Taylor Collins

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R E C U R S I V E M A C R O E C O N O M I C T H E O RY ,L J U N G Q V I S T A N D S A R G E N T ,

3 R D E D I T I O N ,C H A P T E R 1 9

DYNAMIC STACKELBERG PROBLEMS

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BACKGROUND INFORMATION

• A new type of problem

• Optimal decision rules are no longer functions of the natural state

variables

• A large agent and a competitive market

• A rational expectations equilibrium

• Recall Stackelberg problem from Game Theory

• The cost of confirming past expectations

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THE STACKELBERG PROBLEM

• Solving the problem – general idea

• Defining the Stackelberg leader and follower

• Defining the variables:

• Zt is a vector of natural state variables

• Xt is a vector of endogenous variables

• Ut is a vector of government instruments

• Yt is a stacked vector of Zt and Xt

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THE STACKELBERG PROBLEM

• The government’s one period loss function is

• Government wants to maximize

subject to an initial condition for Z0, but not X0

• Government makes policy in light of the model

• The government maximizes (1) by choosing

subject to (2)

(1)

(2)

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PROBLEM S• “The Stackelberg Problem is to maximize (2) by choosing an X0

and a sequence of decision rules, the time t component of which maps the time t history of the state Zt into the time t decision of the Stackelberg leader.”

• The Stackelberg leader commits to a sequence of decisions

• The optimal decision rule is history dependent• Two sources of history dependence

• Government’s ability to commit at time 0• Forward looking ability of the private sector

• Dynamics of Lagrange Multipliers• The multipliers measure the cost today of honoring past government promises• Set multipliers equal to zero at time zero• Multipliers take nonzero values thereafter

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SOLVING THE STACKELBERG PROBLEM

• 4 Step Algorithm

• Solve an optimal linear regulator

• Use stabilizing properties of shadow prices

• Convert Implementation multipliers into state variables

• Solve for X0 and μx0

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STEP 1: SOLVE AN O.L.R.

• Assume X0 is given• This will be corrected for in step 3• With this assumption, the problem has the form of an optimal

linear regulator• The optimal value function has the form

where P solves the Riccati Equation • The linear regulator is

subject to an initial Y0 and the law of motion from (2)• Then, the Bellman Equation is

(3)

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STEP 1: SOLVE AN O.L.R.

• Taking the first order condition of the Bellman equation and solving gives us

• Plugging this back into the Bellman equation gives us

such that ū is optimal, as described by (4)• Rearranging gives us the matrix Riccati Equation

• Denote the solution to this equation as P*

(4)

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STEP 2: USE THE SHADOW PRICE

• Decode the information in P*

• Adapt a method from 5.5 that solves a problem of the form (1),(2)• Attach a sequence of Lagrange multipliersto the

sequence of constraints (2) and form the following Lagrangian

• Partition μt conformably with our partition of Y

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STEP 2: USE THE SHADOW PRICE

• Want to maximize L w.r.t. Ut and Yt+1

• Solving for Ut and plugging into (2) gives us

• Combining this with (5), we can write the system as

(5)

(6)

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STEP 2: USE THE SHADOW PRICE

• We now want to find a stabilizing solution to (6)• ie, a solution that satisfies

• In section 5.5, it is shown that a stabilizing solution satisfies

• Then, the solution replicates itself over time in the sense that

(7)

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STEP 3: CONVERT IMPLEMENTATION MULTIPLIERS

• We now confront the inconsistency of our assumption on Y0• Forces multiplier to be a jump variable

• Focus on partitions of Y and μ• Convert multipliers into state variables• Write the last nx equations of (7) as

• Pay attention to partition of P• Solving this for Xt gives us

(8)

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STEP 3: CONVERT IMPLEMENTATION MULTIPLIERS

• Using these modifications and (4) gives us

• We now have a complete description of the Stackelberg problem

(9)

(9’’)

(9’)

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STEP 4: SOLVE FOR X0 AND

• The value function satisfies

• Now, choose X0 by equating to zero the gradient of V(Y0), w.r.t. X0

• Then, recall (8)

• Finally, the Stackelberg problem is solved by plugging in these initial conditions to (9), (9’), and (9’’) and iterating the process to get

μx0

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CONCLUSION

• Brief Review

• Setup and Goal of problem

• 4 step Algorithm

• Questions, Comments, or Feedback