Demand and Price Analysis, Research Methods/ Statistical Methods,

Huang, Kuo S.

English

Research Papers in Economics

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Measuring Inverse Demand Systems and Consumer Welfare  
 
 
 
 Kuo S. Huang  
  
 
 
 
Economic Research Service 
U.S. Department of Agriculture 
Washington, DC 20036-5831 
 
 
 
 
 
 
 
Poster prepared for presentation at the Agricultural & Applied Economics Association’s 
 2011 AAEA & NAREA Joint Annual Meeting,  
Pittsburgh, Pennsylvania, July 24-26, 2011 
 
 
 
 
The views expressed here are those of the author and cannot be attributed to the 
Economic Research Service or the U.S. Department of Agriculture. 
 
 
 
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Measuring Inverse Demand Systems and Consumer Welfare  
 
An inverse (price-dependent) demand system, in which prices are functions of quantities 
demanded and income, is theoretically sound within the framework of classical demand 
theory. The demand system is important for applied demand analyses and is useful for 
situations in which policy options are directly related to quantity changes, such as 
marketing agreements for restricting availability through the use of a quota or the 
Government acreage control program. These are geared toward controlling supplies 
and/or stabilizing or raising commodity prices. Therefore, quantities rather than prices are 
appropriate instrumental or control variables in a demand system for evaluating some 
policy and program effects. 
 
The major focus of this study is on implementing the conceptual demand relationships 
into empirical modeling and estimation of inve in table 1rse demand systems and 
consumer welfare. An application to U.S. Personal Consumption Expenditures data 
provides an example.  
 
Methodology 
 
The modeling and estimation of an inverse demand system and its consumer welfare measure 
include the following three steps:    
 
Step 1--Apply distance function: d( u, q ) = minp [ ( p' q ) / c( u, p ) ] 
 
Let q be a vector of n quantities demanded, p a vector of the corresponding prices, m the per 
capita income, and u(q) the utility function. A distance function d( u, q ) is dual to a cost function 
c( u, p ) and can be used as a means to explore the properties of an inverse demand relationship 
such as homogeneity, symmetry, and scale aggregation. In addition, the Malmquist-quantity 
index defined as the ratio of two distance functions representing the constant utility quantity 
index for quantity changes can be regarded as a measure of efficiency in quantity metric welfare. 
 
Step 2--Specify inverse demand system: dri / ri =  Σj fij* (dqj* / qj*) + gi (ds / s) 
 
Armed with the concept of a distance function, an empirical inverse demand system can be 
derived from the Hotelling-Wold-identity. Where fij* be the compensated price flexibility of the 
ith commodity with respect to a quantity change in the jth commodity, s is a scale variable 
defined as log s = Σj wj log qj. A reference quantity vector qj*is defined by using the scale 
variable to deflate a quantity as qj* = qj / s.  
 
Step 3--Measure consumer welfare:  CV = p
h (u
0, q
1) ' q
1 - p
0 ' q
0 
 
For measuring consumer welfare, an alternative form of the Malmquist-quantity index is defined 
as the difference between two distance functions and then converted it into Hicksian 
compensating variation in expenditure (CV). Where p
h (u
0, q
1) be a vector of estimated 
compensated inverse demand at given quantity vector q
1 and at the same initial utility level u
0. A 
positive CV implies a requirement of more spending to achieve the same utility level as before   3
the quantity changes and causing a decrease in consumer welfare. By contrast, a negative CV 
implies a reduction in spending and thus a gain in consumer welfare.   
 
Application 
  
The developed methodology is applied to estimate an inverse demand system consisting 
of 11 expenditure categories using the data from U.S. Personal Consumption 
Expenditures covering 1960-2006. The estimated demand system is then applied to 
evaluate the consumer welfare effects of quantity changes in each expenditure category.     
 
Compensated own-price flexibilities. The compensated own-price flexibilities in table 1 
show how much a category price must change to induce consumers to purchase more 
quantity of that category. For example, the compensated own-price flexibilities for food 
consumed at home and energy are -0.5302 and -0.4311, respectively. These estimates are 
relatively larger than most of other categories and explain well the recent soaring prices 
of food and energy, because it takes a large increase in the prices of these basic goods in 
response to a small reduction of their quantities available in the market.  
 
