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A revealed preference test of rationing
Economics Letters 113 (2011) 234–236
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
A revealed preference test of rationing Adrian R. Fleissig a,∗ , Gerald Whitney b a
Department of Economics, California State University, Fullerton, Fullerton, CA 92834, United States
b
Department of Economics and Finance, University of New Orleans, United States
article
info
Article history: Received 5 October 2010 Received in revised form 15 July 2011 Accepted 25 July 2011 Available online 30 July 2011
abstract This paper derives a revealed preference test for utility maximization under rationing and can detect, for which goods rationing is binding without specifying a functional form or imposing rationing constraints prior to estimation. For UK data from 1920–55, we find evidence of utility maximization under rationing with rationing binding for food and other services. Estimated virtual prices exceed observed food prices by 16.5% in 1947 and observed prices of other services by 10.9% in 1952. © 2011 Elsevier B.V. All rights reserved.
JEL classification: C14 E21 Keywords: Afriat Inequalities Generalized axiom of revealed preference Rationing
1. Introduction
2. Rationing and revealed preference
This paper develops a nonparametric procedure to evaluate, for utility maximization, when consumer choices in some periods are subject to single or multiple rationing constraints. Our methodology extends the revealed preference approach of Afriat (1967) and Varian (1982, 1983) and the linear programming procedure of Fleissig and Whitney (2003, 2005). In addition, it gives nonparametric estimates of virtual prices of rationed goods, which provide a measure of the cost associated with rationing as discussed in Hicks (1940), Rothbarth (1941), Neary and Roberts (1980) and Mackay and Whitney (1980). Studies using virtual prices include Huffman and Johnson (2004) to estimate welfare costs for Poland, Hausman’s (1997) price analysis for new brands of cereal and Bettendorf and Buyst (1997) to examine rent control in Belgium. To estimate the effects of dismantling the wartime UK rationing system, Stone (1945, 1954), Stone and Rowe (1954, 1966), Tobin and Houthakker (1951) and Houthakker and Tobin (1952) use a framework, which they assume is consistent with consumer optimizing behavior. We use our new procedure to evaluate their assumption that preferences were consistent with utility maximization after rationing was removed.
The revealed preference approach is frequently used to evaluate consumer demand and Afriat (1967) and Varian (1982) produce conditions, Afriat inequalities and the generalized axiom of revealed preferences (GARP), under which a well-behaved utility function rationalizes the data. To examine utility maximization under rationing, consider the following:
∗
Corresponding author. Tel.: +1 657 278 3816; fax: +1 657 278 1387. E-mail address: [email protected] (A.R. Fleissig).
0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.07.020
Max u(x) s.t pi x ≤ mi
(1)
A i x ≤ bi where pi = (pi1 , . . . , pik ) is a price vector with quantities xi = (xi1 , . . . , xik )′ for the k-goods, Ai is a k by k matrix with each row representing rationing, but not necessarily binding constraints. A special case of rationing by Varian (1983) replaces matrix Ai with a row vector ai and gives an upper bound on a single good. Rationing theorem of Varian (1983). The following conditions are equivalent. (i) The data can be rationalized by a continuous, concave, monotonic, nonsatiated utility function; (ii) There exists numbers, µi , λi ≥ 0 with µi = 0 if ai xi < bi and λi = 0 if pi xi < mi , such that U i ≤ U j + λj pj (xi − xj ) + µj aj (xi − xj ).
A.R. Fleissig, G. Whitney / Economics Letters 113 (2011) 234–236
Now, let xr ,j be a k-element vector of upper bounds that consumers can purchase in period j, x j ≤ x r ,j
for j = 1 to n
(2)
with element xr ,gj > xgj for unrationed goods. This can be represented more generally as: j r ,j
Ax ≤Ax . j j
(3)
The upper bounds are expenditure constraints when Aj is a diagonal matrix with pj on the diagonal. Solving the constrained maximization (1) gives marginal utilities.
(δ U /δ xj ) = λj pj + µj Aj .
(4)
This differs from the single constraint rationing model of Varian (1983) with µj , a k-element row vector of multipliers. Since rationing may not be binding for all goods, µj ≥ 0. Assuming the observed budget constraint is always binding requires λj > 0. Under rationing, marginal utilities of goods in period j may not be a constant proportion (λj ) of price and the Afriat inequalities under rationing are: U i ≤ U j + λj pj (xi − xj ) + µj Aj (xi − xj ).
