B EHAVIORAL M ACROECONOMICS Xavier Gabaix Harvard Computing in Economics and Finance– June 2018 1 / 48
I NTRODUCTION ◮ Normally, (1) we economists use a simplified model of the world, knowing that the model is not literally the true world ◮ (2) But agents in our models are assumed to comprehend the full complexity of their world. ◮ In this paper: (1) applies, but (2) is replaced by (2’): Agents also have a simplified sub-model of their world. ◮ This paper models (2’), and its consequences ◮ Language: “sparsity” ◮ A vector m ∈ R 1,000,000 is sparse if most entries are 0 ◮ The agent pays attention to few dimensions of the world (endogenously) ◮ His attention vector is sparse ◮ He has a low-dimensional (sparse) submodel of the world 2 / 48
R ELATED L ITERATURE ◮ Behavioral economics and finance ◮ A lot of the literature is about modeling tastes or beliefs ◮ There is less on the modeling of rationality itself: O’Donoghue Rabin, k − level models, Crawford et al., Camerer Ho and Chuong, J´ ehiel et al., MacLeod, Hong and Stein, Koszegi and Szeidl, Schwartzstein, Fuster, H´ ebert and Laibson, Bordalo, Gennaioli and Shleifer... ◮ Finance / Macro: ◮ Inattention: Sims 03, Gabaix and Laibson 02, 06, Mankiw Reis 02, Reis 06, Chetty, Kroft Looney 09, Angeletos La’O 10, Ma´ ckowiak and Wiederholt 10, 16, Masatlioglu and Ok 10, Veldkamp 11, Matejka and Sims 11, Caplin, Dean and Martin 11, Woodford 12, Alvarez Lippi, Paciello 13, Koszegi Szeidl 13, Abel, Eberly and Panageas 13, Greenwood Hanson 14, Croce, Lettau Ludvigson 15, Kueng 16, Angeletos and Lian ’17, Farhi Werning ’17 ◮ Early behavioral models: Campbell Mankiw 89 3 / 48
R ELATED L ITERATURE ◮ Learning: Sargent 93, Evans and Honkapohja 01 ◮ Approximation: Krusell Smith 98, Judd 98, Ahn, Kaplan, Moll, Winberry, Wolf 17 ◮ Sparsity in statistics and OR: Tibshirani 96 (Lasso), Cand` es and Tao 06, Donoho 06... ◮ Difficulty of behavioral dynamic programming: Laibson ’97, O’ Donoghue Rabin ’01, Harris and Laibson 01, 13 . ◮ Developments of new tools in computational macro: ◮ Hansen-Sargent ◮ Bounded Rationality: Radner, Sargent 93, Rubinstein 98, Al-Najjar 09, Bolton and Faure-Grimaud 10, Tirole 11, Gul, Pesendorder Strzalecki 12... ◮ Behavioral Public Finance: Chetty, Kroft Looney 09, Congdon Mullainathan Schwartzstein 12, Bernheim Rangel 09, Farhi Gabaix 15. 4 / 48
P ROGRAM : SEEKING UNIFIED BR IN ECONOMICS 1. Micro: “A sparsity-based model of bounded rationality” (QJE 2014): Fairly general and simple device, smax u ( a , x ) subject to b ( a , x ) ≥ 0 a Basic consumer theory: Walrasian demand, Hicksian demand, Slutsky matrix. Competitive equilibrium: Arrow-Debreu, Edgeworth boxes... 2. Macro: “Behavioral Macroeconomics via Sparse Dynamic programming”: Consumption, RBC 3. “A Behavioral New Keynesian model”: monetary and fiscal policy 4. Public economics: “Optimal taxation with behavioral agents” (with E. Farhi) Ongoing work: 5. Finance: in the works. 6. Game theory: “Some game theory with sparsity-based bounded rationality”. Sparse Nash equilibrium 5 / 48
