TA Sessions in Macroeconomic Theory I. Diogo Baerlocher

Transcription

1 TA Sessions in Macroeconomic Theory I Diogo Baerlocher Fall 206

2 TA SESSION Contents. Constrained Optimization 2. Robinson Crusoe 2. Constrained Optimization The general problem of constrained optimization we face is () max x,...,x n f(x,..., x n ) subject to: g (x,..., x n ) = b g 2 (x,..., x n ) = b 2 g k (x,..., x n ) = b k. Next theorem states that the critical points of the Lagrangean k L = f(x,..., x n ) λ i [g i (x,..., x n ) b i ], provide the vectors (x, λ ) that optimizes the problem above. i= Theorem 8.4 (Simon and Blume): Suppose that f, g,..., g k C functions of n variables. Let x R be the local maximizer of f on the constraint set defined above. Also, suppose that, at x. the non-degenerate constraint qualification is satisfied. Then, there exists λ R such that: () L(x,λ ) x i = 0 i (2) λ i [g i(x ) b i ] = 0 i (3) λ 0,..., λ k 0 Date: Aug 24.

3 2 TA SESSION (4) g (x ) b,..., g k (x ) b k. The non-degenerate constraint qualification (NDCQ) guarantees that the constraint set has well defined tangent plane everywhere by requiring the Jacobian matrix of the binding constraints to have maximum rank at x. Next, we want to make sure the First-Order Necessary Conditions from the theorem above are sufficient conditions as well. Additionally, we want to guarantee the critical points are global maximizer, not only local maximizer. Theorem 2.22 (Simon and Blume): Let D be convex, open subset of R. Let f : D R be C pseudo-concave function on D. Let g,..., g k : D R be C quasi-convex function. Consider the problem () and suppose that NDCQ holds. if there is (x, λ ) such that () L(x,λ ) x i = 0 i (2) λ i [g i(x ) b i ] = 0 i (3) λ 0,..., λ k 0 (4) g (x ) b,..., g k (x ) b k. Then x is a global maximizer on the constraint set. We say g( ) is quasi-convex if f( ) = g( ) is quasi-concave. Quasi-concave functions are a class of functions that hold the ordinal properties of concave function such that a R C + a {x D : f(x) a}, is a convex set. Pseudo-Concave functions are a subset of quasi-concave functions that retain the property that critical points are automatically global maxima. That is, they possess two of the nice properties of concave functions: () Critical points characterize global maximum points; (2) Level set is convex. 2. Robinson Crusoe Our next step is to show that the properties imposed on the utility and production function in the Robinson Crusoe s problem allows us to guarantee that the First-Order Necessary Conditions from the derived lagrangean provide a sequence of consumption and capital that maximizes Robinson s lifetime utility. The problem is (2) U = max {c t,k t+ } T subject to: T β t u(c t )

4 TA SESSION 3 (i) c t + k t+ f(k t ) + ( δ)k t (ii) c t 0, k t+ 0 (iii) Given k 0 > 0 where u( ) and f( ) are concave functions. Therefore, since the summation of concave function is also concave, the lifetime utility of Robinson is concave, and consequently, pseudo-concave. Now, we need to show that the constraints are quasi-convex functions. Constraint (ii) is trivial. Let s rewrite constraint (i) as g(k t, k t+, c t ) = c t + k t+ f(k t ) ( δ)k t+, which implies we need to show that h( ) = g( ) is quasi-concave. Let x = ( k t, k t+, c t ) and ˆx = (ˆk t, ˆk t+, ĉ t ) be in D such that h( x a and h(ˆx) a, then h[λ x + ( λ)ˆx] a if is convex. Therefore, C + a {x D : h(x) a}, h[λ x + ( λ)ˆx] = f[λ k t + ( λ)ˆk t ] + ( δ)[λ k t + ( λ)ˆk t ] λ k t+ ( λ)ˆk t+ λ c t ( λ)ĉ t = f[λ k t + ( λ)ˆk t ] + λ[( δ) k t k t+ c t ] + ( λ)[( δ)ˆk t ˆk t+ ĉ t ] λf( k t ) + ( λ)f(ˆk t ) + λ[( δ) k t k t+ c t ] + ( λ)[( δ)ˆk t ˆk t+ ĉ t ] = λ[f( k t ) + ( δ) k t k t+ c t ] + ( λ)[f(ˆk t ) + ( δ)ˆk t ˆk t+ ĉ t ] = λh( x) + ( λ)h(ˆx) a. Since the objective function is pseudo-concave and the constraints are quasi-convex functions the sequence of consumption, next-period capital and lagrange multipliers λ, µ, γ that satisfy the First-Order Necessary Conditions from T T T T L = β t u(c t ) λ t [c t + k t+ f(k t ) ( δ)k t ] + µ t c t + γ t k t+, maximize U.

5 TA SESSION 2 Contents. Week 2: Review 2. No Ponzi Game and Transversality Condition 3. Borrowing in Infinite Horizon and Fluctuating Income 3. Week 2: Review This week we went over the first-order necessary conditions of the Robinson Crusoe s problem in finite time, which lead to the Euler Equation. The Euler Equation is a necessary condition for the sequence of capital be a global maximizers of lifetime utility. If we want to say more about the solution we need to make more assumptions about the utility function and the production function. Usually we would like to have homothetic utility function (level curves have tangents with same slope along a ray crossing the origin). Utility functions with Constant Relative Risk Aversion (CRRA) are often used. CRRA functions yield constant intertemporal elasticities of substitution. Additionally, we saw that taking the limits of the finite horizon problem s solution is not necessarily the solution for the infinite horizon problem. We need the objective function to be concave. Given this condition, we can characterize the solution of the infinite horizon problem by the Euler Equation of the finite horizon problem and the Transversality Condition (and other constraints). Infinite horizon formulations are intuitive and useful to draw conclusion about behavior of variables in the Balanced Growth Path, i.e., a situation where the endogenous variables grow at constant rate. Lastly, we saw that we need to impose the so called No Ponzi Game condition to have a solution in the borrowing problem with infinite horizon. 2. No Ponzi Game and Transversality Condition We want to show that the No Ponzi Game condition is equivalent to the Transversality Condition. We start with the finite horizon problem: () U = max {c t,b t+ } T subject to: T β t u(c t ) Date: Sept 2.

