1) Construct the representative household's utility maximisation decentralised problem such that a4 = 24-1e#+Et, whereby & N 0, 02); find the optimal rules attendant on consumption, labour and capital.?

2) All else equal, exhibit the intermediate goods firm's profit maximisation problem and calculate the optimal rules attendant on labour and capital.?

3) All else equal, take the representative household's budget constraint and plug the implicit profit equation into the same, cancelling out terms.?

4) All else equal, display the laws of motion for consumption (i.e. first and third household FOCs), labour supply (i.e. second household FOC), labour demand (i.e. first firm FOC), capital return (i.e. second firm FOC), the aggregate resources (i.e. explicit household budget constraint), output (i.e. production function) and technology (i.e. productivity equation) and express consumption, capital and output as trend non-stationary variables, that is, as X= 24.14 for X = C, K, Y and r=c, k, y. NB. FOC stands for first order condition.

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5.22 RBC model elaboration 5.22a) Such established, the utility function, expressing household or consumer prefer- ences, is homogeneous of first degree, continuous, increasing in consumption, decreasing in labour and concave in both; such function actually appertains to the class of the HRRA utility functions. The social planner maximises utility with consumption, labour and capital as arguments subject to a budget constraint such that demand for consumption and capital net of its depreciated value be supplied by featuring factor augmenting productivity multiplying extant capital and labour at their respective output shares. In effect, such supply delineates the imperfect complements or Cobb Douglas production function, a declension of the CES production function family, as mentioned, homogeneous of first degree, continuous, increasing in both capital and labour and concave in the same two. Qua twice continuously differentiable function by the mentioned properties it spotlights these limit laws: lim, 10k, afik. D = ∞o; limg, 1- = 0. Such properties are cognised Ək, 1 af(k. 1) as the Inada conditions, due to Ken-Ichi Inada. The law of motion for factor augmenting or Hicks neutral productivity, juxta John Hicks, that being technology, is finally represented by an autoregressive process of first order, non-stationary, drifted and subject to white noise perturbations, denominated random walk with drift. Reminisce stationarity; a time series process is stationary if and only if retains a finite mean, variance and covariance: E (at) = μ max E₂ H+01 {Ct₂4, Kr} t=0 C₁ + K₁ - (1-6) K₁-1=a₂kl²-ª a = a₁-1e+, whereby lim 3¹xcKe = 0. 1+%, s.t. ~. ~N(0, ²), Notice the last constraint, where A, is the budget constraint's shadow price or upcoming Lagrangian multiplier: the transversality condition, in that presently valued priced capital in the anterior future must be worthless in order to have eventually optimised. The problem's dynamic Lagrangian equation, wherethrough parameter 3 € (0, 1) discounts the stream values of future constrained utility, thence is: = =B² (C₁-₁)=0C = ₁; L₁ = Eo = EΣ {[12-1 | + №e [asK²_,il?¯ª − Ci − Ki + (1 − 6) Kie-a]} - 1+01, The first order conditions are the stationary points in relation to the Lagrangian equation's first derivatives with respect to consumption, labour and capital: ac₁ ac ac₁ = # [-? +\@ (1−a)K}"]= = 0 ⇒ 1¹ = Xeαe (1 − a) Kikº; Əe ƏL —№₁‚ß² + E₂3²+¹ \£÷1 [aª₁+1K²¬¹l+î + (1 − 8)] = 0 Əkt ⇒ A=X+1 [00+1K²¬¹l+ + (1 − 6)]. The combination of the first and the third and the first and the second engenders laws of motion or optimal rules for consumption and labour. The presence of four endogenous variables yet speaks to the need for four reflective equations. Capital's law of motion is then naturally the representative household's budget constraint, which can be hereby interpreted as being the aggregate or economic resources constraint; the law of motion for technology is instead lastly its aforementioned autoregressive first order process. C=E₁3C [+1K¹+(1-5)] (consumption or consumption Euler equation); P¹ = Ca (1a) K (labour or labour clearance); C₂ + K₁ − (1 − 8) K₁-1 = a+K²_₁²" (capital or aggregate resources); a₁a₁-1 (technology). 5.22 RBC model elaboration 5.22a) Such established, the utility function, expressing household or consumer prefer- ences, is homogeneous of first degree, continuous, increasing in consumption, decreasing in labour and concave in both; such function actually appertains to the class of the HRRA utility functions. The social planner maximises utility with consumption, labour and capital as arguments subject to a budget constraint such that demand for consumption and capital net of its depreciated value be supplied by featuring factor augmenting productivity multiplying extant capital and labour at their respective output shares. In effect, such supply delineates the imperfect complements or Cobb Douglas production function, a declension of the CES production function family, as mentioned, homogeneous of first degree, continuous, increasing in both capital and labour and concave in the same two. Qua twice continuously differentiable function by the mentioned properties it spotlights these limit laws: lim, 10k, afik. D = ∞o; limg, 1- = 0. Such properties are cognised Ək, 1 af(k. 1) as the Inada conditions, due to Ken-Ichi Inada. The law of motion for factor augmenting or Hicks neutral productivity, juxta John Hicks, that being technology, is finally represented by an autoregressive process of first order, non-stationary, drifted and subject to white noise perturbations, denominated random walk with drift. Reminisce stationarity; a time series process is stationary if and only if retains a finite mean, variance and covariance: E (at) = μ max E₂ H+01 {Ct₂4, Kr} t=0 C₁ + K₁ - (1-6) K₁-1=a₂kl²-ª a = a₁-1e+, whereby lim 3¹xcKe = 0. 1+%, s.t. ~. ~N(0, ²), Notice the last constraint, where A, is the budget constraint's shadow price or upcoming Lagrangian multiplier: the transversality condition, in that presently valued priced capital in the anterior future must be worthless in order to have eventually optimised. The problem's dynamic Lagrangian equation, wherethrough parameter 3 € (0, 1) discounts the stream values of future constrained utility, thence is: = =B² (C₁-₁)=0C = ₁; L₁ = Eo = EΣ {[12-1 | + №e [asK²_,il?¯ª − Ci − Ki + (1 − 6) Kie-a]} - 1+01, The first order conditions are the stationary points in relation to the Lagrangian equation's first derivatives with respect to consumption, labour and capital: ac₁ ac ac₁ = # [-? +\@ (1−a)K}"]= = 0 ⇒ 1¹ = Xeαe (1 − a) Kikº; Əe ƏL —№₁‚ß² + E₂3²+¹ \£÷1 [aª₁+1K²¬¹l+î + (1 − 8)] = 0 Əkt ⇒ A=X+1 [00+1K²¬¹l+ + (1 − 6)]. The combination of the first and the third and the first and the second engenders laws of motion or optimal rules for consumption and labour. The presence of four endogenous variables yet speaks to the need for four reflective equations. Capital's law of motion is then naturally the representative household's budget constraint, which can be hereby interpreted as being the aggregate or economic resources constraint; the law of motion for technology is instead lastly its aforementioned autoregressive first order process. C=E₁3C [+1K¹+(1-5)] (consumption or consumption Euler equation); P¹ = Ca (1a) K (labour or labour clearance); C₂ + K₁ − (1 − 8) K₁-1 = a+K²_₁²" (capital or aggregate resources); a₁a₁-1 (technology).

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