Cobb-Douglas indicators (cd_*)

Table of contents

• Cobb-Douglas indicators (cd_*)
  • Overview
  • Parameters
    • Standard Sector Parameters
      • cd_o
      • cd_e
      • cd_m
    • Household Capital Sector Parameters
      • cd_oH
      • cd_eH
      • cd_mH
    • Raw Capital Sector Parameters
      • cd_oR
      • cd_eR
      • cd_mR
    • Trade Parameters
      • cd_df
      • cd_ff
  • Calibration
  • Usage in Model
  • Economic Interpretation
  • References

Overview

The cd_* parameters are switch parameters that determine whether a Cobb-Douglas or Constant Elasticity of Substitution (CES) production function is used at each tier of production. A value of 1 indicates Cobb-Douglas, while 0 indicates CES.

These parameters are automatically inferred from the supplied elasticities of substitution. When the elasticity of substitution is close to 1, the Cobb-Douglas specification is used for numerical stability.

Parameters

Standard Sector Parameters

cd_o

SYM Declaration:

parameter cd_o(sectors,regions)    '1 if cd, 0 if ces'

Definition: Indicator for the KLEM (Capital-Labor-Energy-Materials) tier of production in standard sectors. Determines whether inputs at this tier are combined using Cobb-Douglas or CES.

cd_e

SYM Declaration:

parameter cd_e(sectors,regions)    '1 if cd, 0 if ces'

Definition: Indicator for the Energy tier of production in standard sectors. Determines whether energy inputs are combined using Cobb-Douglas or CES.

cd_m

SYM Declaration:

parameter cd_m(sectors,regions)    '1 if cd, 0 if ces'

Definition: Indicator for the Materials tier of production in standard sectors. Determines whether material inputs are combined using Cobb-Douglas or CES.

Household Capital Sector Parameters

cd_oH

SYM Declaration:

parameter cd_oH(regions)    '1 if cd, 0 if ces'

Definition: Indicator for the KLEM tier in household capital sector (sector Z).

cd_eH

SYM Declaration:

parameter cd_eH(regions)    '1 if cd, 0 if ces'

Definition: Indicator for the Energy tier in household capital sector.

cd_mH

SYM Declaration:

parameter cd_mH(regions)    '1 if cd, 0 if ces'

Definition: Indicator for the Materials tier in household capital sector.

Raw Capital Sector Parameters

cd_oR

SYM Declaration:

parameter cd_oR(regions)    '1 if cd, 0 if ces'

Definition: Indicator for the KLEM tier in raw capital sector (sector Y).

cd_eR

SYM Declaration:

parameter cd_eR(regions)    '1 if cd, 0 if ces'

Definition: Indicator for the Energy tier in raw capital sector.

cd_mR

SYM Declaration:

parameter cd_mR(regions)    '1 if cd, 0 if ces'

Definition: Indicator for the Materials tier in raw capital sector.

Trade Parameters

cd_df

SYM Declaration:

parameter cd_df(goods_o,regions)    '1 if Cobb-Douglas, 0 if CES'

Definition: Indicator for the domestic-foreign aggregation tier. Determines how domestic and aggregated foreign goods are combined.

cd_ff

SYM Declaration:

parameter cd_ff(goods_o,dest)    '1 if cd, 0 if ces'

Definition: Indicator for the foreign-foreign aggregation tier. Determines how imports from different foreign regions are combined.

Calibration

The cd_* parameters are automatically calibrated based on the corresponding sigma_* (elasticity of substitution) parameters:

def set_cd_parameters(self):
    """
    cd_* parameters are all switches to flip between the
    Constant Elasticity of Substitution (CES) production function
    and the Cobb-Douglas production function.

    A value of 1 means Cobb-Douglas and 0 means CES.

    These parameters are all inferred from the supplied elasticities of
    substitution. If the elasticities of substitution are close enough to 1
    then the elasticity of substitution is effectively set to 1 by
    working with the Cobb-Douglas function.
    """

The mapping between sigma_* and cd_* parameters:

Sigma Parameter	CD Parameter
sigma_o	cd_o
sigma_e	cd_e
sigma_m	cd_m
sigma_oH	cd_oH
sigma_eH	cd_eH
sigma_mH	cd_mH
sigma_oR	cd_oR
sigma_eR	cd_eR
sigma_mR	cd_mR
sigma_df	cd_df
sigma_ff	cd_ff

Usage in Model

The cd_* parameters are used to switch between production function forms. For example, in the price equation for the KLEM tier:

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

When cd_o = 1 (Cobb-Douglas), the price is a simple weighted average. When cd_o = 0 (CES), the full CES aggregation formula is used.

Economic Interpretation

• Cobb-Douglas ($\sigma = 1$): Unit elasticity of substitution; constant cost shares
• CES ($\sigma \neq 1$): Variable elasticity allowing for different substitution patterns
  • $\sigma < 1$: Complements (inputs used together)
  • $\sigma > 1$: Substitutes (inputs can replace each other)

References

• Substitution elasticities (sigma_*)
• Delta parameters