6 2 TA SESSION 2 (i) c t + b t+ y t + ( + r t )b t (ii) c t 0, b t+ 0 (iii) Given b 0 > 0 such that u( ) is increasing, concave and satisfies the Inada conditions such that c t > 0 and the budget constraint binds. The FONCs for this problem are: t (2) (i) β t u (c t ) λ t = 0 (ii) λ t + ( + r t+ )λ t+ = 0 t T (iii) φ T b T + = 0 (iv) φ T λ T = 0 that yield the following Euler Equation: (v) λ t 0; c t + b t+ = y t + ( + r t )b t (3) u (c t ) = β( + r t+ )u (c t+ ). Also, from the FONCs we have the boundary condition: (4) β T u (c T )b T + = 0. According to proposition 3.4 in Krusell s lecture notes, the sequences that satisfy the conditions above taken to infinity will maximize the infinite horizon problem (since u is concave). Note that it means satisfy the Euler Equation, the budget constraint and the Transversality Condition (5) lim T βt u (c T )b T + = 0. However, note that from equation (3): β T u (c T ) T u (c 0 ) =, + r s which we can plug in (4) and find the equivalent Transversality Condition: (6) lim T which is the No Ponzi condition. ( T s= + r s s= ) b T + = 0,

7 TA SESSION Borrowing in Infinite Horizon and Fluctuating Income We now have the representative agent problem in Infinite Horizon. We will assume the log utility function and that the household receives a unit of income when period t is even and no income when t is odd. Some the household solves the following problem: (7) U = max {c t,b t+ } subject to: (i) β t ln(c t ) c t + b t+ y t + ( + r t )b t (ii) c t 0 (iii) Given b 0 = 0 and {y t } (iv) No Ponzi Game Condition as in Equation (6) Therefore, the Euler equation is = β( + r t+ ). c t c t+ Solving the finite horizon problem recursively we find the intertemporal budget constraint: (8) ( T t s= + r s ) c t = ( T t s= + r s ) y t + ( T s= + r s ) b T + where we take limits in both side to find the solution for the infinite horizon case. Note the with the No Ponzi Game condition we get rid of the last term in the right-hand side: (9) ( t s= + r s ) c t = ( t s= + r s which means that the present value of the consumption must be equal to the present value of income. Now, let s add the assumption that + r = /β such that c t+ = c t = c. Therefore we can rewrite Equation (9) as (0) c such that β t = β t y t = c = β β 2 = + β. β 2t ) y t,

8 TA SESSION 3 Contents. Week 3: Review 2. Ricardian Equivalence in a Pure-Exchange Economy. Week 3: Review This week we started to work with general equilibrium models, specifically with an economy only with households with fluctuating income that exchange their endowments. We learned how to define a sequential equilibrium, i.e, an equilibrium where agents trade every period. We also saw how to add the government parametrically in the model economy. i.e., assuming that the stream of government spending is exogenous. 2. Ricardian Equivalence in a Pure-Exchange Economy Let s analyze a pure exchange economy with two types of households distributed in a continuum of size. Households of type A receive income equal to in even periods while households of type B receive income equal to in odd periods. There is a government spending g t and choosing how to finance the spending: either by tax or debt. The type-i household intertemporal problem is () U = max {c t,b t+ } T subject to: (i) T β t ln(c i t) c i t + b i t+ y i t τ i t + ( + r t )b i t (ii) c i t 0, b i t+ 0 (iii) Given b i 0 = 0. The government must choose {τt A, τt B, B t+ } T that satisfies the budget constraint (2) g t + B t+ = τ A t + τ B t + ( + r t )B t. Therefore a competitive equilibrium can be defined as follows: Date: Sept 9.

9 TA SESSION 3 2 Definition: A competitive equilibrium is a sequence of prices {rt } T t=, households choices {c A t, c B t, b A t+, bb t+ }T, government policies {g t, τt A, τt B, Bt+ }T such that: () Given prices and taxes, households choices maximize problem (); (2) Given prices and expenditure plan {g t } T, government policies satisfy budget constraint (2); (3) The market for goods clears: c A t + c B t + g t = yt A + yt B ; (4) The market for bonds clears: b A t+ + bb t+ + B t+ = 0. Since the Inada conditions hold for the log utility and it is an increasing function of consumption we know that c i t 0 will never hold in equality and the budget constraint will never hold in inequality. Therefore we have the following Lagrangian from the optimization of household i: T T L = β t ln c i t λ t [c i t + b i t+ yt i ( + r t )b i t + τt i ], that yields the following FONCs: (3) (i) β t /c i t λ t = 0 (ii) λ t + ( + r t+ )λ t+ = 0 t T (iii) φ T b i T + = 0 (iv) φ T λ T = 0 that yield the following Euler Equation: (v) λ t 0; c i t + b i t+ = y i t + ( + r t )b i t τ i t (4) c i t+ = β( + r t+ )c i t. Note that this condition will hold for both types of households and that β and interest rates are common to them. Therefore c A t+ c A t = β( + r t+ ) = cb t+ c B t ca t+ c A t = ca t+ g t+ c A t g t, where we used the good s market clearing. Let s assume the the spending is constant over time such that g t+ = g t = ḡ, which implies in perfect smoothing by the households c i t+ = ci t = c i. Note that this implies that ( + r t ) = /β. Solving the budget constraint recursively we have the intertemporal budget constraint (5) ( T t s= + r s ) c i t = ( T t s= + r s ) y i t ( T t s= + r s where we can plug in the results for consumption and interest rates such that ) τ i t

10 TA SESSION 3 3 T β t c i = T β t yt i T β t τt i c i = β β T + T β t yt i β β T + T β t τt i. The optimal consumption depends on the present value of all taxes levied rather than the level of taxes in specific time, that is, the timing of taxes doesn t matter. Ricardian Equivalence: Let the sequence {rt, c A t, c B t, b A t+, bb t+, g t, τt A, τt B, Bt+ }T be an equilibrium. Then {rt, c A t, c B t, b A t+, bb t+, g t, τ t A, τ t B, B t+ }T is also an equilibrium if T β t τ T t i = β t τt i i, and the sequence { B t+ }T satisfies τ A t + τ B t + ( + r t )B t = τ A t + τ B t + ( + r t ) B t Therefore, let s guess τ A t = τ B t = τ for all t and B t+ = 0 for all t as policies that satisfy the equilibrium conditions. Then we have (6) c A = + β τ and cb = β + β τ. Also, note that on t = 0 b A = β and b B = β + β + β, from the budget constraint at time zero, while in time t = we have b A 2 = bb 2 = 0. Therefore we have the optimal choice of bonds given by (7) b A t+ = β + β and b B t+ = β + β b A t+ = b B t+ = 0 if time is even; if time is odd. This choices of consumption and bonds satisfy the budget constraints and the Euler equation, therefore they are optimal. Additionally, substitute the optimal consumption in the the good s market clearing then τ = ḡ/2 which is in line with the choice of B t+ = 0 since it satisfies both the government budget constraint and the bond s market clearing. Equilibrium:: A sequence of prices r = /β for all t, household s choices given by equation (6) and (7), and policies τ = ḡ/2 and B t+ = 0 for all t is an competitive equilibrium.