C ONCLUSION : M ICRO BEHAVIOR FOR MACRO 1. Agents react more to near, rather than future, shocks. 2. Pervasive failure of the high-frequency Euler equation (Carroll 97) 3. High MPC to tax rebates, especially amongst misoptimizing households (Parker 15, Kueng 16). 4. Agents accumulate a too small buffer of savings, as they’re (partially) myopic to the risk of income fluctuations (Lusardi 11) 5. Agents start saving “too late” for retirement (falls at age 45; controversial) 6. Very low sensitivity to interest rates (Hall 88) (so rat IES should be low). However, people seem OK with non-smooth consumption profiles (so rat IES should be high). 7. When choosing their portfolio, agents are pay less attention to the hedging demand motive. 8. Plain introspection: we don’t solve the full macro equilibrium in our head 6 / 48
C ONCLUSION : A GGREGATE MACRO 1. The Lucas critique has zero (or only partial) bite 2. RBC model: has more internal propagation + less sensitivity to the interest rate 3. The agent is a hybrid of Lucas neoclassical agent and a present-looking old Keynesian agent. 3.1 Like Lucas agent: Has general methodology + sensitivity to important parameters 3.2 Like old Keynesian agent: Has more common-sense behavior So this may be a useful synthesis. 4. Macro policy (“A behavioral New Keynesian model”) 4.1 Forward guidance by central bank is less powerful with behavioral agents 4.2 Fiscal policy is more powerful, as they’re non Ricardian 4.3 Economy is more stable, even at the ZLB 7 / 48
I NTRODUCTION R OADMAP ◮ The static smax operator, from “A sparsity-based model of bounded rationality” (2014) ◮ Application 1: Behavioral version of Varian ◮ Application 2: Measuring inattention in health care plans ◮ Sparse dynamic programming ◮ Application 1: Consumption-savings problems ◮ Application 2: Finite-horizon life-cycle model ◮ Application 3: RBC (not today) ◮ New Keynesian model ◮ Monetary & Fiscal policy ◮ ZLB, paradoxes 8 / 48
A TTENTION - AUGMENTED DECISION UTILITY ◮ Primitive u ( a , x ) ◮ Form attention-augmented decision utility : u ( a , x , m ) u ( a , x , m ) : = u ( a , m 1 x 1 , ..., m n x n ) = u ( a , m ⊙ x ) with m ⊙ x : = ( m i x i ) i = 1... n ◮ This gives a ( x , m ) : = arg max a u ( a , x , m ) . ◮ Sensitivity of action to attention: a m i : = ∂ a ( m , x ) = − u − 1 aa · u am i ∂ m i � x , m d � evaluated at m = m d , a d = a . ◮ Example 1: if u ( a , x ) = − 1 2 ( a − ∑ i b i x i ) 2 , then 2 ( a − ∑ i b i m i x i ) 2 , u ( a , x , m ) = − 1 a ( x , m ) = ∑ b i m i x i , a m i = b i x i i 9 / 48
T HE S PARSE BR A LGORITHM : M OTIVATION ◮ Agent will take action a ( x , m ) = arg max a u ( a , x , m ) and experience utility v ( x , m ) = u ( a ( x , m ) , x ) ◮ So it is sensible to allocate attention m as: m E [ v ( x , m )] − C ( m ) max ◮ Smax agent simplifies the problem: ◮ replaces utility by linear-quadratic approximation, and removes correlations in x ◮ Chooses optimal attention m in that simplified model ◮ Chooses optimal action with a ( x , m ) action with correct utility. ◮ First part: with ι = ( 1, ..., 1 ) = full attention, � E � x � 2 � E [ v ( x , m )] = v ( ι ) − 1 2 ( m − ι ) ′ Λ ( m − ι ) + o ◮ Also, take C ( m ) = κ ∑ i m α i . Agent does: m ∈ [ 0,1 ] n − 1 ( 1 − m i ) 2 Λ ii − κ ∑ m α 2 ∑ max i i i 10 / 48
