Adjustment cost parameters (phi, phi_y, phi_z)

Table of contents

Overview

The adjustment cost parameters control the costs incurred when firms adjust their capital stock through investment. These costs represent installation, learning, and organizational disruption costs that firms face when changing their capital levels. Higher adjustment costs slow down the response of investment to changes in Tobin’s Q.

Parameters

phi

SYM Declaration:

parameter phi(sec_std,regions)    'adjustment cost parameter std sector'

Definition: The adjustment cost parameter for standard production sectors. This parameter determines how costly it is for firms in each sector to adjust their capital stock through investment.

Calibration: Default values are sector-dependent:

  • Most sectors: 4
  • Electricity sector: 15
  • Some specialized sectors: 30

The values are defined in the PHI dictionary in the parameters module:

PHI: Dict[int, List[float]] = {
    22: [4, 4, 4, 4, 4, 15, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 30, 1, 1, 30, 4],
    # ... other model configurations
}

Usage in Model: Used in the investment equations:

INV = ( (1+.5*(TOB-1))*(TOB-1)*CAP/phi )*forei
    + ( (1+.5*(TPA-1))*(TPA-1)*CAP/phi )*(1-forei)

JNV = ( INV + (phi/2)*(base_jk^2)*CAP )/(1+phi*base_jk)

Also used in the rental rate of capital equation:

ROR = ... - (1-TITC)*exp(PRII-PRID)*(phi/2)*(JNV/CAP)^2

phi_y

SYM Declaration:

parameter phi_y(regions)    'adjustment cost parameter capital sector'

Definition: The adjustment cost parameter for the raw capital producing sector (sector Y). This parameter governs the cost of adjusting capital stock in the sector that produces capital goods.

Calibration: Default value: 4.0 (same across all regions)

PHI_Y: float = 4.0

Usage in Model: Used in the raw capital sector investment equations:

INVY = JNVY * ( 1 + (phi_y/2) * JNVY/CAPY )
JNVY = CAPY * ( fore_i_y*TOBY + (1-fore_i_y)*TPAY - 1 ) / phi_y

Also in the rental rate equation:

RORY = ... - exp(PRII-PRID)*(phi_y/2)*(JNVY/CAPY)^2

phi_z

SYM Declaration:

parameter phi_z(regions)    'adjustment cost parameter household capital'

Definition: The adjustment cost parameter for the household capital sector (sector Z). This parameter governs the cost of adjusting household capital stock (e.g., housing, durable goods).

Calibration: Default value: 4.0 (same across all regions)

PHI_Z: float = 4.0

Usage in Model: Used in the household capital sector investment equations:

INVZ = JNVZ * ( 1 + (phi_z/2) * JNVZ/CAPZ )
JNVZ = CAPZ * ( fore_i_z*TOBZ + (1-fore_i_z)*TPAZ - 1 ) / phi_z

Also in the rental rate equation:

RORZ = ... - (1-TITCH)*exp(PRII-PRID)*(phi_z/2)*(JNVZ/CAPZ)^2

Economic Interpretation

The adjustment cost function follows a convex quadratic form. The cost of investment is:

\[\text{Investment Cost} = I \cdot \left(1 + \frac{\phi}{2} \cdot \frac{I}{K}\right)\]

Where:

  • $I$ is gross investment
  • $K$ is the existing capital stock
  • $\phi$ is the adjustment cost parameter

Higher values of $\phi$ imply:

  1. Greater costs for rapid capital adjustment
  2. Slower response of investment to Tobin’s Q changes
  3. More gradual convergence to steady-state capital levels

References