11 TA SESSION 4 Contents. Week 4: Review 2. Arrow-Debreu Economies 2.. Pure Exchange Economy 2.2. Neoclassical Growth Model Infinite Horizon, Representative agent and pollution externality associated 3. Week 4: Review This week we introduced production in the economies previously studied which lead us to the Neoclassical Growth Model. With this framework in hand we went over the definition of sequential competitive equilibrium for this economy when we have different distributions of endowments: Firms own the capital and Households own the capital. Intuitively, the different relies on who is going to invest in capital. However, even though the structure of the problem changes, the outcome is the same, i.e., both economies will lead to the same equilibrium conditions. We also learned how we can transform the sequential problem in a Date-0 problem by combining the sequence of budget constraints in one intertemporal budget constraint. Moreover, when we define consumption in different periods as different goods we can see the Date-0 problem as a single period general equilibrium problem in the fashion of Arrow- Debreu. Since this is a general problem set up we went over its elements and learned how to define Arrow-Debreu Date-0 equilibria. Lastly, we learned that if preferences are convex, then any competitive equilibrium is equivalent to a type i-identical equilibrium. 2. Arrow-Debreu Economies The goal of this recitation is to go over examples of Arrow-Debreu economies depending on the assumptions over production, property rights, time horizon and preferences. We start with a simple pure exchange economy. 2.. Pure Exchange Economy. Let s specify what are the 8 elements of an Arrow- Debreu economy in the context of a pure exchange equilibrium: A. People: There two types of agents distributed uniformly in a continuum of size λ i. Date: Sept 6.

12 TA SESSION 4 2 B. Commodities: There is only one commodity being trade in this economy each period: the final good. However, consumption each period is regarded as a different commodity. C. Commodity Space: Then, the commodity space is L = {R T + ; is the Euclidean norm}. D. Consumption Set: The set of (technically) feasible consumption goods each type-i consumer faces is X i = {x i L : x i 0}. E. Preferences: The preferences are given by the lifetime utility function T βt u i (c i t), where u c > 0, u cc < 0 and Inada conditions hold. F. Firms: There are no firms J = 0 G. Technology: It is a convention to assume that in the pure exchange economy the technology satisfies the free disposable conditions such that the production set is Y j = { R T + + }. H. Endowments: Household have the following exogenous endowment sequence: w A = (w A o,..., w A T ); w B = (w B o,..., w B T ). Then we can define the competitive equilibrium as follows: Definition: An competitive equilibrium is an allocation (x A, x B ) and a price vector p R T + + such that: () given p, x i X i maximizes T βt u i (x i ) subject to p x i p w i forall i; (2) Market clears: λ A 0 x A t (m)dm + λ B 2.2. Neoclassical Growth Model. 0 λ A λ B x B t (m)dm = wt A (m)dm + wt B (m)dm Firms own capital. When the firms own capital they make the decision about investment and there is not trade of capital. Therefore capital is not a commodity. The only commodity being traded is the final good. households own firms and receive dividends according to the share θ of firm they hold. Then we have the following commodity space, consumption set and production set: Commodity Space: L = {R T + ; is the Euclidean norm}; Consumption Set: X = {c L : c 0}; Production Set: Y = {y L : y t 0 and {k t+ } T such that y t + [k t+ ( δ)k t ] f(k t ), given k 0 }. Definition: A competitive equilibrium is an allocation (c, y ) and prices p R T + + such that:

13 TA SESSION 4 3 () given prices, c X maximizes T βt u(c t ) such that p c J j= θj p y j ; (2) given prices, y j Y j is such that p y j p y j y j Y j ; (3) Market clears: λ 0 c (m)dm = J j= yj Households own capital and sell capital services to firms. In this case there are two goods being traded: final goods (which can be used to consumption or investment) and capital services. Therefore Commodity Space: L = {R 2(T +) ; is the Euclidean norm}; Consumption Set: X = {x L where x t (x t, x 2t ) : x t 0, x 2t 0 and {c t, k t+ } T such that c t + [k t+ ( δ)k t ] x t and x 2t k t, given k 0.}; Production Set: Y = {y L where y t (y t, y 2t ) : 0 y t f(y 2t ) such that y 2t 0} Infinite Horizon, Representative agent and pollution externality associated. Consider a situation where there is a continuum of identical, infinitely lived agents uniformly distributed on the [0, ] interval. They have the period utility u(c t, Y t ) depending on consumption and aggregate income: u c > 0 and u Y < 0. The technology of production is given by F (k t, l t ) using capital and land as inputs. Capital depreciates at rate δ. Assume households own k 0 and L =, and sell their services to the firm. Write down the Commodity Space, Consumption Set and Production set, and define the Arrow-Debreu Date-0 Equilibrium. Note that there are three commodities being traded: final good, land services and capital services. Then Commodity Space: L = {R 3( ) ; is the supnorm}; Consumption Set: X = {x L where x t (x t, x 2t, x 3t ) : x t 0, x 2t 0, x 3t 0 and {c t, k t+ } such that c t + [k t+ ( δ)k t ] x t, x 2t and x 3t k t, given k 0.}; Production Set: Y = {y L where y t (y t, y 2t, y 3t ) : 0 y t F (y 2t, y 3t ) such that y 2t 0 y 3t 0}.

14 TA SESSION 4 4 Definition: An Arrow-Debreu Data-0 equilibrium is a price vector p = {p t } where p t (p t, p 2t, p 3t ) and allocation (x, {y j } J j= ) which satisfies () given p and {y j } J j=, x X maximizes U(x, T j=0 yj ) subject to p tx t (p 2tx 2t + p 3tx 3t ) (2) given p, (p ty j t p 2ty j 2t p 3ty j 3t ) (p ty j t p 2ty j 2t p 3ty j 3t ) y j Y j (3) Markets clear: (a) Final good: x t = J j= yj t ; (b) Land Services: x 2t = J j= yj 2t ; (c) Capital Services: x 3t = J j= yj 3t

