The Zero Lower Bound and Unconventional Policy

Level: Graduate | Field: Monetary Economics

1. The effective lower bound

Because cash pays a zero nominal return, the nominal interest rate cannot fall much below zero: \(i_t\ge i_{\text{ELB}}\approx0\). The bound binds when the natural real rate becomes very negative, for instance after a financial crisis or a collapse in demand. U.S. policy rates were at the bound in 2008–2015 and 2020–2022; the euro area, Japan, Switzerland and Sweden pushed policy rates slightly below zero. The secular decline in the neutral real rate (Lecture 7) makes the bound bind more often.

2. A liquidity trap in the New Keynesian model

Following Eggertsson and Woodford (2003), suppose the natural rate falls to \(r^n_L<0\) and stays there with probability \(\mu\) each period, returning permanently to \(r^*>0\) with probability \(1-\mu\). After the shock ends, policy delivers \(\pi=\tilde y=0\). While the shock lasts, \(i=0\) and the equilibrium is constant:

which gives

The denominator is positive but can be small: falling output lowers inflation, which raises the real rate, which lowers output further. Small changes in persistence or price flexibility generate very large recessions — a deflationary spiral. Paradoxically, more price flexibility (a larger \(\kappa\)) makes things worse (Eggertsson and Krugman, 2012). Fiscal multipliers are large at the bound because government spending raises expected inflation and lowers real rates (Christiano, Eichenbaum and Rebelo, 2011).

3. Forward guidance

The optimal commitment policy keeps rates at zero after the natural rate recovers and lets inflation overshoot. The promise raises expected inflation and output in the trap, lowering real rates today (Krugman, 1998; Eggertsson and Woodford, 2003). The policy is time inconsistent: when the trap ends, the central bank would prefer to raise rates. Central banks used Delphic guidance (forecasts of future policy) and Odyssean guidance (commitments), with calendar-based and threshold-based formulations (Campbell et al., 2012).

4. The forward-guidance puzzle

In the standard model, iterating the Euler equation forward gives \(\tilde y_t=-\frac1\sigma\sum_{k\ge0}E_t r_{t+k}\): a one-time cut in the real rate at any horizon \(T\) raises output today by the same amount, and through the Phillips curve the effect on today's inflation grows with \(T\). Promises about the distant future are implausibly powerful (Del Negro, Giannoni and Patterson, 2012). Resolutions include:

• Discounted Euler equations from incomplete markets and borrowing constraints, \(\tilde y_t=ME_t\tilde y_{t+1}-\frac1\sigma r_t\) with \(M<1\) (McKay, Nakamura and Steinsson, 2016);
• Bounded rationality and cognitive discounting (Gabaix, 2020);
• Finite planning horizons and level-\(k\) thinking (Woodford, 2019; Farhi and Werning, 2019);
• Imperfect credibility of promises.

5. Quantitative easing and balance-sheet policy

Large-scale asset purchases replace long-term bonds held by the private sector with reserves. In a frictionless world this is irrelevant (Wallace, 1981). With segmented markets or preferred habitats (Vayanos and Vila, 2021), purchases lower term premia; with constrained intermediaries (Gertler and Karadi, 2011), purchases of private assets ease credit spreads. Purchases also signal that rates will stay low (Bauer and Rudebusch, 2014). Event studies of the 2008–09 programmes find declines in 10-year yields of tens of basis points per programme (Gagnon et al., 2011), with smaller effects later. In 2022 the Fed began quantitative tightening, shrinking its balance sheet.

6. Negative rates and other tools

Negative policy rates in Europe and Japan (2014–2024) lowered market rates, but pass-through to deposit rates is limited, which may squeeze bank margins and lending (the "reversal rate", Brunnermeier and Koby, 2018). Other options include yield-curve control (Bank of Japan, 2016–2024; Reserve Bank of Australia, 2020–21), a higher inflation target (Blanchard, Dell'Ariccia and Mauro, 2010) and make-up strategies such as price-level targeting (Lecture 7).

7. Computation

The lower bound makes the model nonlinear. Dynare handles occasionally binding constraints with the OccBin toolkit (Guerrieri and Iacoviello, 2015), using the occbin_constraints block and occbin_solver, and solves deterministic paths with perfect_foresight_setup and perfect_foresight_solver. The two-state model of Section 2 and the perfect-foresight experiments of Section 4 have closed forms and are solved directly in MATLAB in the exercises.

beta = 0.99; sigma = 1; kappa = 0.02; mu = 0.8; rL = -0.01; rstar = 1 / beta - 1;
% TODO: solve the two linear equations for (x_L, pi_L), first with x_E = pi_E = 0, then with forward guidance

beta = 0.99; sigma = 1; kappa = 0.02; mu = 0.8; rL = -0.01; rstar = 1 / beta - 1;
% in the trap (i = 0):  x_L = mu x_L + (1-mu) x_E + (1/sigma)(mu pi_L + (1-mu) pi_E + rL)
%                       pi_L = kappa x_L + beta (mu pi_L + (1-mu) pi_E)
A = [1 - mu, -mu / sigma; -kappa, 1 - beta * mu];
fprintf('                         output gap (%%)   inflation (%%, annualised)\n');
for fg = [0 1]
  xE = fg * rstar / sigma; piE = kappa * xE;                % exit quarter with i = 0 instead of r*
  rhs = [(1 - mu) * xE + ((1 - mu) * piE + rL) / sigma; beta * (1 - mu) * piE];
  sol = A \ rhs;
  if fg == 0, nm = 'no forward guidance  '; base = sol; else, nm = '1 qtr of guidance    '; end
  fprintf('%s      %7.2f            %7.2f\n', nm, 100 * sol(1), 400 * sol(2));
end
fprintf('expected trap length: %.1f quarters; guidance removes %.0f%% of the output loss\n', 1 / (1 - mu), ...
        100 * (1 - sol(1) / base(1)));

output gap (%)   inflation (%, annualised)
no forward guidance          -8.13              -3.13
1 qtr of guidance            -6.33              -2.36
expected trap length: 5.0 quarters; guidance removes 22% of the output loss

sigma = 1; kappa = 0.1717; beta = 0.99; dr = -0.25;
% TODO: for each T and M in {1, 0.97}: backward recursion over t = T, T-1, ..., 0

sigma = 1; kappa = 0.1717; beta = 0.99; dr = -0.25;
fprintf('   T    standard Euler: x0    pi0 (ann.)   |   discounted Euler (M = 0.97): x0   pi0 (ann.)\n');
for T = [0 4 8 20 40]
  out = zeros(1, 4);
  for j = 1:2
    M = 1 - 0.03 * (j == 2); x = 0; p = 0;
    for t = T:-1:0
      r = dr * (t == T);
      x = M * x - r / sigma; p = beta * p + kappa * x;
    end
    out(2*j-1:2*j) = [x 4 * p];
  end
  fprintf('%4d        %6.3f      %7.3f      |                    %6.3f      %7.3f\n', T, out);
end

T    standard Euler: x0    pi0 (ann.)   |   discounted Euler (M = 0.97): x0   pi0 (ann.)
 0         0.250        0.172      |                     0.250        0.172
 4         0.250        0.842      |                     0.221        0.792
 8         0.250        1.485      |                     0.196        1.316
20         0.250        3.267      |                     0.136        2.423
40         0.250        5.799      |                     0.074        3.223