Q2. What is the probability measure for each fixed z Z mapping?
For each fixed z Z mapping Q(z, ·) : Z R is a probability measure, and for each fixed B Z mapping Q(·,B) :Z R is a measurable function.
Q3. What is the effect of the welfare theorems?
By the welfare theorems, the uniqueness of the multipliers entails that an optimal allocation is just supported by a unique price system.
Q4. What are the conditions that are required to establish that function v is of class C1?
To establish that function v is of class C1, the authors require differentiability of the return function U , and some regularity conditions for boundary solutions.
Q5. What are the conditions transported from the static theory?
The authors consider two conditions transported from the static theory: A boundary restriction on the policy function over the domain of definition and a constraint qualification (cf. [20]).
Q6. What are the conditions that ensure this contraction property?
In their case, the following conditions ensure this contraction property: (i) Function U(x, y, z) and its partial derivatives DjU(x, y, z), j = 1,2, are uniformly bounded on × Z, and (ii) sup(x,y,z) ∫ Z G(x, y, z )μ(z, dz ) < 1.
Q7. What is the utility function for a given agent?
Uaut for all s, (P3) Us [Uaut,Umax] for all s,where the value function V (U0) assigns the maximum utility to an agent (say agent 2) over all possible utility levels U0 of the other agent.
Q8. What is the recursive formulation of the pure currency model?
The recursive formulation of the pure currency model isv(m, z) = max c, m 0{ u(c, z) + ∫ Z v(m , z )μ(dz ) }s.t. m (1 + ) + c m y 0, c m 0.
Q9. What is the non-negativity condition of Proposition 3.2?
The non-negativity conditions of Proposition 3.2 allow for a simple extension of the transversality condition to unbounded domains, e.g., see Benveniste and Scheinkman [3].
Q10. What is the way to show that the value function is unbounded?
The authors show that in the above growth model studied in Section 3.4.2, the derivative of the value function may become unbounded as the stock of capital approaches zero – even if the derivatives of the utility and production functions are bounded.
Q11. What is the sequence of choice variables xt t 0?
The sequence of choice variables {xt }t 0 belongs to a set X Rn, and the realizations of the exogenous stochastic process {zt }t 0 lie in a space Z. Let X be the Borel -algebra of X and Z the -algebra of Z.
Q12. What is the definition of the stochastic optimization problem?
A contingency plan {xt }t 0 is feasible if xt+1 :Zt X is a measurable function and xt+1(zt ) (xt (zt 1), zt ), for zt Zt and t = 0,1, . . . .The stochastic optimization problem can be defined as follows:
Q13. What is the decomposition of the value function into the fundamental value?
The decomposition given in Theorem 3.1 of the derivative of the value function into the fundamental value and a bubble term is common in asset pricing, where B0 can be identified with the bubble term of some existing assets.
Q14. What is the meaning of qz1(Mv)(y, z)?
The interpretation of (3) is as follows: For z Z fixed, qz1(Mv)(y, z) if and only if there is a measurable mapping z qz(z ) with qz(z ) 1v(y, z ) almost everywhere (a.e.) in the measure Q(z, ·) such that qz = ∫ Z qz(z )Q(z, dz ).
Q15. What is the value function of the matrix of partial derivatives?
Then for each (x, z) there exists some optimal solution y H(x, z) with y int(X) such that the rank of the matrix of partial derivatives {D2gi(x, y, z): i The author(x, y, z)} is equal to s(x, y, z).
Q16. What is the definition of a function of class C1?
D1: For every x int(X) and z Z function U(·,·, z) is of class C1 on some open neighborhood N(x,y) of every point (x, y) with y H(x, z).
Q17. What is the way to define end-of-period asset holdings?
End-of-period asset holdings can be defined in a rather arbitrary way, as the agent can be endowed with new securities at the beginning of each period so as to replicate the optimal path {xt+1(zt )}t 0.