15 TA SESSION 5 Contents. Week 5: Review 2. Negeshi-Mantel Algorithm - An example. Week 5: Review This week we went over the Fundamental Welfare theorems. The first welfare theorem ensures that, if preferences are locally non-satisfied, allocations yielded by competitive equilibrium are Pareto optimum, which happens if and only if these allocations solve the weighted social planner problem. Second theorem states that, under certain conditions, for all Pareto optimum allocation there is a vector of prices which together with this allocation characterizes a Quasi-Price Equilibrium with Transfers. Therefore, we can use Negeshi- Mantel Algorithm to find the competitive allocations and prices by solving a weighted social planner problem and setting transfers equal to zero. Negeshi-Mantel Algorithm is very useful when we have heterogeneity in preferences or endowments. The question remaining is: when can we ignore heterogeneity? We saw that when firms have constant returns to scale we can work with a representative firm. For households, we need the indirect function to have a Gorman Form, i.e., the indirect utility is linear on wealth and its slope is individual-independent. 2. Negeshi-Mantel Algorithm - An example We will use question 4 in problem 4 to illustrate an application of Negeshi-Mantel algorithm. In this problem we have two types of agents with different preferences and different endowments. A way to solve for equilibrium prices and allocation is to follow the algorithm: () Solve the social planner problem for arbitrary weights (2) Use the first order conditions from the problem to back out prices (3) Compute the transfers as the dat-0 difference between spending and wealth as a function of weights Date: Sept 23.

16 (4) Solve for the weights such that transfers equal zero. a. The household problem is TA SESSION 5 2 () max {c i t,ci 2t,bi t+ } subject to: (i) β t [ln c i t + φ i c i 2t] c i t + p 2t c i 2t + b i t+ w i t + ( + i t )b i t (ii) c i t 0, c i t 0 (iii) Given b i 0 (iv) No Ponzi Game Note that solving the budget constraint recursively we have the following intertemporal budget constraint: ( t s= + i s ) (c i t + p 2t c i 2t) = ( t s= + i s ) w i t The Firms problem is to choose x t to maximize p 2t Λx t x t. p t (c i t + p 2t c i 2t) = p t wt. i Definition: A Date-0 competitive equilibrium is a sequence of prices {p t, p 2t }, household allocations {c i t, ci 2t } and firms allocations {x t} such that () Given prices, household allocations maximize the following problem: subject to: max {c i t,ci 2t } β t [ln c i t + φ i c i 2t] (i) p t (c i t + p 2t c i 2t) = (ii) c i t 0, c i t 0 p t wt i For all i = A, B. (2) Given prices, firms allocations maximize profits. (3) Market Clearing: (a) Good : c A t + cb t + x t = w A t + wb 2t (b) Good 2: c A 2t + cb 2t = Λx t

17 TA SESSION 5 3 b. The Social Planner s problem is: subject to: max α {c A t,ca 2t,cB t } β t [ln c A t + φ i c A 2t] + ( α) β t ln c B t (i) c A t + cb t + x t = w A t + wb 2t (ii) c A 2t = Λx t (iii) c i t 0, c i t 0 i = A, B c. The Lagrangian from the Social Planner s problem is L = α β t [ln c A t + φ i c A 2t] + ( α) β t ln c B t λ t [c A t + c B t + c A 2t/Λ ]+ [µ t c A t + γ t c A 2t + η t c B t] Given the log utility for good and that parameters are such that c 2t > 0, we can get rid of non-negativity lagrange multipliers. Using the first order conditions we have the following allocations: c t = Λφ A c A 2t = Λ αφ A c B t = α αλφ A We have that in equilibrium the ratio of marginal utility between two different goods must be equal to the ratio of their prices. Then, p t = β t and p 2t = /Λ, where we have used the fact that consumption is constant over time. We then compute the transfers as a function of weights: τ(α) A = p t (c A t + p 2t c A 2t wt A ) = β t ( ( + Λφ A )α αλφ A τ(α) A = ( + ΛφA )α ( β)αλφ A β 2, w A t )

18 and solve for α when transfers equal zero. The result is α = TA SESSION β + β( + Λφ A ).

19 TA SESSION 6 Contents. Week 6: Review 2. Markov Chains 3. Leisure and Aggregation 2. Week 6: Review This week we introduced uncertainty to the economy. We saw that goods can be distinguished by their physical characteristics, time and state of nature. Therefore, the consumption in t will depend of the history of events until time t. We learned that in the Debreu approach approach of dealing with uncertainty (which is basically a date-0 approach) the structure doesn t change much once agents will buy contingent claims, i.e., they will sign contracts contingent to events history for each history in each period. An important result so far is that households fully insure themselves if there is no aggregate uncertainty and the markets are complete. Also, there will be full risk sharing, i.e., individual consumption will depend on aggregate income, not on individual incomes. 2. Markov Chains Recall that π(s t ) is the probability of history s t = (s t, s t,..., s 0 ) to happen. Therefore we can solve recursively for π(s t ) such that π(s t ) = P r(s t s t )P r(s t s t 2 )... P r(s s 0 )π(s 0 ). Usually we want to add some structure to the distribution of events so we can simplify the above equation. A straightforward simplification is to say that the events are IID (Independent and identically distributed). Then P r(s t s t ) = P r(s t ) for all t, i.e., the next event doesn t depend on the history of events. However, this a strong assumption. A weaker assumption is that the history of events s t has the Markov Property. Date: Sept 30. See Ljungqvist and Sargent (202, p.29)

20 TA SESSION 6 2 Definition: A stochastic process s t is said to have the Markov Property if for all t and all k, P r(s t+ s t, s t,..., s t k ) = P r(s t+ s t ). Definition: A time-invariant Markov Chain characterizes a Markov Process s t with the following triple of objects: () State Space: S (2) Transition Matrix: P (3) Initial Distribution: π 0 The elements of P are P ij = P r(s t+ = e j s t = e i ) t. Note that P r(s t+2 = e s t = e ) = P r(s t+2 = e s t+ = e )P r(s t+ = e s t = e ) + P r(s t+2 = e s t+ = e 2 )P r(s t+ = e 2 s t = e ) = where P (2) P P + P 2 P 2 = P (2), is the element [, ] of matrix P P. Generally: P r(s t+ = e j s t = e i ) = P (k) ij. Definition: We say that S, P, π 0 is a Stationary Markov Chain if the initial distribution is such that π 0 = π 0 P, i.e, π 0 is an eigenvector associated with P. 3. Leisure and Aggregation 2 In this economy there are I types of households who value leisure. Firms own capital and households own firms. Therefore, there are two markets in this economy: final good and labor. a. b. Commodity Space: L = {R : supnorm is the metric} Consumption Set: X i = {x i L : x i 0, 0 xi 2 } Production Set: Y j = {y j L : y j 0,j 2 0 and {Kj t+ } such that yj t + (K j t+ ( δ)kj t ) AKjθ t y j θ 2t, K j t+ 0 and given Kj 0 } 2 Question 2, Midterm, Fall 200

