# Step 283: a central bank with an effective lower bound

**Why.** Step 282 ended with two price closures and no monetary policy. At the investment response
estimated from OECD data, preservation is 49% when the price level is held and 11% when prices fall with
the demand gap. "Held" assumed that something holds the price level. Here a central bank tries to do so,
with an interest-rate rule that cannot go below zero.

The model is `scripts/closure_model_v283.py`, a copy of step 279; with the new options off it reproduces
step 279 exactly (tested). The grid is `scripts/monetary_v283.py`, with output in
`data/processed/monetary_v283.json`. All 432 cases use the revised specification with the symmetric demand
closure: prices respond to the demand gap, and investment responds to Tobin's q. Illustrative; not a
forecast.

## The rule

- **Policy rate.** i_t = max(0, 0.7·i_(t−1) + 0.3·[3% + 1.5·π_(t−1) + 0.5·gap_(t−1)]). Here π is the
  model's inflation, measured around its zero baseline, and gap is planned demand less capacity, as a
  share of output.
- **What it moves:**
  - the government bond rate (the risk premium is added on top), with the public debt repricing a seventh
    a year, an average maturity of about seven years;
  - the firms' overdraft rate;
  - planned investment: planned I/K changes by −s·(i − 3%).
- **The investment sensitivity s.** Planned I/K is about 0.1 in the model. Each value of s corresponds to
  a response of investment to a 1-point rate cut:

  | s | Investment response to a 1-point cut | Source |
  |---:|---|---|
  | 0.15 | about +1.5% | US local projections with Romer–Romer shocks; the RBA's MARTIN model gives about 1.3% |
  | **0.6 (central)** | about +6% | the direct loan-rate channel in German firm surveys (ifo), over two years |
  | 1.5 | about +15% | the total first-year effect of euro-area high-frequency shocks on corporate investment |

  The model generates the demand feedback itself, so the direct channel is the central value.

## Results

Each cell gives the number of the 432 cases in which both displaced cohorts are preserved, with fiscal
crises in brackets. The rule is the central one; Phillips 0.5 as in step 279.

| Investment decay | No central bank | s 0.15 | **s 0.6** | s 1.5 | Rate at the bound (s 0.6) |
|---|---:|---:|---:|---:|---:|
| 0 (investment holds) | 378 (0) | 271 (70) | **298 (25)** | 294 (12) | 0% of cases |
| 0.35 (Israel since 2000) | 341 (8) | 284 (58) | **291 (12)** | 289 (12) | 6% |
| 1.00 (full slump) | 150 (204) | 167 (228) | **234 (130)** | 276 (19) | 57% |
| **1.27 (OECD estimate)** | 49 (310) | 65 (314) | **131 (290)** | 251 (90) | 85% |

With a flatter Phillips curve (0.25), the central rule gives 290, 286, 248 and 207 cases. Without a central
bank the same rows give 341, 326, 206 and 114.

Sensitivity of the rule, at s 0.6 and decay 1.27:

| Rule | Preserved |
|---|---:|
| Central (smoothing 0.7, debt reprices over seven years) | 131 |
| Debt reprices over three years | 120 |
| No smoothing, whole debt reprices every year | 122 |

Without smoothing the rule is unstable where demand runs ahead: 53 runs stop at decay 0.

**What it shows.**

- **A central bank narrows the uncertainty but does not remove it.** Across the four investment worlds,
  preservation without a central bank ranges from 11% to 88%. With the central rule it ranges from 30% to
  69%.
- **It costs something where demand runs ahead.** When investment holds, inflation of about 3% a year used
  to erode the public debt. The rule raises the rate against it, and preservation falls from 88% to 69%.
  Higher rates also raise interest payments to bond holders, which in this model is extra demand. With a
  weak investment channel this partly offsets the tightening.
- **At the estimated slump the lower bound binds.** The rate reaches zero in 85% of cases, and the price
  level still falls, by a median of about 40% over 20 years. Preservation rises from 11% to 30%, not to
  the 49% that a held price level would give.
- **What decides the slump case is how strongly investment responds to the rate.** At the strong end
  (s 1.5), preservation reaches 58%; at the weak end (0.15), 15%.

**Correction to step 282.** The statement "monetary policy that prevents falling prices keeps preservation
at 49% instead of 11%" assumed that policy can hold the price level. With a lower bound at zero it cannot
in most slump cases. With the central rule the figure is 30% (15%–58% by the investment sensitivity).

## Limits

- **A simple rule.** It is annual, has one target and no forward guidance, quantitative easing or exchange
  rate.
- **Investment sensitivity is borrowed** from studies of ordinary monetary shocks, not estimated for
  Israel.
- **The equity discount rate does not move with the policy rate.** With present-value equity, moving it
  one for one would triple Tobin's q at the bound.
- **Interest income is spent like other income.** Owners consume 90% of income, interest included, so
  higher rates add demand. With a lower propensity to consume out of interest, tightening would bite
  harder.
- **No fiscal–monetary coordination.** The central bank does not buy government debt.