Flashcards
Macroeconomics I
Cards in this deck (38)
- 1Bellman Equation\[V(y) = \underset{x}{max} F(x,y) + \beta v(y)\]
- 2Technologies in Neoclassical GrowthInvestment: \[k_(t+1) = (1-\delta)k_t + s_t\]
Production: \[y = f(k,l)\]
Output: \[c_t + i_t \le y_t\] - 3Arrow-Debreu EquilibriumTime-Zero Trading
\[\sum_{t=0}^{\infty}p_t * c_t \le [\sum_{t=0}^{\infty}p_t * e_t \forall i\] - 4Sequential Trading Equilibrium\[c_t + q_t*a_{t+1} \le e_t + a_t\]
\[a_{t+1} \le A_t\]
for each agent i - 5Natural Borrowing Constraint\[A_t = \sum_{t=\tau}^{\infty} p_t * e_t\]
- 6Arrow-Debreu Market Clearance\[c_t = e_t\] for each agent i
- 7Sequential Market Clearance\[c_t = e_t\] for each agent
\[a_{t+1} =0\] across all agents - 8Neoclassical Firm Optimization\[\underset{y_t, k_t, n_t}{max} \sum_{t=0}^{\infty} p_t[y_t - r_t*k_t - w_t*l_t\]
s.t. \[y_t \le f(k_t,l_t)\] - 9Neoclassical Household Optimization\[\underset{c_t, k_{t+1}}{max} sum_{t=0}^{\infty} beta^t*c^t
s.t:
\[sum_{t=0}^{\infty} p_t[c_t+i_t] \le sum_{t=0}^{\infty} p_t[r_t*k_t + w_t*l_t] + \pi \]
\[k_{t+1} = k_t(1-delta_ + i_t\]
\[k_0\] given
\[c_t \ge 0, k_{t+1} \ge 0\] - 10Neoclassical Market Clearance\[y_t = c_t + i_t\]
\[n_t^s = n_t^d\]
\[k_t^s = k_t^s\] - 11\[\Gamma (x)\]Constraint space of state variable \[(\{(c_t, k_{t+1}) in R^2: c_t +k_{t+1} <= f(k_t)\}\]
- 12\[\Pi (x)\]Set of feasible plans: \[\{\bar{x}: x_{t+1} \in
\gamma( x_t) \forall t\}\] - 13Assumptions to map Sequential Problem to Bellman Equation (i.e. the solution to the SP is a solution to the FE)1. \[\gamma (x)\] is nonempty
2. \[\lim_{t \to \infty} \sum_{t=0}^T beta^t F(x_t, x_{t+1}\] exists for all \[x_0 \in X\] and \[\bar{x} \in \Pi(x_0)\] - 14Assumption to map Bellman Equation to Sequential Problem\[lim_{n \to \infty} beta^n v(x_n) =0 \forall x_0 \in X \ and \ x_bar \in Pi(x_0)\]
- 15[\G(x)\]Policy correspondence: \[\{y \in \gamma (x) \subseteq X: v(x) = F(x,y) + v(y)\}\]
- 16Assumptions to make the Bellman Equation a Fixed Point Problem1. X is convex; \[\gamma: X \to X\] is compact-valued, continuous, and non-empty
2. \[F:X x X \to R\] is continuous and bounded; \[\beta \in (0,1)\] - 17Contraction mapping\[(S,\rho)\] is a metric space and \[T: S\to S\]. T is a contraction mapping with mod \[\beta\] if
\[\rho(Tx, Ty) \le \rho\beta(x,y)\] - 18Blackwell's Sufficient Conditions1. Monotonic: \[f(x) \le g(x) \implies Tf(x) \le Tg(x)\]