21 TA SESSION 6 3 Definition: A date-0 competitive equilibrium is a sequence of household choices {c i t, h i t },i I, firms allocations {K j t+, H j t },j J and prices {p t, wt } such that () Given prices and dividends, household allocations maximize βt u(c i t, h i t) subject to p t c i t p t wt h i t + θ ij p t d j t, j J and c i t 0, 0 h i t for all i I. (2) Given prices, firms maximize p t d j t subject to d j t = AKjθ t H j θ t wt H j t Kj t+ + ( δ)kj t, and K j t+ 0 and Hj t for all j J. (3) Market Clearing: i I λi h t = j J Hj t and λ i c i t = AK jθ t H j θ t K j t+ + ( δ)kj t. i I i I c. The Lagrangian is [ ] L = β t u(c i t, h i t) λ p t c i t p t wt h i t a i 0, where a i 0 = j J θij p t d j t, and we ignore the non-negativity constraints given the functional form of utility. The FONCs lead to the following equilibrium equations c i t = β t c 0 /p t and h t = ( η)c t ηw t. Using this conditions in the intertemporal budget constraint we find that both consumption and hours worked equilibrium equation have the Gorman form: and h i 0 = c i 0 = η( β) ( β)( η) w 0 p t wt + η( β)a i 0, p t wt ( η)( β) w 0 a i 0.

22 TA SESSION 7 Contents. Week 7: Review 2. Asset Pricing. Week 7: Review We realized that the Date-0 (Debreu) approach for dealing with uncertainty make agents to choose allocations for more state histories than they would have to choosing sequentially. Therefore, we move towards the Arrow (sequential) approach where we introduced the Arrow securities. Arrow securities are contingent claims that will pay units of consumption for the realization of given state. This kind of asset allows households to transfer wealth, not only across time, but also across states of nature. This led us to a discuss of different ways to define the No Ponzi Game and to show that if markets are complete, i.e., agents can buy and sell securities for all states, the date-0 and sequential approach are equivalent and the price of Arrow securities are equal to the date-0 prices. This way to price Arrow Securities is an example of Asset Pricing. 2. Asset Pricing I m gonna use question 2 in the midterm of Fall 202 to illustrate the idea of Asset Pricing. So, assume a pure exchange economy with no aggregate uncertainty but idiosyncratic shocks. Also, let the number of state be finite and follow an IID. a. The budget constraint with a complete set of Arrow securities is () c t (s t ) + q t (s t, s t+ )a t+ (s t, s t+ ) a t (s t ) + w t (s t ). s t+ S b. The budget constraint for the risk-free-only case is: c t (s t ) + b t+ (s t ) w t (s t ) + ( + r t (s t ))b t (s t ). Date: Oct 7.

23 c. Households maximize TA SESSION 7 2 β t π(s t ) ln c(s t ), s t S t subject to (), no-negativity constraints and no Ponzi game condition. necessary conditions will lead to q t (s t, s t+ ) = β π(st, s t+ ) π(s t ) c(s t ) c(s t, s t+ ). The first order Note that, since preferences are logarithmic and there is no aggregate uncertainty, then the households we share the risks such that c(s t ) = c(s t+ ) = w, where w is the aggregate endowment. Therefore, q t (s t, s t+ ) = β π(st, s t+ ) π(s t ) = βπ(s t+ s t ) = βπ(s t+ ), where the third equality comes from the fact that the states are IID. d. Let s first understand how to price an general asset. At date zero, the price of an general asset must the present value of the dividends it pays for all periods and all states. Then, P j 0 = s t p t (s t )d j t (st ), where p are the Arrow-Debreu prices and d are dividends. If the left-hand side is bigger than the right-hand side, households would want to sell j assets. If the opposite happens households want to buy infinitely many j assets. Therefore, the equality above must be the equilibrium condition. We make the same argument for the price of a general asset at time t: P j t (st τ=t s ) = τ s t p τ (s τ )d j τ (s τ ) p t (s t, ) where the summation over s τ s t means the sum over all possible histories for s τ given s t happened already. For example, suppose we want to price a risk-free bond that pays one See Ljungqvist and Sargent (202, p.266).

24 TA SESSION 7 3 unit of consumption one period ahead and nothing afterwards for all states. Then Pt risk-free (s t s ) = t+ s t p t+(s t+ ) s p t (s t = t+ S p t+(s t, s t+ ) ) p t (s t = q t (s t, s t+ ), ) s t+ S where in the last equality we used the fact that the ratio of Arrow-Debreu prices is the price of an Arrow security as shown in class. For this question we need to price a risk-free asset that pays one unit of consumption only two periods ahead nothing before, nothing after. The price of this two-period bond is P two-period t (s t s ) = t+2 s t p t+2(s t+2 ) p t (s t = ) s t+ S p t+ (s t, s t+ ) p t (s t ) s t+2 S p t+2 (s t+, s t+2 ) p t+ (s t, s t+ ) s t+ S s t+2 S p t+2(s t, s t+, s t+2 ) p t (s t = ) = s t+ S Then, using the result from previous item: P two-period t (s t ) = βπ(s t+ ) s t+ {,2} q t (s t, s t+ ) s t+2 {,2} (βπ) 2 + 2β 2 π( π) + β 2 ( π) 2 = β 2. s t+2 S βπ(s t+2 ) = q t (s t+, s t+2 ).

25 TA SESSION 8 Contents. Week 8: Review 2. Good Times and Bad Times. Week 8: Review This week we discussed about complete and incomplete markets. We reviewed how, with complete markets, households will full insure since ratio of marginal utility for all agents are the same for all possible state histories. We also saw how incomplete markets may arise either endogenously from private information, moral hazard, adverse selection and limited enforcement constraints, or exogenously by simply allowing households to have bonds as financial securities. In such situations households save more than they would with complete markets to protect themselves from bad states of nature; we call these extra savings precautionary savings. Lastly, we went over a general equilibrium model with uncertainty: the stochastic Neoclassical Growth Model where we introduced a financial sector which sell Arrow securities to households and buy capital and rent it to firms. 2. Good Times and Bad Times In this recitation we will go over one example about how to apply Negeshi-Mantel Algorithm to find the competitive allocation in a Dynamics Stochastic General Equilibrium model. We use question 2 in the second midterm of Fall 200. In this question aggregate labor productivity and individual labor supply are state contingent. a. The date-0 budget constraint is: p t (s t )c i t(s t ) w t (s t )h i t(s t ), s t S t s t S t and the non-negativity constraints are c i t(s t ) 0 and 0 h i t(s t ) for all t and s t S t. Date: Oct 4.