Compensated cross-price flexibilities. The compensated cross-price flexibility in table 1, for 
example, between the price of food consumed at home and the quantity of clothing is -0.0478 
which implies that the two expenditure categories are substitutes. A marginal 10-percent increase 
in the quantity of clothing is associated with a 0.478 percent decrease in the price of food to 
induce consumers to purchase the same quantity of food. In contrast, the compensated cross-
price flexibility between the price of food and the quantity of medical care is positive at 0.3017 
indicating a complementary relationship between the two categories.   
 
Consumer welfare effects of quantity changes. The estimates of the demand system are then 
applied to the analysis of consumer welfare effects in response to quantity changes in various 
expenditure categories. Among the calculated welfare effects in table 2, a 10-percent decrease in 
the quantities of either food consumed at home or energy would increase per capita annual 
expenditures or incur consumer welfare losses by $1,491 or $798, respectively.  
 
Conclusion 
 
The concept of a distance function is useful for empirical modeling of an inverse demand 
system and consumer welfare measurement. The proposed differential-form inverse 
demand system has linear parameters for easy estimation, and the estimates can be 
interpreted directly as price flexibilities. The compensating variation in expenditures 
reflecting the quantity changes in distance functions is a proper measure for representing 
the efficiency in quantity metric welfare. The developed procedures are used to estimate 
an inverse demand system consisting of 11 U.S. expenditure categories and show the 
compensated price flexibilities and the consumer welfare effects of reduced quantity on a 
specific expenditure category.  
 
 
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Table 1--Compensated price flexibilities for U.S. personal consumption expenditures, 1960-2006
Price -- Reference quantity -- Scale Constant RMS
F.home F.away Energy Clothing O.nondur. Cars Furniture O.durable Transport Medical O.service errors, %
Food at home -0.5302 -0.0046 -0.0550 -0.0478 0.0915 -0.0114 0.0373 0.0960 -0.0199 0.3017 0.1424 -1.0166 -0.0166 1.15
0.1077 0.0374 0.0939 0.0426 0.0542 0.0334 0.0369 0.0210 0.0265 0.0866 0.1504 0.1972 0.0060
Food aw ay home -0.0074 -0.2104 0.1392 -0.1030 0.1325 -0.0313 -0.0734 0.0228 0.0191 0.1041 0.0076 -0.7354 0.0030 0.70
0.0607 0.0583 0.0650 0.0432 0.0492 0.0245 0.0326 0.0141 0.0218 0.0572 0.0868 0.1191 0.0035
Energy -0.0977 0.1522 -0.4311 0.1166 0.0901 -0.0979 0.1356 0.1250 -0.0211 0.4257 -0.3975 -1.7354 0.0126 7.31
0.1669 0.0711 0.2513 0.0860 0.1099 0.0676 0.0755 0.0454 0.0530 0.1844 0.2972 0.5064 0.0170
Clothing -0.0910 -0.1207 0.1250 -0.5977 -0.0763 -0.0205 0.1105 -0.0048 -0.0153 0.2136 0.4772 -0.2858 -0.0459 1.13
0.0811 0.0506 0.0921 0.0798 0.0712 0.0350 0.0443 0.0201 0.0290 0.0792 0.1290 0.1807 0.0055
Other nondurable 0.0929 0.0828 0.0515 -0.0407 -0.4272 0.0172 -0.0362 0.0499 0.0073 -0.1020 0.3044 -1.2077 0.0073 1.02
0.0550 0.0308 0.0628 0.0379 0.0603 0.0239 0.0306 0.0141 0.0193 0.0549 0.0918 0.1257 0.0038
Cars and parts -0.0168 -0.0282 -0.0808 -0.0158 0.0248 -0.0084 0.0341 0.0619 0.0014 0.0822 -0.0545 -1.3631 0.0045 2.59
0.0490 0.0221 0.0558 0.0270 0.0345 0.0339 0.0245 0.0189 0.0259 0.0629 0.0706 0.2276 0.0068
Furniture 0.0675 -0.0818 0.1382 0.1052 -0.0645 0.0422 -0.1833 0.0278 0.0268 -0.2982 0.2200 -0.7136 -0.0131 1.33
0.0668 0.0363 0.0770 0.0421 0.0545 0.0302 0.0473 0.0178 0.0240 0.0695 0.1118 0.1665 0.0051
Other durable 0.3318 0.0486 0.2433 -0.0087 0.1700 0.1461 0.0531 -0.7368 0.1102 0.0824 -0.4399 0.2398 -0.0200 2.03
0.0725 0.0300 0.0884 0.0365 0.0480 0.0446 0.0340 0.0330 0.0335 0.0900 0.1398 0.2742 0.0080
Transportation -0.0387 0.0229 -0.0231 -0.0157 0.0140 0.0019 0.0288 0.0620 -0.0406 0.0409 -0.0525 -1.3721 0.0059 2.61
0.0516 0.0261 0.0580 0.0296 0.0370 0.0344 0.0258 0.0189 0.0293 0.0656 0.0654 0.2244 0.0067
Medical care 0.1665 0.0354 0.1322 0.0619 -0.0554 0.0309 -0.0909 0.0131 0.0116 -0.2088 -0.0967 -1.1016 0.0268 1.04
0.0478 0.0194 0.0573 0.0229 0.0298 0.0237 0.0212 0.0144 0.0186 0.0731 0.0764 0.1573 0.0048
Other services 0.0327 0.0011 -0.0514 0.0576 0.0689 -0.0085 0.0279 -0.0292 -0.0062 -0.0218 -0.0710 -0.9533 0.0048 0.95
0.0345 0.0123 0.0384 0.0156 0.0208 0.0111 0.0142 0.0093 0.0077 0.0272 0.0760 0.0917 0.0028
Expenditure share 0.0841 0.0518 0.0261 0.0442 0.0828 0.0574 0.0464 0.0243 0.0432 0.1524 0.3874 1.0000
Note: For each pair of estimates, the upper part is the estimated compensated price flexibility, and the low er part is the standard error.  
         RMS = Root-mean-square errors.    
   5
 