2. Discounting: There exists \[\beta \in (0,1)\] s.t.
\[[T(f+a)(x)]\le (Tf)(x) + \beta a\] - 19Properties of v and g1. V is strictly increasing
2. V is strictly concave
3. G is continuous and single-valued
4. V is differentiable - 20Assumptions for V to be strictly increasing1. For each y, \[F(x,y)\] is strictly increasing in X
2. \[x\le x^\prime\] implies \[\gamma (x) \subseteq \gamma (x^\prime)\] - 21Assumptions for concavity of V1. F is strictly concave
2. \[\gamma\] is convex - 22Neoclassical Growth assumptions on F1. Continuous
2. \[f(0) = 0\] and there exists \[\bar{x} >0\] s.t. \[f(\bar{x}) = \bar{x}\] (fixed point)
3. Strictly increasing
4. weakly concave
5. Continuously Differentiable - 23Neoclassical Growth assumptions on u1. u(0) is finite
2. Continuous
3. Strictly Increasing
4. Strictly concave
5. Continuously Differentiable - 24Guess and Verify Method1. Assume functional form.
2. Substitute into FE
3. Take FOCs and find Policy Function.
4. Substitute back into FE for closed-form constants.
5. Get closed form Policy Function - 25Von-Neumann-Morgenstern Preferences\[U(c^i) = \sum_{t=0}^{\infty} \sum_{s\inS} \beta^t \pi(s^t) u(c_t^i(s^t))\]
- 26Arrow Securities\[a_{t+1}^i (s^t, s_{t+1}) \in R\]
The number of units of consumption an agent purchases in a given history contingent on a future realization of a state. - 27Aggregate State Budget Constraint\[c_t^i(s^t) + \sum_{s_{t+1}\in S} q_t a^i_{t+1} \le e_t^i(s^t) + a_t^i(s^t)\]
- 28Stochastic natural debt limit\[-a_{t+1}^i(s^t, s_{t+1}) \le A^i(s^t, s_{t+1})\]
- 29Stochastic Sequential Trading EquilibriumAllocations of consumptions and assets, and arrow prices such that agents optimize subject to budget constraint and debt limit, and markets clear:
\[c^i(s^t) = e^i(s^t)\]
\[a_{t+1}^i(s^t, s_{t+1}) = 0\] - 30Stochastic Contracting Mapping Assumptions1. Gamma is non-empty and measurable given some probability space.
2. F is measurable and bounded, and the limit of the sum of its lifetime expectations exists (i.e. there is a finite lifetime utility value) - 31Big K, Little k TrickK as aggregate capital stock, k as individual
Price systems as functions of K s.t.
\[V(k,K) = u(R(K)k + w(K)n - k^\prime) + \beta V(k^\prime, K^\prime)\] s.t. \[K^\prime = G(K)\] - 32Recursive Competitive EquilibriumSet of pricing functions, law of motion for aggregate capital, household value functions and decision rule, and firm decision rules such that:
1. Given prices and the law of motion, the value and policy function solve the optimization problem.
2. Given prices, firms optimize
3. Markets clear: Capital and labor supply and demand, and consistency between policy function and law of motion. - 33Rational Expectations\[g(K,K) = G(K)\]
Consistency Condition - 34Stochastic NGMStochastic parameters are state variables, and choice variables must be defined in terms of them.
- 35Intratemporal Labor Decision\[u_c(c, l)F_n(k,n) = u_l(c,l)\]
- 36Search Model\[v_0(w) = max\{v_1(w), b + \beta \int v_o(w^\prime) dF(w^\prime)\}\]
\[v_1(w) = \frac{w}{1-\beta}\] - 37Reservation Wage\[v_1(w^*) = b + \beta \int v_0(w^\prime)dF(w^\prime)\]
\[w^* - b = \frac{\beta}{1-\beta}\int_{w^*}^{\Bar{w}} (w^\prime - w^*) dF(w^\prime)\] - 38Value functions with Separations\[v_1(w) = w + \beta(1-\delta) v_1(w) + \beta\delta[b+ E[V_0(w)]]\]
\[v_0(w) = max\{v_1(w), b + E[v_0(w)]\}\]
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- Graduate-level Macroeconomics Flash Cards
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- Max Schnidman
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