Hi, I would like to get help for Advanced Macroeconomics. I need solution for the following problem. Output with handwritten or typed solutions would be great. Thank you!

Consider the Ramsey model in continuous time with technological progress. A(t) is technology with growth rate g, L(t) is population (or size of the representative household) with growth rate n and we normalize L(0) = 1. The production function is Cobb-Douglas, Y = F(K, AL) = K^ (AL)^(1) , with 0 < < 1. There is no depreciation, so aggregate capital evolves according to K= Y (t) C(t), where C(t) is aggregate consumption. Households have CRRA instantaneous utility and maximize lifetime utility

0e(n)t1(L(t)C(t))1dt

Denote consumption per units of effective labor as c(t) = C(t)/(A(t)L(t)) . The Euler equation that describes the optimal consumption path is c(t)c(t)=r(t)g

(a) On an optimal consumption path, what is the rate of growth of per-capita consumption C(t)/L(t)?

(b) Show and explain why the assumption n > (1 )g is necessary for lifetime utility to remain finite.

(c) Show that in a steady state equilibrium,r > n + g, where ris the real interest rate in the steady state. Can such an equilibrium be dynamically inefficient?

(d) Derive the law of motion for capital per effective units of labor, k(t) =A(t)L(t)K(t) . That is, derive an equation for k (t) in terms of k(t), c(t) and the parameters of the model.

(e) What is the golden-rule level of capital per effective

labor, kGR, that is, the level of

capital per effective labor that maximizes steady state consumption c? Your solution should be an expression involving only g, n and . Is kGRincreasing in g?

(f) In the steady-state equilibrium, is the level of capital per effective labor, k, smaller than, equal to, or larger than kGR? Prove your answer mathematically.