S PARSE MAX : Q UICK VERSION ◮ Proposition : One solves smax a ; m u ( a , x , m ) as follows. Step 1: Choose attention: � � m ∗ a ′ i = A α ( E m i u aa a m i / κ ) Step 2: Choose action: a s = arg max a u ( a , x , m ∗ ) . � a d , m d � ◮ a m i = − u − 1 aa u am i , and u aa evaluated at ( a , m ) = � � with a d = arg max a u a , m d , x 2 ( 1 − m ) 2 | v | + κ m α ◮ A α ( v ) : = arg min m 1 A 0 ( σ 2 ) A 1 ( σ 2 ) A 2 ( σ 2 ) 1 1 1 σ 2 σ 2 σ 2 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 11 / 48
A PPLICATION 2 ( a − ∑ i b i m i x i ) 2 , ◮ Quadratic example: u ( a , x , m ) = − 1 10 6 a r = ∑ b i x i : Non-Sparse Action i = 1 a s = ∑ b i m i x i : Sparse action i � � b 2 i σ 2 m i = A α x i / κ ◮ Sims ’03: no source-dependent inattention a Sims = m ∑ b i x i + η i ◮ Others (Woodford, Ma´ ckowiak and Wiederholt) have source-dependent inattention ◮ smax a ; m u ( a , x , m ) applies easily beyond the linear-quadratic framework 12 / 48
S PARSE MAX WITH CONSTRAINTS : M OTIVATION ◮ Example : Consumption problem : max c u ( c ) s.t. p · c ≤ w p i = p d i + x i p s i = p d i + m i x i ◮ How to satisfy the budget constraint? ◮ The “trade-off” intuition holds with perceived prices: u ′ = p s c 1 1 u ′ p s c 2 2 ◮ So u ′ ( c ) = λ p s . To satisfy budget: pick λ such that. p · c ( λ ) = w . 13 / 48
S PARSE M AX WITH C ONSTRAINTS smax u ( a , x ) subject to b ( a , x ) ≥ 0 a Define u ( a , x , m ) : = u ( a , m ⊙ x ) , b ( a , x , m ) : = b ( a , m ⊙ x ) , L ( a , x , m ) : = u ( a , x , m ) + λ d · b ( a , x , m ) , with λ d multiplier when m = 0. Definition (Sparse max with constraints) Step 1. Allocate attention: with a m i : = − L − 1 aa L am i m i = A ( − E [ a m i L aa a m i ] / κ ) Step 2: Find action. Solve in a , λ : u a ( a , x , m ∗ ) + λ b a ( a , x , m ∗ ) = 0 (perceived model m for FOC) b ( a , x , ι ) = 0 (actual model ι for budget) with ι = ( 1, ..., 1 ) = full rationality. 14 / 48
B ASIC CONSUMPTION PROBLEM : M ARSHALLIAN DEMAND ◮ I use s for sparse, r for rational / traditional. ◮ Proposition . Demand is: c s ( p , w ) = c r � p s , w ′ � where w ′ solves p · c r ( p s , w ′ ) = w . ◮ Quasi-linear utility: Take u ( c ) = v ( c 1 , ..., c n − 1 ) + c n , with v concave. Demand for good i < n is independent of wealth and is: c s i ( p ) = c r i ( p s ) . ◮ If traditional demand is linear in wealth: c r ( p s , w ) c s ( p , w ) = p · c r ( p s , 1 ) ◮ Budget is w = $ 100. I have $105 in my supermarket cart. ◮ Demand linear in wealth: So, I lower my consumption by 5%. ◮ General case: I lower it by − d c = ∂ c s ∂ w ( p s , w ) × $ 5 15 / 48
S PARSITY MODEL WITH BUDGET CONSTRAINT ◮ Cobb-Douglas preferences: u ( c ) = ∑ n i = 1 α i ln c i , i ( p , w ) = α i w c s p j p s ∑ j α j i p s j ◮ CES demand: u ( c ) = ∑ i c 1 − 1 / η / ( 1 − 1 / η ) i w i ) − η c s i ( p , w ) = ( p s � � − η p s ∑ j p j j 16 / 48
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