(5)
Virtual prices p∗ j are those prices that when combined with observed prices and quantities for the rationed goods will satisfy the Afriat inequalities U i ≤ U j + λj p∗ (xi − xj ) j
(6)
with prices of unrationed goods identical to pj . U i ≤ U j + λj (pj + µj Aj /λj )(xi − xj ).
(7)
Here, µj is a vector with zeros for non-rationed goods and nonzeros for rationed goods giving the virtual price p∗j . p∗j = pj + µj Aj /λj .
(8)
The virtual price of the gth rationed good in period j, A is a diagonal matrix with pj on the diagonal, is: j
p∗gj = pgj + pgj µgj /λj = pgj (1 + µgj /λj ).
(9)
These virtual prices are not necessarily unique because other values of µgj and λj can satisfy GARP. Virtual prices can be used to measure changes in welfare using the compensating variation of Hicks (1943). Neary and Roberts (1980) show that the difference between virtual and actual prices gives a measure of the maximum amount a consumer would be willing to pay to have one extra unit of the rationed good. Virtual prices with estimated cross price elasticities of substitution can help assess the effect on other goods of relaxing rationing constraints. In applications, the researcher may not know for which goods rationing is binding; so µgj ≥ 0. The aim is to find values for µgj for which rationing is binding when there is a feasible solution for (5). Following Diewert (1973), Varian (1983) and Fleissig and Whitney (2003, 2005), this is a linear programming problem with ∑n ∑h a weighted objective function j=1 g =1 w g µgj , where w g are the weights. LP 1. Rationing for multiple goods Min Z =
n − h −
wg µgj
j=1 g =1
subject to U i ≤ U j + λj pj (xi − xj ) + µj Aj (xi − xj )
(10)
λi > 0 µgj ≥ 0. If a feasible solution exists, we conclude that the data can be rationalized by a utility function with binding rationing on goods
235
Table 1 Testing GARPa . Period
GARP violations
Years of violations
1920–38 1920–55
0 2
n/a 1947 and 1952
a GARP evaluated using transitive closure procedure (Varian, 1982).
where µgi > 0. No feasible solution implies that relaxing the rationing constraint fails to eliminate GARP violations. This solution differs from the quantity perturbation approaches of Varian (1985), Swofford and Whitney (1994) and Jones and de Peretti (2005). 3. Application using UK data A relatively complex UK rationing system across goods was imposed in January 1940. The gradual elimination of rationing started after World War II and many goods were no longer restricted by 1948 with the last restrictions on meat removed in 1954. Eliminating rationing was a concern for policy makers who used estimates such as Stone’s (1954) range of estimates of how much 1952 expenditures on different categories of goods would change if all war-time rationing were removed. We evaluate rationing using UK annual (1920–55) expenditure and price data from Stone and Rowe (1954, 1966) on goods and services: food, alcoholic drink, tobacco, rents, fuel and light, clothing, durable household goods, transportation and communications, other goods and other services. Price indices were obtained by dividing current by constant expenditures. Constant expenditures are converted into per capita terms using the adult equivalent population with weights: 1 for males 15 years and older, .9 for females 15 years and older, .675 for those between 5 and 14 years old and .28 for those under 5 years. Since 1938, Stone’s formula was applied using age distribution census data from Mitchell and Deane (1962). There were no GARP violations over the 1920–38 pre-war period, which implies that there exists a nonsatiated utility function that rationalizes the data (Table 1). When evaluating the period 1920–55, two GARP violations occur for 1947 and 1952 and imply the data are not consistent with utility maximization. To evaluate if the violations can be attributed to rationing, rationing is allowed only on all goods from 1939–54 with µgj = 0 for the other years. The LP procedure found a feasible solution and detects that food and other services were rationed (Table 2). The results show that if consumption of food and other services could have been increased, while decreasing consumption of at least one of the remaining eight goods, there exists a feasible solution for the Afriat inequalities. Therefore, rationing is a potential explanation for the violations and we conclude that preferences remain unchanged over the period. This supports the post-war estimates of Houthakker and Tobin (1952), Stone (1954) and Stone and Rowe (1954, 1966). An important result is detecting food rationing, which, according to Houthakker and Tobin (1952) and Stone and Rowe (1954), reached a height in 1947. Virtual prices are calculated and show that food prices would have had to increase by 16.5% while the price of other services would have had to increase by 10.9% for current supplies to meet unrationed demand, thus avoiding short term shortages. Following Neary and Roberts (1980), these estimates can be interpreted as the willingness to pay 16.5% more to obtain an additional unit of the food category and 10.9% more to obtain one more unit of other services. 4. Conclusions The new nonparametric procedure can detect rationing of many goods over multiple periods without having to specify a functional
236
A.R. Fleissig, G. Whitney / Economics Letters 113 (2011) 234–236 Table 2 Virtual prices. Period
1920–55
Year where
Ration good
Percentage difference price and virtual price (%)
1947 1952
Food Other services
16.5 10.9
µgj > 0
LP was solved using Excel add-in Large-Scale Engine.