26 TA SESSION 8 2 b. The sequential constraint with complete markets is: c i t(s t ) + q t (s t, s t+ )a i t+(s t, s t+ ) w t (s t )h i t(s t ) + a i t(s t ), s t+ S for all t and s t S t. The non-negativity constraints are c i t(s t ) 0 and 0 h i t(s t ) for all t and s t S t. Lastly, the No Ponzi Game condition is lim q T (s T, s T + )a i T +(s T, s T + ) 0, T s T S T for all s T + S. c. The sequential constraint with incomplete markets is: c i t(s t ) + q t (s t )b i t+(s t ) w t (s t )h i t(s t ) + b i t(s t ), for all t and s t S t. The non-negativity constraints are c i t(s t ) 0 and 0 h i t(s t ) for all t and s t S t. The No Ponzi Game condition is lim q T (s T )b i T +(s T ) 0, T s T S T for all s T + S. Will Consumers be better off in complete markets? The intuitive answer for this question is: yes; at least they cannot be worse off. Recall that with a complete set of Arrow Securities any household can mimic a risk-free bond by buying the same amount of claims for all possible states. Moreover, this portfolio will cost the same as the risk-free bond. Therefore, in the worst-case scenario, households will be as good with complete markets as in incomplete markets. Additionally, complete markets add the possibility to households to transfer income also across states, not only time. To see this let T = and s be IID. Also, assume the uncertainty happens before households make decisions. Therefore they maximize ln c 0 + β s S π(s) ln c (s). The first order conditions for the consumption decision will be the same for the complete and incomplete markets. The first order conditions for the asset will change Complete Markets Incomplete Markets λ 0 q(s) + λ (s) = 0 λ 0 q + s S λ (s) = 0.

27 TA SESSION 8 3 Note that for the case of incomplete markets the condition is not state dependent. Therefore, the household cannot smooth consumption across states. (See Krusell s lecture notes, page 74). 2. We follow Negeshi-Mantel algorithm to solve for competitive equilibrium. First we find the Social Planner allocation as function of weights, then find prices and transfers also as function of weights. Lastly we solve for weights when transfers are zero. The social planner problem is to choose consumption and supply of labor for each household type for all periods and states that maximizes α β t π(s t ) ln c t (s t ) + ( α) β t π(s t ) ln c 2 t (s t ), s t S t s t S t constrained by the resource constraint c t (s t ) + c 2 t (s t ) = A t (s t )[h t (s t ) + h 2 t (s t )] = A t (s t ), for all periods and states, and the non-negativity constraints. The first order conditions for the social planner maximization problem leads to αc 2 t (s t ) = ( + α)c t (s t ), which together with the resource constraint yields c t (s t ) = αa t (s t ) and c 2 t (s t ) = ( α)a t (s t ), for all t 0 and all s t S t. We will have prices for trading across time and state for final good and labor. We will set p 0 (s = G) =. First, note that the ratio of prices is the ratio of marginal utilities such that: β t π(s t )c i 0 (s 0) π(s 0 )c i t (st ) Same rationale works for prices across states: Then, = p t(s t ) p 0 (s 0 ). π(s 0 = B)c i 0 (s 0 = G) π(s 0 = G)c i t (s 0 = B) = p 0(s 0 = B) p 0 (s 0 = G) = αa G αa B = Ag A B. p t (s t = G) = β t π(s t ) π(s t ) π(s t ) π(s t 2 )... π(s ) c i 0 (s 0 = G) π(s 0 ) c i t (s t = G) p 0(s 0 = G) = β t π(s t s t )π(s t s t 2 )... π(s s 0 ) = β t 0.5 t,

28 and TA SESSION 8 4 p t (s t = G) = β t π(s t ) π(s t ) π(s t ) π(s t 2 )... π(s ) c i 0 (s 0 = B) π(s 0 ) c i t (s t = G) p 0(s 0 = B) = (β/2) 2 AB A G A G A B. We then use the price across states to find p t (s t = G) = (β/2) 2 and p t (s t = B) = AG A B (β/2)2. Moreover, from the zero profit condition: w t (s t ) = p t (s t )A t (s t ), which implies p t (s t = G) = A G (β/2) 2 and p t (s t = B) = A G (β/2) 2. We define the transfer for the type- agent as τ (α) = p t (s t )c t (s t ) w t (s t )h t (s t ), s t S t s t S t where first we plug in the consumption allocations and use the fact that there are 2 t+ histories in time t, in which half are good times and half bad times: τ (α) = 0.5p t (s t = G)2 t+ αa G + 0.5p t (s t = B)2 t+ αa B 0.5w t (s t = G)2 t+ ( ɛ) + 0.5w t (s t = B)2 t+ ɛ. Plugging the prices in and rearranging: τ (α) = (2α )A G β t. Then, to make the transfer equal to zero α = 0.5. Applying the result in the allocation of consumers we have c t (s t ) = c 2 t (s t ) = 0.5A t (s t ).

29 TA SESSION 9 Contents. Review.. Balanced Growth Path 2.2. Arrow Debreu Economies 2.3. Stochastic Economies 3. Review In this first part of the course we went over the foundations of Neoclassical Growth model. We started the course with a social planner problem Robinson Crusoe, which is basically an inter-temporal allocation problem. There are no prices in this framework and the dynamics of the problem depends heavily on the storage technology (investment), i.e., the ability to send resources to the future. Within this environment we learned about transversality conditions, balanced growth paths and steady states. Then we dropped the hypothesis of a social planner and moved towards the equilibrium through market exchanges, where households will trade endowments and use bonds to smooth their consumption across periods, that is a pure exchange economy. Adding firms we have a production economy. Therefore, writing down households and firms problems constrained by sequences of budget and technologies constraints, market clearing conditions and prices given, we define a sequential competitive equilibrium. Nonetheless, one of our main goals is to show that markets will allocate resources as good as a social planner (given some conditions). For this we find useful to write down our problem in a Arrow-Debreu General Equilibrium framework, where we see goods in different periods as different goods and use the inter-temporal budget constraint instead of the sequential constraints. This is a alternative way to define competitive equilibrium called Date-0 since all decisions (contracts) are made in the first period. Within this framework we discussed the first and second welfare theorems showing that competitive equilibrium will be Pareto optimum. This makes room for the Negeshi-Mantel algorithm which help us to pin down competitive allocations from the social planner maximization problem. Date: Oct 2.

