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Three Policy Options for Crown Financial Policy - WP 03/30

Part B:  Application to Loss Functions

Appendix Table 1 below reproduces Table 1 in the main text for ease of reference.

Appendix Table 1 – Available policy options with three objectives
Policy Options Distortionary taxation Time-consistency Agency cost
1 H H H
2 H H L
3 H L H
4 L H H
5 H L L
6 L H L
7 L L H
8 L L L

The value of each policy option is the value of the correct assignment less the probability-weighted value of losses where the assignments are incorrect (refer equation 5 in Appendix I). Given that each of the three issues may be economically significant or insignificant, each policy option gives rise to one combination where the assignment of high and low weights is correct and seven combinations where the assignments are incorrect, i.e. seven loss functions may be defined for each policy option. The loss matrix, L, is presented in Appendix Table 2 below.

Appendix Table 2 – Matrix of loss functions with three objectives

Loss
Functions

 Policy Options
  1 2 3 4 5 6 7 8
1 - TP,TP,FN TP,FN,TP FN,TP,TP TP,FN,FN FN,TP,FN FN,FN,TP FN,FN,FN
2 TP,TP,FP - TP,FN,FP FN,TP,FP TP,FN,TN FN,TP,TN FN,FN,FP FN,FN,TN
3 TP,FP,TP TP,FP,FN - FN,FP,TP TP,TN,FN FN,FP,FN FN,TN,TP FN,TN,FN
4 FP,TP,TP FP,TP,FN FP,FN,TP - FP,FN,FN TN,TP,FN TN,FN,TP TN,FN,FN
5 TP,FP,FP TP,FP,TN TP,TN,FP FN,FP,FP - FN,FP,TN FN,TN,FP FN,TN,TN
6 FP,TP,FP FP,TP,TN FP,FN,FP TN,TP,FP FP,FN,TN - TN,FN,FP TN,FN,TN
7 FP,FP,TP FP,FP,FN FP,TN,TP TN,FP,TP FP,TN,FN TN,FP,FN - TN,TN,FN
8 FP,FP,FP FP,FP,TN FP,TN,FP TN,FP,FP FP,TN,TN TN,FP,TN TN,TN,FP -

The following makes a qualitative assessment of the average size of losses, the variability or uncertainty of losses and the worst-case scenario for the 56 loss functions (= 8 policy options x 7 loss functions per policy).

1.  High weight on all issues (H,H,H)

1. High weight on all issues (H,H,H)
LTP,TP,FP = VTP,TP,TN - VTP,TP,FP: The loss when agency cost is insignificant but receive high weight, while distortionary taxation and time-consistency are significant and correctly receive high weight.  Policy erroneously seeks to restrict operating surpluses, limit the build up of fungible assets, and seeks to discipline government expenditure by maintaining fairly high debt levels.
  Ave: H High expected loss because the high weight on agency costs substantially inhibits tax smoothing and jeopardises time-consistency by maintaining fairly high debt levels
  Var: H Variance of losses high because the adverse impact of limited tax smoothing depends on whether economy suffers adverse shocks.  Similarly, risk of time-inconsistency depends on adverse shocks occurring.
  Worst: H Same reasoning as above.
LTP,FP,TP = VTP,TN,TP - VTP,FP,TP:  The loss when time-consistency is insignificant but receives high weight, while distortionary taxation and agency cost is significant and correctly receive high weight.  Policy erroneously targets low net debt to avoid incentives for unexpected inflation and other forms of debt repudiation.
  Ave: L Low expected loss because agency cost and tax smoothing issues counteract each other in any case.  Agency costs may be somewhat higher due to debt target being too low, but impact limited because distortionary taxation also implies debt restrictions in cases of unjustified risk premia.
  Var: L  
  Worst: L  
LFP,TP,TP = VTN,TP,TP – VFP,TP,TP: The loss when distortionary taxation is insignificant but receives high weight, while time-consistency and agency cost is significant and correctly receive high weight.  Policy erroneously seeks to hedge risks through build up of assets (via operating surpluses) and leverage (up to level that attracts unjustified risk premia) and may seek to build up positive CNW buffer (if negative correlation between taxes and consumption).
  Ave: M The pressure to build up financial assets results in higher agency costs than under the correct policy.  Implementation risk and potential distortions of citizens’ investment portfolios due to capital income taxation (Coleman, 1997) add to losses.  A false positive error favouring tax smoothing is also likely to be persistent as difficult to observe the error ex post.  But, fundamentally, the high weight on agency costs limits the size of losses incurred on tax smoothing.
  Var: L Magnitude of losses not highly dependent on state of the economy or other factors.
  Worst: H High agency cost, poor implementation of tax smoothing, and significant distortion to private investment.
LTP,FP,FP = VTP,TN,TN - VTP,FP,FP:  The loss when both time-consistency and agency cost is insignificant but receive high weight, while distortionary taxation is significant and correctly receives high weight. 
  Ave: H Same reasons as for first case, LTP,TP,FP
  Var: H
  Worst: H
LFP,TP,FP = VTN,TP,TN – VFP,TP,FP:  The loss when both distortionary taxation and agency cost is insignificant but receive high weight, while time-consistency is significant and correctly receives high weight.
  Ave: L The high weights on tax smoothing and agency cost counteract each other.  Implementation risk and private sector distortion would be limited.  No agency costs incurred.  Some additional risk of time-inconsistency as the high weight on agency costs may limit run down of debt.
  Var: L  
  Worst: L  

LFP,FP,TP

 = VTN,TN,TP – VFP,FP,TP:  The loss when both distortionary taxation and time-consistency are insignificant but receive high weight, while agency cost is significant and correctly receives high weight.
  Ave: M Same as for LFP,TP,TP
   Var: L
  Worst: H
LFP,FP,FP = VTN,TN,TN – VFP,FP,FP: The loss when all three issues are insignificant but receive high weight.
  Ave: L The high weight on all issues means they counteract each other.  Potential exists for implementation risk on tax smoothing and distortions to citizen’s but losses would be limited.
  Var: L  
  Worst: L  
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