form or to impose constraints on goods prior to estimation. Applied to UK data, our test finds that there exists a utility function under rationing from 1920 to 1955 with rationing binding for food and other services. Estimated virtual prices show that food prices would have had to increase by 16.5% in 1947 and services by 10.9% in 1952 to make unrationed demand equal to the rationed quantities. References Afriat, S., 1967. The construction of a utility function from expenditure data. International Economic Review 8, 67–77. Bettendorf, L., Buyst, E., 1997. Rent control and virtual prices: a case study for interwar belgium. Journal of Economic History 57, 654–673. Diewert, W., 1973. Afriat and revealed preference theory. Review of Economic Studies 40, 419–426. Fleissig, A., Whitney, G., 2003. A new PC based test for varian’s weak separability test. Journal of Business and Economic Statistics 21, 133–144. Fleissig, A., Whitney, G., 2005. Testing for the significance of violations of afriat’s inequalities. Journal of Business and Economic Statistics 23, 355–362. Hausman, J., 1997. Valuation of new goods under perfect and imperfect competition. In: Bresnahan, Gordon (Eds.), The Economics of New Goods. University of Chicago Press. Hicks, J., 1940. The valuation of the social income. Economica 105–124. Hicks, J., 1943. The four consumers’ surpluses. Review of Economic Studies 11, 37–41. Huffman, S., Johnson, S., 2004. Impacts of economic reform in poland: incidence and welfare changes within a consistent framework. Review of Economics and Statistics 86, 626–636.
Houthakker, H., Tobin, J., 1952. Estimates of the free demand for rationed foodstuffs. Economic Journal 62, 103–118. Jones, B., de Peretti, P., 2005. A comparison of two methods for testing the utility maximization hypothesis when quantity data are measured with errors. Macroeconomic Dynamics 9, 612–629. Mitchell, B., Deane, P., 1962. Abstract of British Historical Statistics. University Press. Mackay, R., Whitney, G., 1980. The comparative statics of conditional demands: theory and some applications. Econometrica 48, 1727–1744. Neary, J., Roberts, K., 1980. The theory of household behavior under rationing. European Economic Review 13, 25–42. Rothbarth, E., 1941. The measurement of changes in real income under conditions of rationing. Review of Economic Studies 100–107. Stone, R., 1945. The analysis of market demand. Journal of Royal Statistical Society 108, 1–98. Stone, R., 1954. Linear expenditure systems and demand analysis: an application to the pattern of british demand. Economic Journal 64, 511–527. Stone, R., Rowe, D., 1954. The Measurement of Consumers’ Expenditure and Behaviour in the United Kingdom, 1920–1938. Cambridge University Press. Stone, R., Rowe, D., 1966. The Measurement of Consumers’ Expenditure and Behaviour in the United Kingdom 1920–1938. Cambridge University Press. Swofford, J., Whitney, G., 1994. A revealed preference test for weakly separable utility maximization with incomplete adjustment. Journal of Econometrics 60, 235–249. Tobin, J., Houthakker, H., 1951. The effects of rationing on demand elasticities. Review of Economic Studies 18, 140–153. Varian, H., 1982. The nonparametric approach to demand analysis. Econometrica 50, 945–973. Varian, H., 1983. Nonparametric tests of consumer behavior. Review of Economic Studies 50, 99–110. Varian, H., 1985. Non-parametric analysis of optimizing behavior with measurement error. Journal of Econometrics 445–458.
