Chapter 11 Ljungqvist and Sargent, Fiscal Policies in the non-stochastic growth model

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Chapter 11 Ljungqvist and Sargent, Fiscal
Policies in the non-stochastic growth model
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The model

• Preferences
• Resource constraint
• Investment identity

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Household problem

• Household present value budget constraint
• Some Household FONCs
• Government budget constraint

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User Cost of Capital

• Rewrite the household budget constraint as
• Resources are finite thus

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Deriving the user cost of capital

• Transversality condition imposes the following
  restriction on allocations
• User cost of capital formula

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Firms problem

• Perfectly Competitive Firms operating Constant
  returns to scale production technology.

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Government

• Government budget constraint

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Competitive Equilibrium

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A useful equilibrium condition

• Suppose initially that labor supply is inelastic
  
• Then we can derive

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Steps in computing a dynamic equilibrium

1. Step one derive a steady-state equilibrium. A
   steadystate equilibrium is a particular
   competitive equilibrium in which all forcing
   variables are constant. Note that 11.3.3
   implies steady state capital stock is given by

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What about other variables?

• Use steadystate analogues of

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Finding the dynamic path

• Step 2 Solve for dynamic path (several different
  ways to do this).
• Need
• initial capital stock k0
• Terminal capital stock

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Finding dynamic path using shooting algorithm

• Given these objects the shooting algoritm
  consists of the following steps

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Role of lumpsum taxes

• With lump-sum taxes we can implement the shooting
  algorithm for given settings of the forcing
  variables and then adjust the level of lumpsum
  taxation to insure that the government budget
  constraint is satisfied.
• If lump-sum taxes are not available need to add
  an additional loop to the shooting algorithm to
  insure that government budget constraint is
  satisifed.

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Solving more complicated systems

• For more complicated systems with money and or
  other endogenous state variables it is more
  convenient to solve for the equilibrium using a
  nonlinear equation solver.

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• As before need initial and terminal capital stock
  k0,ks.
• Need to solve

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Nonlinear equation solver

• Choose T large (100, 150 years)
• Assume that in period T the economy is in a
  steadystate.
• Dynare uses this approach. It implements a
  version of Newtons algorithm to solve the system
  of nonlinear equations.

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What are the effects of an increase in the
investment tax credit?

• Suppose the economy is initially in a
  steadystate.
• Investment tax credit is increased permanently in
  period zero.
• What is the initial response of capital,
  consumption?
• How do capital and consumption compare in the new
  steadystate with their values in period zero?

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What are the effects of an anticipated increase
in the investment tax credit?

• Suppose the investment tax credit occurs in
  period 10 instead of period 1.
• In future periods the return in capital will be
  high.
• Takes time to accumulate capital.
• Makes sense to start accumulating capital now.
• What about consumption?

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PERMANENT INCREASE IN ?i
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TEMPORARY INCREASE IN ?i
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What are the effects of government policy on
allocations and prices?

• Ricardian Equivalence. If only lumpsum taxes are
  used to finance government purchases the timing
  of lumpsum taxes doesnt matter. The lumpsum tax
  only enters the present value budget constraints.
  It doesnt enter any of the marginal conditions.
  (allocations and prices dont depend on the
  timing of l.s. taxes)

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More effects of government policy on allocations

• When labor supply is inelastic the tax rate on
  labor is not distorting. A constant consumption
  tax is also not distorting.
• If the consumption tax varies over time it is
  distorting.
• Taxes on capital and investment tax credits are
  distorting even when constant and the timing of
  these taxes matters.

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Generalizing the model to allow for a labor
supply decision.

• Endogenous labor supply adds an additional
  equation

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Endogenous labor, steadystate and dynamics

• Steadystate
• Now solve for n, k and c.
• Dynamics Use previous two equations resource
  constraint
• to solve for ct, nt, kt1.

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Technological progress.

• Suppose that the production function is
• where
• No steadystate! Instead define a balanced growth
  path with constant µ.
• Existence of a balanced growth path imposes
  restrictions on preferences when labor is
  endogenous. See below.

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Solving for the dynamic competitive equilibrium

• Two strategies for proceeding.
• 1) leave the growth in.
• 2) Transform the economy and use the previous
  notion of steady-state equilibrium.
• The transformations with growing inelastic
  labor input are

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New dynamic equations for capital and consumption
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Restrictions on preferences with endogenous labor

• A period utility function of the following form
  is consistent with balanced growth
• Where ht is hours per worker! Assume aggregate
  labor input is given by htnt
• An example of a form of preferences that is not
  consistent with balanced growth

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Where we are going next

• Applications
• Hayashi and Prescott (2002)
• Chen, Imrogoroglu, Imrohoroglu (2006).
• Homework 1 The effects of temporary and
  permanent increases in government purchases on
  output and interest rates. (Dynare).