted by 8K 0 and g 0, respectively (assumed to be exogenous and constant as well). The capital stock accumulates following (1) K(t) = I(t) 8K(t), (2) where X(t) designates the time derivative of a given variable X(t), and 8 > 0 represents the depreciation rate of capital. The initial aggregate capital stock K(0) = Ko > 0 is given. Finally, the household's savings rate is exogenous and given by s (0, 1). The flow of aggregate savings at time t is denoted by S(t). 1. (a) Show that under some conditions on the value of the technology parameters a > 0 and > 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 a and under which the above assumptions are satis- fied. [2.5 points] (b) Demonstrate that under the appropriate conditions highlighted in question (a), we have the equality: Y(t) = k(K(t), L(t), t) K(t) + F (K(t), L(t), t) L(t), (3) where FK and F denotes partial derivatives of F with respect to K and L. [2.5 points] From now on, assume the conditions highlighted in question (a) are satisfied.