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
A revealed preference test of rationing Adrian R. Fleissig a,∗ , Gerald Whitney b a
Department of Economics, California State University, Fullerton, Fullerton, CA 92834, United States
b
Department of Economics and Finance, University of New Orleans, United States
article
info
Article history: Received 5 October 2010 Received in revised form 15 July 2011 Accepted 25 July 2011 Available online 30 July 2011
abstract This paper derives a revealed preference test for utility maximization under rationing and can detect, for which goods rationing is binding without specifying a functional form or imposing rationing constraints prior to estimation. For UK data from 1920–55, we find evidence of utility maximization under rationing with rationing binding for food and other services. Estimated virtual prices exceed observed food prices by 16.5% in 1947 and observed prices of other services by 10.9% in 1952. © 2011 Elsevier B.V. All rights reserved.
JEL classification: C14 E21 Keywords: Afriat Inequalities Generalized axiom of revealed preference Rationing
1. Introduction
2. Rationing and revealed preference
This paper develops a nonparametric procedure to evaluate, for utility maximization, when consumer choices in some periods are subject to single or multiple rationing constraints. Our methodology extends the revealed preference approach of Afriat (1967) and Varian (1982, 1983) and the linear programming procedure of Fleissig and Whitney (2003, 2005). In addition, it gives nonparametric estimates of virtual prices of rationed goods, which provide a measure of the cost associated with rationing as discussed in Hicks (1940), Rothbarth (1941), Neary and Roberts (1980) and Mackay and Whitney (1980). Studies using virtual prices include Huffman and Johnson (2004) to estimate welfare costs for Poland, Hausman’s (1997) price analysis for new brands of cereal and Bettendorf and Buyst (1997) to examine rent control in Belgium. To estimate the effects of dismantling the wartime UK rationing system, Stone (1945, 1954), Stone and Rowe (1954, 1966), Tobin and Houthakker (1951) and Houthakker and Tobin (1952) use a framework, which they assume is consistent with consumer optimizing behavior. We use our new procedure to evaluate their assumption that preferences were consistent with utility maximization after rationing was removed.
The revealed preference approach is frequently used to evaluate consumer demand and Afriat (1967) and Varian (1982) produce conditions, Afriat inequalities and the generalized axiom of revealed preferences (GARP), under which a well-behaved utility function rationalizes the data. To examine utility maximization under rationing, consider the following:
∗
Corresponding author. Tel.: +1 657 278 3816; fax: +1 657 278 1387. E-mail address: [email protected] (A.R. Fleissig).
0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.07.020
Max u(x) s.t pi x ≤ mi
(1)
A i x ≤ bi where pi = (pi1 , . . . , pik ) is a price vector with quantities xi = (xi1 , . . . , xik )′ for the k-goods, Ai is a k by k matrix with each row representing rationing, but not necessarily binding constraints. A special case of rationing by Varian (1983) replaces matrix Ai with a row vector ai and gives an upper bound on a single good. Rationing theorem of Varian (1983). The following conditions are equivalent. (i) The data can be rationalized by a continuous, concave, monotonic, nonsatiated utility function; (ii) There exists numbers, µi , λi ≥ 0 with µi = 0 if ai xi < bi and λi = 0 if pi xi < mi , such that U i ≤ U j + λj pj (xi − xj ) + µj aj (xi − xj ).
A.R. Fleissig, G. Whitney / Economics Letters 113 (2011) 234–236
Now, let xr ,j be a k-element vector of upper bounds that consumers can purchase in period j, x j ≤ x r ,j
for j = 1 to n
(2)
with element xr ,gj > xgj for unrationed goods. This can be represented more generally as: j r ,j
Ax ≤Ax . j j
(3)
The upper bounds are expenditure constraints when Aj is a diagonal matrix with pj on the diagonal. Solving the constrained maximization (1) gives marginal utilities.
(δ U /δ xj ) = λj pj + µj Aj .
(4)
This differs from the single constraint rationing model of Varian (1983) with µj , a k-element row vector of multipliers. Since rationing may not be binding for all goods, µj ≥ 0. Assuming the observed budget constraint is always binding requires λj > 0. Under rationing, marginal utilities of goods in period j may not be a constant proportion (λj ) of price and the Afriat inequalities under rationing are: U i ≤ U j + λj pj (xi − xj ) + µj Aj (xi − xj ).