 
Table 2--Consumer welfare effects of reduced quantity on a specific expenditure category 
Expenditure Reduced quantity on a specific expenditure category by 1 percent
category F.home F.away Energy Clothing O.nondur. Cars Furniture O.durable Transport Medical O.service
Compensating variation (CV) -- Increased expenditures (dollars) 
Food at home 5.61 1.39 2.50 2.25 -0.18 1.69 0.24 -1.73 1.55 -3.56 5.58
Food away home 1.06 -7.67 -1.59 2.07 -1.09 1.12 1.64 -0.08 0.19 0.12 3.99
Energy 3.87 -0.99 0.42 -0.63 0.85 3.13 -0.87 -1.31 1.53 -2.56 16.40
Clothing 1.32 1.56 -1.28 2.54 1.15 0.42 -1.12 0.14 0.32 -1.96 -4.29
Other nondurable 0.21 -0.50 0.14 2.30 1.33 1.27 2.25 -0.50 1.10 6.99 3.37
Cars and parts 1.94 1.46 2.15 1.12 1.30 -12.23 0.43 -0.43 0.85 1.86 8.19
Furniture -0.10 1.52 -1.34 -0.94 1.58 -0.02 -7.25 -0.13 0.05 5.20 0.53
Other durable -2.38 -0.41 -1.72 -0.01 -1.29 -1.08 -0.43 3.13 -0.82 -0.81 2.38
Transportation 1.68 0.53 0.96 0.83 1.09 0.84 0.38 -0.31 -8.75 1.84 6.07
Medical care -3.71 1.09 -4.03 -0.67 7.38 1.62 7.14 0.69 1.81 -12.41 25.14
Other services 5.06 5.15 10.29 -1.65 1.08 6.74 1.74 5.59 5.05 17.81 -17.05
Total 14.57 3.11 6.49 7.21 13.20 3.52 4.15 5.04 2.89 12.52 50.32
Expenditure 2428 1525 1588 1150 2444 1479 1278 677 1093 5028 10662
Change (percent) 0.60 0.20 0.41 0.63 0.54 0.24 0.32 0.74 0.26 0.25 0.47 

Measuring Inverse Demand Systems and Consumer Welfare

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