Delta parameters (input weights)

Table of contents

Overview

The delta_* parameters represent input weights in the nested CES/Cobb-Douglas production functions used throughout the G-Cubed model. These weights determine the relative importance of different inputs at each tier of production.

The delta parameters are calibrated from Input-Output (IO) tables and represent cost shares in the base year calibration.

Standard Sector Parameters

delta_o

SYM Declaration:

parameter delta_o(factors,sec_std,regions)    'input weights, KLEM tier'

Definition: Input weights for the KLEM (Capital-Labor-Energy-Materials) tier in standard production sectors. The factors dimension includes:

  • k: Capital
  • l: Labor
  • e: Energy aggregate
  • m: Materials aggregate

Usage in Model: Used in factor demand equations:

LAB = exp(SHL+SHY)^(sigma_o-1) * delta_o(l) * OUP * exp(PRP-WAG)^sigma_o
ENT = exp(SHEF+SHY)^(sigma_o-1) * delta_o(e) * OUP * exp(PRP-PRE)^sigma_o
OIN = exp(SHY)^(sigma_o-1) * delta_o(m) * OUP * exp(PRP-POI)^sigma_o

And in the price aggregation:

PRP = cd_o*( delta_o(k)*PRK + delta_o(l)*WAG + delta_o(e)*PRE + delta_o(m)*POI - ... )
    + (1-cd_o)*ln( delta_o(k)*exp(PRK-SHK)^(1-sigma_o) + ... ) / (1-sigma_o*(1-cd_o))

delta_e

SYM Declaration:

parameter delta_e(regions,sec_std,goods_e)    'input weights, individ goods'

Definition: Input weights for individual energy goods within the Energy tier of production. Determines the relative importance of different energy sources (coal, oil, gas, electricity, etc.) in each sector’s energy bundle.

delta_m

SYM Declaration:

parameter delta_m(regions,sec_std,goods_m)    'input weights, individ goods'

Definition: Input weights for individual material goods within the Materials tier of production. Determines the relative importance of different material inputs in each sector’s materials bundle.

Household Capital Sector Parameters (Sector Z)

delta_oH

SYM Declaration:

parameter delta_oH(factors,regions)    'input weights, KLEM tier'

Definition: Input weights for the KLEM tier in the household capital sector. Calibrated from the Consumption (C) column of the IO table.

delta_eH

SYM Declaration:

parameter delta_eH(goods_e,regions)    'input weights, individ goods'

Definition: Input weights for energy goods in household capital production. Calibrated from the energy sectors of the Consumption column.

delta_mH

SYM Declaration:

parameter delta_mH(goods_m,regions)    'input weights, individ goods'

Definition: Input weights for material goods in household capital production. Calibrated from the materials sectors of the Consumption column.

Raw Capital Sector Parameters (Sector Y)

delta_oR

SYM Declaration:

parameter delta_oR(factors,regions)    'input weights, KLEM tier'

Definition: Input weights for the KLEM tier in the raw capital sector. Calibrated from the Investment (I) column of the IO table.

delta_eR

SYM Declaration:

parameter delta_eR(goods_e,regions)    'input weights on individual energy goods'

Definition: Input weights for energy goods in raw capital production. Calibrated from the energy sectors of the Investment column.

delta_mR

SYM Declaration:

parameter delta_mR(goods_m,regions)    'input weights on individual materials goods'

Definition: Input weights for material goods in raw capital production. Calibrated from the materials sectors of the Investment column.

Government Sector Parameters (Sector G)

delta_oG

SYM Declaration:

parameter delta_oG(factors,regions)    'gov KLEM budget shares'

Definition: Budget shares for the KLEM tier in government expenditure.

delta_eG

SYM Declaration:

parameter delta_eG(goods_e,regions)    'gov budget shares on energy goods'

Definition: Budget shares for energy goods in government expenditure.

delta_mG

SYM Declaration:

parameter delta_mG(goods_m,regions)    'gov budget shares on materials goods'

Definition: Budget shares for material goods in government expenditure.

Trade Parameters

delta_dom

SYM Declaration:

parameter delta_dom(goods_o,regions)    'domestic weight in CES function combining domestic and foreign sources'

Definition: The weight on domestic production in the CES aggregation of domestic and imported goods. Represents the home bias in consumption/production.

Calibration: Calculated as the ratio of domestic production (OUP) to composite supply (OUY) in the calibration year:

delta_dom = OUP / OUY

delta_ff

SYM Declaration:

parameter delta_ff(goods_o,dest,orig)    'foreign source weights in FF'

Definition: Weights on imports from different foreign regions in the foreign goods aggregation tier. Determines the relative importance of imports from each trading partner.

Calibration

All delta parameters are calibrated from IO tables and trade flow data:

Parameter IO Table Source
delta_o, delta_e, delta_m Intermediate demand matrix
delta_oH, delta_eH, delta_mH Consumption (C) column
delta_oR, delta_eR, delta_mR Investment (I) column
delta_oG, delta_eG, delta_mG Government (G) column
delta_dom OUP/OUY ratio
delta_ff Bilateral trade flows

Economic Interpretation

The delta parameters represent cost shares in the Cobb-Douglas case and distribution parameters in the CES case. They satisfy:

\[\sum_i \delta_i = 1\]

for each tier of production.

Higher delta values indicate:

  • Greater importance of that input in production
  • Higher cost share in total costs
  • Larger quantity demanded relative to other inputs

References