(5)
Virtual prices p∗ j are those prices that when combined with observed prices and quantities for the rationed goods will satisfy the Afriat inequalities U i ≤ U j + λj p∗ (xi − xj ) j
(6)
with prices of unrationed goods identical to pj . U i ≤ U j + λj (pj + µj Aj /λj )(xi − xj ).
(7)
Here, µj is a vector with zeros for non-rationed goods and nonzeros for rationed goods giving the virtual price p∗j . p∗j = pj + µj Aj /λj .
(8)
The virtual price of the gth rationed good in period j, A is a diagonal matrix with pj on the diagonal, is: j
p∗gj = pgj + pgj µgj /λj = pgj (1 + µgj /λj ).
(9)
These virtual prices are not necessarily unique because other values of µgj and λj can satisfy GARP. Virtual prices can be used to measure changes in welfare using the compensating variation of Hicks (1943). Neary and Roberts (1980) show that the difference between virtual and actual prices gives a measure of the maximum amount a consumer would be willing to pay to have one extra unit of the rationed good. Virtual prices with estimated cross price elasticities of substitution can help assess the effect on other goods of relaxing rationing constraints. In applications, the researcher may not know for which goods rationing is binding; so µgj ≥ 0. The aim is to find values for µgj for which rationing is binding when there is a feasible solution for (5). Following Diewert (1973), Varian (1983) and Fleissig and Whitney (2003, 2005), this is a linear programming problem with ∑n ∑h a weighted objective function j=1 g =1 w g µgj , where w g are the weights. LP 1. Rationing for multiple goods Min Z =
n − h −
wg µgj
j=1 g =1
subject to U i ≤ U j + λj pj (xi − xj ) + µj Aj (xi − xj )
(10)
λi > 0 µgj ≥ 0. If a feasible solution exists, we conclude that the data can be rationalized by a utility function with binding rationing on goods
235
Table 1 Testing GARPa . Period
GARP violations
Years of violations
1920–38 1920–55
0 2
n/a 1947 and 1952
a GARP evaluated using transitive closure procedure (Varian, 1982).
where µgi > 0. No feasible solution implies that relaxing the rationing constraint fails to eliminate GARP violations. This solution differs from the quantity perturbation approaches of Varian (1985), Swofford and Whitney (1994) and Jones and de Peretti (2005). 3. Application using UK data A relatively complex UK rationing system across goods was imposed in January 1940. The gradual elimination of rationing started after World War II and many goods were no longer restricted by 1948 with the last restrictions on meat removed in 1954. Eliminating rationing was a concern for policy makers who used estimates such as Stone’s (1954) range of estimates of how much 1952 expenditures on different categories of goods would change if all war-time rationing were removed. We evaluate rationing using UK annual (1920–55) expenditure and price data from Stone and Rowe (1954, 1966) on goods and services: food, alcoholic drink, tobacco, rents, fuel and light, clothing, durable household goods, transportation and communications, other goods and other services. Price indices were obtained by dividing current by constant expenditures. Constant expenditures are converted into per capita terms using the adult equivalent population with weights: 1 for males 15 years and older, .9 for females 15 years and older, .675 for those between 5 and 14 years old and .28 for those under 5 years. Since 1938, Stone’s formula was applied using age distribution census data from Mitchell and Deane (1962). There were no GARP violations over the 1920–38 pre-war period, which implies that there exists a nonsatiated utility function that rationalizes the data (Table 1). When evaluating the period 1920–55, two GARP violations occur for 1947 and 1952 and imply the data are not consistent with utility maximization. To evaluate if the violations can be attributed to rationing, rationing is allowed only on all goods from 1939–54 with µgj = 0 for the other years. The LP procedure found a feasible solution and detects that food and other services were rationed (Table 2). The results show that if consumption of food and other services could have been increased, while decreasing consumption of at least one of the remaining eight goods, there exists a feasible solution for the Afriat inequalities. Therefore, rationing is a potential explanation for the violations and we conclude that preferences remain unchanged over the period. This supports the post-war estimates of Houthakker and Tobin (1952), Stone (1954) and Stone and Rowe (1954, 1966). An important result is detecting food rationing, which, according to Houthakker and Tobin (1952) and Stone and Rowe (1954), reached a height in 1947. Virtual prices are calculated and show that food prices would have had to increase by 16.5% while the price of other services would have had to increase by 10.9% for current supplies to meet unrationed demand, thus avoiding short term shortages. Following Neary and Roberts (1980), these estimates can be interpreted as the willingness to pay 16.5% more to obtain an additional unit of the food category and 10.9% more to obtain one more unit of other services. 4. Conclusions The new nonparametric procedure can detect rationing of many goods over multiple periods without having to specify a functional
236
A.R. Fleissig, G. Whitney / Economics Letters 113 (2011) 234–236 Table 2 Virtual prices. Period
1920–55
Year where
Ration good
Percentage difference price and virtual price (%)
1947 1952
Food Other services
16.5 10.9
µgj > 0
LP was solved using Excel add-in Large-Scale Engine.
