(50 points) Consider the two-period economy with investment discussed in class. Forthe sake of simplicity, assume that the utility is time separable and that the per-periodutility is equal to:u(C) = log CAssume also that the technology to produce the consumption goods has constant returns to scale. Moreover, capital is the only input to production. Hence the currentperiod technology can be represented as:Y = zK1where Y , K, and z stand for output, capital and total factor productivity in the currentperiod. The future period technology is akin to the current period technology. There isno government expenditure in the current and future period so that G = 0 and G0 = 0.Finally there is full capital depreciation, namely δ = 1.(a) (10 points) Write down the social planner’s problem. State the variables withrespect to which the social planner maximises.(b) (10 points) Derive the first order conditions to the planner problem. Rearrangethere condition to find the conditions characterising the Pareto efficient allocation.Interpret them.(c) (10 points) Find the solution to the planner problem. In other words, expresscurrent and future consumption and investment as a function of the exogenousvariables (K, z and z0) and the parameters of the model (β).(d) (8 points) Find the real interest rate that decentralises the Pareto efficient allocation as a competitive equilibrium.(e) (12 points) Suppose the economy experiences an increase in the current level ofcapital. Analyse the equilibrium effects of this shock on current and future consumption, investment and the real interest rate. Explain your answer intuitively.

 (50 points) Consider the two-period economy with investment discussed in class. For
the sake of simplicity, assume that the utility is time separable and that the per-period
utility is equal to:
u(C) = log C
Assume also that the technology to produce the consumption goods has constant returns to scale. Moreover, capital is the only input to production. Hence the current
period technology can be represented as:
Y = zK
1
where Y , K, and z stand for output, capital and total factor productivity in the current
period. The future period technology is akin to the current period technology. There is
no government expenditure in the current and future period so that G = 0 and G0 = 0.
Finally there is full capital depreciation, namely δ = 1.
(a) (10 points) Write down the social planner’s problem. State the variables with
respect to which the social planner maximises.
(b) (10 points) Derive the first order conditions to the planner problem. Rearrange
there condition to find the conditions characterising the Pareto efficient allocation.
Interpret them.
(c) (10 points) Find the solution to the planner problem. In other words, express
current and future consumption and investment as a function of the exogenous
variables (K, z and z
0
) and the parameters of the model (β).
(d) (8 points) Find the real interest rate that decentralises the Pareto efficient allocation as a competitive equilibrium.
(e) (12 points) Suppose the economy experiences an increase in the current level of
capital. Analyse the equilibrium effects of this shock on current and future consumption, investment and the real interest rate. Explain your answer intuitively.