[20 points] Consider a continuous-time Ramsey growth model with population growth rate n > 0 and technology progress with rate g > 0. Capital depreciates at rate 8 > 0. Assume that the household maximizes the sum of the utilities of all household members (as discussed in class). Each household member's instantaneous utility function is given by u : R+ R such that -1 u(c) 1 0 cl-o - for all c> 0 where 0 > 0 is a given parameter. The rate of time preference is given by p > 0. Each member is endowed with 1 unit of labor. The number of household and the population size at time 0 are both normalized to be 1. Let Bo > O be the household's total asset at time 0. The household earns labor income and asset income (as well as dividend if any). (a) Solve the household's life-time utility maximization problem. Find the growth rate of consumption per capita in terms of interest rate. (b) Suppose the representative firm's production function is homogeneous of degree 1. Let f the production function in intensive form, which satisfies the Inada conditions. Solve the firm's profit maximization problem to find the relationship between the profit- maximizing inputs use and the interest rate of capital. (c) Use the market clearing condition to derive the Euler equations for consumption and capital per unit of effective labor. (d) Using a phase diagram, describe how the steady state (balanced growth path) changes when the depreciation rate 8 > 0 changes and when the population growth rate changes. (e) Consider a Solow model with the same parameter values as in the Ramsey model and with the saving rate s* which maximizes the steady-state consumption per unit of effective labor. How do the steady states in the two models compare? Would the steady-state consumption per unit of effective labor larger or smaller in the Ramsey model? [20 points] Consider a continuous-time Ramsey growth model with population growth rate n > 0 and technology progress with rate g > 0. Capital depreciates at rate 8 > 0. Assume that the household maximizes the sum of the utilities of all household members (as discussed in class). Each household member's instantaneous utility function is given by u : R+ R such that -1 u(c) 1 0 cl-o - for all c> 0 where 0 > 0 is a given parameter. The rate of time preference is given by p > 0. Each member is endowed with 1 unit of labor. The number of household and the population size at time 0 are both normalized to be 1. Let Bo > O be the household's total asset at time 0. The household earns labor income and asset income (as well as dividend if any). (a) Solve the household's life-time utility maximization problem. Find the growth rate of consumption per capita in terms of interest rate. (b) Suppose the representative firm's production function is homogeneous of degree 1. Let f the production function in intensive form, which satisfies the Inada conditions. Solve the firm's profit maximization problem to find the relationship between the profit- maximizing inputs use and the interest rate of capital. (c) Use the market clearing condition to derive the Euler equations for consumption and capital per unit of effective labor. (d) Using a phase diagram, describe how the steady state (balanced growth path) changes when the depreciation rate 8 > 0 changes and when the population growth rate changes. (e) Consider a Solow model with the same parameter values as in the Ramsey model and with the saving rate s* which maximizes the steady-state consumption per unit of effective labor. How do the steady states in the two models compare? Would the steady-state consumption per unit of effective labor larger or smaller in the Ramsey model