form or to impose constraints on goods prior to estimation. Applied to UK data, our test finds that there exists a utility function under rationing from 1920 to 1955 with rationing binding for food and other services. Estimated virtual prices show that food prices would have had to increase by 16.5% in 1947 and services by 10.9% in 1952 to make unrationed demand equal to the rationed quantities. References Afriat, S., 1967. The construction of a utility function from expenditure data. International Economic Review 8, 67–77. Bettendorf, L., Buyst, E., 1997. Rent control and virtual prices: a case study for interwar belgium. Journal of Economic History 57, 654–673. Diewert, W., 1973. Afriat and revealed preference theory. Review of Economic Studies 40, 419–426. Fleissig, A., Whitney, G., 2003. A new PC based test for varian’s weak separability test. Journal of Business and Economic Statistics 21, 133–144. Fleissig, A., Whitney, G., 2005. Testing for the significance of violations of afriat’s inequalities. Journal of Business and Economic Statistics 23, 355–362. Hausman, J., 1997. Valuation of new goods under perfect and imperfect competition. In: Bresnahan, Gordon (Eds.), The Economics of New Goods. University of Chicago Press. Hicks, J., 1940. The valuation of the social income. Economica 105–124. Hicks, J., 1943. The four consumers’ surpluses. Review of Economic Studies 11, 37–41. Huffman, S., Johnson, S., 2004. Impacts of economic reform in poland: incidence and welfare changes within a consistent framework. Review of Economics and Statistics 86, 626–636.
Houthakker, H., Tobin, J., 1952. Estimates of the free demand for rationed foodstuffs. Economic Journal 62, 103–118. Jones, B., de Peretti, P., 2005. A comparison of two methods for testing the utility maximization hypothesis when quantity data are measured with errors. Macroeconomic Dynamics 9, 612–629. Mitchell, B., Deane, P., 1962. Abstract of British Historical Statistics. University Press. Mackay, R., Whitney, G., 1980. The comparative statics of conditional demands: theory and some applications. Econometrica 48, 1727–1744. Neary, J., Roberts, K., 1980. The theory of household behavior under rationing. European Economic Review 13, 25–42. Rothbarth, E., 1941. The measurement of changes in real income under conditions of rationing. Review of Economic Studies 100–107. Stone, R., 1945. The analysis of market demand. Journal of Royal Statistical Society 108, 1–98. Stone, R., 1954. Linear expenditure systems and demand analysis: an application to the pattern of british demand. Economic Journal 64, 511–527. Stone, R., Rowe, D., 1954. The Measurement of Consumers’ Expenditure and Behaviour in the United Kingdom, 1920–1938. Cambridge University Press. Stone, R., Rowe, D., 1966. The Measurement of Consumers’ Expenditure and Behaviour in the United Kingdom 1920–1938. Cambridge University Press. Swofford, J., Whitney, G., 1994. A revealed preference test for weakly separable utility maximization with incomplete adjustment. Journal of Econometrics 60, 235–249. Tobin, J., Houthakker, H., 1951. The effects of rationing on demand elasticities. Review of Economic Studies 18, 140–153. Varian, H., 1982. The nonparametric approach to demand analysis. Econometrica 50, 945–973. Varian, H., 1983. Nonparametric tests of consumer behavior. Review of Economic Studies 50, 99–110. Varian, H., 1985. Non-parametric analysis of optimizing behavior with measurement error. Journal of Econometrics 445–458.