30 TA SESSION 9 2 Another issue macroeconomists need to deal with regards aggregation: when can we use representative agents? We then discussed the result that representative households can be assumed when indirect utilities are in the Gorman form. For the firms, we need the technology to have constant returns to scale. Finally we introduced uncertainty to our model. The date-0 approach to deal with uncertainty is called the Debreu approach and lead to some important results such the full insurance and full risk sharing. On the other hand, we can use the sequential approach introducing the Arrow Securities to the sequence of constraints. Within this framework, we learned about the basics of asset pricing. When markets are complete, agents can protect themselves from any state of nature, i.e., they can transfer wealth across states. However, in incomplete markets that is not possible. By introducing uncertainty in a model with households and firms we have the stochastic Neoclassical Growth model... Balanced Growth Path [Spring 200, Midterm, Question ]. The Social planner chooses {c t, e t, k t+ } to maximize βt u(c t ) subject to c t + k t+ + e t = k α t e ( α)φ + ( δ)k t, non-negativity conditions and given initial capital allocations. The first order conditions are β t u (c t ) = λ t λ t = λ t+ [αkt+ α e( α)φ t + ( δ)] λ t = λ t+ [( α)φkt+e α [( α)φ ] t ] Note that if c t+ /c t = + γ c is constant, then λ t /λ t+ is constant. Consequently the terms within square brackets are constants. This implies ( + γ k ) = ( + γ e ) φ, and + γ k = ( + γ e ) [ φ( α)]/α. Those two equations hold in equality only if γ k = γ e = 0 given φ <. The fact that physical and human capital are constants over time together with the resource constraint implies that consumption is also constant of time. Then, we have a steady state in this economy..2. Arrow Debreu Economies [Spring 2007, Midterm, Question ]. Note that there are 4 goods being traded: () final good from firms to households, (2) labor services

31 TA SESSION 9 3 from households to firms, (3) capital services from households to financial sector, and (4) capital services from financial sector to firms. Therefore our commodities space have four dimensions: L = {R 4 : is the Euclidean Norm} and consumption and production sets are subsets of it X = {x L : x 0, 0 x 2, 0 x 3 k 0, k 0 : x c = k } Y B = {y B L : 0 y B y Bθ 4 y B θ 2, y B 4 0, y B 2 0} Y F = {y F L : y F 4 yf 3 + γ, yf 3 0, y F 4 0}..3. Stochastic Economies [Spring 200, Midterm 2, Question ]. The social planner optimization will lead to two sets of allocations: c t (s t ) = α α and c2 t (s t ) = w t α α, if we have interior solution. If corner solution c t (s t ) = w t and c 2 t (s t ) = 0. Let s stick with the interior solution. Then we have prices p t (s t ) = (β/2) t. Find transfer and solving for α when transfers are 0 will lead to α = γ +γ. Then c t (s t ) = γ. Note that if γ < ɛ we always have interior solution. Then c t (s t ) = γ and c 2 t (s t ) = w t γ. To find the value of this insurance contract we need the price q t (s = R) = p t+(s = R) p t (s = R) = β/2. Additionally we want to find a t+ (s = R) that insures the same income in both states, then we want a such that γ( + ɛ) = γ( ɛ) + a, such that a t+ (s = R) = 2γɛ. Therefore, the value of the insurance contract is q t (s = R)a t+ (s = R) = βγɛ.

32 TA SESSION 0 Contents. Weeks 9 & 0: Review 2. Overlapping Generation Models 2.. Pareto Optimality 2.2. Seignorage 2. Weeks 9 & 0: Review The two last weeks we talked about deviations from the Neoclassical Growth model. The first deviation is the fairly used Overlapping Generation Model. This model provides an intuitive way to model household behavior where different generations interact in one period and choices of one generation will affect future generations. Even though intuitive, the downside of these models are that they are not necessarily Pareto optimal. Additionally, these models allow equilibrium with valued currency. The second deviation is the introduction of monopolistically competitive environments where we have different intermediate goods used to produce a final good or preferences for different goods. The aggregation of either production or preferences is done by the Spence-Dixit-Stiglitz aggregator. Another important feature in this kind of model is the fixed cost of production. 2. Overlapping Generation Models 2.. Pareto Optimality. We start with an example presented by Acemoglu (2009, p. 328) to illustrate why competitive equilibria in OLG models are not necessarily Pareto optimal. There is an infinity number of households. Each household is endowed with one unit of an unique good. Therefore, there are infinitely many goods as well. Household i draws utility from his own good and the good to next index individual only, i.e., u i = c i i + c i i+. One can show that a vector of prices such that p j = for all j will lead to a competitive equilibrium in autarky. Date: Oct 28.

33 TA SESSION 0 2 Nonetheless, this CE is not Pareto optimum. We could set an arbitrary individual i such that she will consume her endowment and the endowment of individual i +. All households indexed by i < i would keep consuming their own endowment, while individuals i > i would consume the endowment of individuals i+. Therefore, household i is better off and all households i i are indifferent. This allocation Pareto dominates the CE, which implies that the First Welfare Theorem doesn t apply in this situation. Why? The assumption that the value of endowments must be bounded is violated in this case, since there are infinitely many households receiving one unit of endowment and prices are constant, the value of endowments goes to infinity, what makes the allocation proposed above feasible. We can translate this example to the case of OLG if we let i mean generation Seignorage. We will use question in Midterm 2 of Fall 200 to review the idea of monetary equilibrium in the OLG models. a. Definition: A monetary sequence competitive equilibrium is a sequence of young household choices {(c t (t), c t+ (t), m t (t))} t=, old alive choice c (0), government policies {M t, T t } t=, and prices {q t } t= such that: () Given prices and policies, young household choices maximize subject to ln(c t (t) + α) + ln(c t+ (t) + γ), c t (t) + q t m t (t) w, c t+ (t) w 2 + q t+ [m t (t) + T t+ ], T t+ = (z )m t (t), and the non-negativity conditions for all t. (2) The old alive t = chooses consumption such that c (0) = w 2 + q M 0. (3) Government budget is such that T t = M t M t for all t. (4) Market clears: c t (t) + c t (t ) = w + w 2 and m t (t) = M t for all t. b. The point here is that the transfer is not lump-sum, therefore the young agent will take in consideration that more savings will lead to a bigger transfer. The first order conditions for the young agent lead to the following difference equation: q t+ = q t (w 2 + γ) z[w + α 2q t m t (t)].

34 TA SESSION 0 3 c. A stationary equilibrium means that q t M t is constant over time. Therefore, we can rewrite the previous equation as S t+ = S t(w 2 + γ) w + α 2S t = f(s t ). One can show that f(0) = 0, f (S t ) > 0, f (0) < and f (S t ) > 0. Therefore, there is a unique stationary equilibrium S and for any S 0 > S, lim t S t =, which means that real cash balances will exceed the supply of goods. d. Solving for S such that S = f(s ) implies S = (w + α) (w 2 + γ) 2M t = p t [(w + α) (w 2 + γ)] 2 e. The result here is different because the transfer is not lump-sum. With the lump-sum transfer, the result would be ( + z)m t = p t [(w + α) z(w 2 + γ)].

35 TA SESSION 2 Contents. Weeks & 2: Dynamic Programming.. A Reinterpretation of Dynamic Problems.2. The General Formulation 2.3. Numerical Solution 3.4. Guess and Verify 3. Weeks & 2: Dynamic Programming.. A Reinterpretation of Dynamic Problems. Let s write our usual social planner planner problem: max β t u(c t ) {c t,k t+ } subject to c t + k t+ f(k t ) c t 0, k t+ 0, k 0 given. Given Inada conditions hold we can rewrite it as () max {k t+ } β t u(f(k t ) k t+ ) subject to c t 0, k t+ 0, k 0 given. Now, let V : R + R be a function that returns the maximized date zero utility given k 0. We call V (k 0 ) a value function. Note that in problem () we want to find a infinite sequence of variables (k t ) by solving one problem. We will rewrite this as sequence of infinite (similar) problems with only one variable. Note that we can use Bellman s Principle of Optimality Date: Nov. An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.

36 TA SESSION 2 2 to show the following: V (k 0 ) = max {k t+ } β t u(f(k t ) k t+ ) = max {k t+ } u(f(k 0 ) k ) + max k max k [ max {k t+ } t= u(f(k 0 ) k ) + β u(f(k 0 ) k ) + max β t u(f(k t ) k t+ ) = t= β t u(f(k t ) k t+ ) = t= {k t+ } t= t= ] β t u(f(k t ) k t+ ) = max k {u(f(k 0 ) k ) + βv (k )}. Intuitively, the planner chooses only the next period capital in each time, instead of choosing all the streaming of savings. The non-negativity constraints become a set where the choice of k is in: k 0 and 0 c 0 f(k 0 ) k which implies k f(k 0 ). Moreover, note that differently from the previous approach, here the unknown is the function V ( ), therefore we have a functional equation: V (k 0 ) = max {u(f(k 0) k ) + βv (k )}. 0 k f(k 0 ) The study of dynamic optimization problems through the analysis of such functional equations is called Dynamic Programming The General Formulation. The Dynamic Programming formulation has two key components: state variables (x t ) and control variables (a t ). The state variables represent the variables given in the beginning of the period, while control variables are the actions taken by individuals. In the example above, k 0 is a state variable and k is the control variable. The formulation also has an instantaneous function u(x t, a t ), a transition equation x t+ = f(x t, a t ) and feasibility constraints on control variables Γ(x t ). In the planner s problem has the utility function as instantaneous function, the law of motion of capital as the transition equation and the real line [0, f(k 0 )] as the feasibility constraints for k. When maximizing the period problem the individual will choose controls as a function of the states α(x t ). We call this optimal decision rule policy rule. Then we can rewrite the problem as V (k 0 ) = u(f(k 0 ) α(k 0 )) + βv (α(k 0 )). 2 Note that for finite horizon, the derivation above is not quite right since V (k) would have a summation for T periods. Therefore, we would need to index our value function by the period its is being evaluated. For this motive, there is not so much gain in using dynamic programming for Finite Horizon s problems.

37 TA SESSION 2 3 Therefore, V ( ) and α( ) are unknown functions. Lastly, we say that we have a Stationary Dynamic Programming problem when the functions Γ, f, u are constant overtime, what implies that V and α are constant functions overtime. Given the stationarity of the problem, we have infinitely many similar problems where the time subscripts are meaningless, except when distinguishing present period and next period variables. Then it is common practice to rewrite k t = k and k t+ = k..3. Numerical Solution. Note that V (k) = Ψ(V (k)), where Ψ : C C is a mapping that takes continuous, bounded functions into continuous, bounded functions. Therefore, we have a fixed point to find. Let s say we have a initial function V 0 to start with. Then if V 0 = ΨV 0 then we are done. If not, we call V = ΨV 0 and keep going V 2 = ΨV = Ψ 2 V 0,... V n = Ψ n V 0. Then we have a sequence {Ψ i (V 0 )} n i=. The question is: does this sequence converge? If yes, lim i n Ψ i (V 0 ) = V where V such that V = ΨV. To guarantee that we have a solution for any starting function V 0 we need Ψ to be a contraction mapping: Theorem (Contraction Mapping). Let (C, d) be complete metric space and Ψ : C C be a contraction mapping. Then, () ΨV = V has exactly one solution V C (2) for any V 0 C and i = 0,, 2,... d(ψ i (V 0 ), V ) β i d(v 0, V ) 0 < β < The easiest way is to check if Ψ satisfies Blackwell s Sufficient conditions: Theorem 2 (Blackwell s Sufficient Conditions for Contraction Mapping). Let (C, d) be a space 3 of continuous, bounded functions R n R with d(f, g) = sup x D C f(x) g(x). If Ψ : C C satisfies: () Monotonicity: V (x) W (x) Ψ(V (x)) Ψ(W (x)); (2) Discounting: Ψ(V + α) αβ + Ψ(V ). Then, Ψ is a contraction mapping..4. Guess and Verify. We can also find the functional forms V and α by guessing. Let s use question 2 in Fall 202 Final exam to illustrate this idea. In the problem the planner wants to maximize βt ln c t subject to c t + k t+ Ak t and the non-negativity conditions. 3 A complete metric space.

38 TA SESSION 2 4 Figure. Value function iteration 0 and 00 iterations a. The Bellman equation associated to the problem is V (k) = max {ln[ak 0 k Ak k ] + βv (k )}, where we used Inada conditions to have the resource constraints in equality. b. To use Benveniste-Scheinkman we assume V is differentiable and take the first order condition for the maximization above, which implies: Ak k = βv k (k ). Then, we use the policy function to rewrite the Bellman equation as and find the derivate with respest to k: V k (k) = A α k(k) Ak k + βv k (k )α k (k) = V (k) = ln[ak α(k)] + βv (α(k)), V k (k) = Plugging the result in the first step we have [ A Ak k + ] Ak k + βv k(k ) α k (k) A Ak k. A Ak k = β Ak k. Ak α k (k) = β A Aα k (k) α k (α k (k)). This is a Functional Euler Equation. c. Guess that the planner saves a constant fraction of income α(k) = θak. Then Ak θak = β A θa 2 k θ 2 A 2 k Ak θak = β A Aθ[Ak θak]. Therefore β = θ. So α(k) = βak, i.e., the planner saves a constant fraction of income.