Monotone comparative statics

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Monotone comparative statics is a sub-field of comparative statics that focuses on the conditions under which endogenous variables undergo monotone changes (that is, either increasing or decreasing) when there is a change in the exogenous parameters. Traditionally, comparative results in economics are obtained using the Implicit Function Theorem, an approach that requires the concavity and differentiability of the objective function as well as the interiority and uniqueness of the optimal solution. The methods of monotone comparative statics typically dispense with these assumptions. It focuses on the main property underpinning monotone comparative statics, which is a form of complementarity between the endogenous variable and exogenous parameter. Roughly speaking, a maximization problem displays complementarity if a higher value of the exogenous parameter increases the marginal return of the endogenous variable. This guarantees that the set of solutions to the optimization problem is increasing with respect to the exogenous parameter.

Basic results[edit]

Motivation[edit]

Let and let be a family of functions parameterized by , where is a partially ordered set (or poset, for short). How does the correspondence vary with ?

Standard comparative statics approach: Assume that set is a compact interval and is a continuously differentiable, strictly quasiconcave function of . If is the unique maximizer of , it suffices to show that for any , which guarantees that is increasing in . This guarantees that the optimum has shifted to the right, i.e., . This approach makes various assumptions, most notably the quasiconcavity of .

One-dimensional optimization problems[edit]

While it is clear what it means for a unique optimal solution to be increasing, it is not immediately clear what it means for the correspondence to be increasing in . The standard definition adopted by the literature is the following.

Definition (strong set order):[1] Let and be subsets of . Set dominates in the strong set order () if for any in and in , we have in and in .

In particular, if and , then if and only if . The correspondence is said to be increasing if whenever .

The notion of complementarity between exogenous and endogenous variables is formally captured by single crossing differences.

Definition (single crossing function): Let . Then is a single crossing function if for any we have .

Definition (single crossing differences):[2] The family of functions , , obey single crossing differences (or satisfy the single crossing property) if for all , function is a single crossing function.

Obviously, an increasing function is a single crossing function and, if is increasing in (in the above definition, for any ), we say that obey increasing differences. Unlike increasing differences, single crossing differences is an ordinal property, i.e., if obey single crossing differences, then so do , where for some function that is strictly increasing in .

Theorem 1:[3] Define . The family obey single crossing differences if and only if for all , we have for any .

Proof: Assume and , and . We have to show that and . We only need to consider the case where . Since , we obtain , which guarantees that . Furthermore, so that . If not, which implies (by single crossing differences) that , contradicting the optimality of at . To show the necessity of single crossing differences, set , where . Then for any guarantees that, if , then . Q.E.D.

Application (monopoly output and changes in costs): A monopolist chooses to maximise its profit , where is the inverse demand function and is the constant marginal cost. Note that obey single crossing differences. Indeed, take any and assume that ; for any such that , we obtain . By Theorem 1, the profit-maximizing output decreases as the marginal cost of output increases, i.e., as decreases.

Interval dominance order[edit]

Single crossing differences is not a necessary condition for the optimal solution to be increasing with respect to a parameter. In fact, the condition is necessary only for to be increasing in for any . Once the sets are restricted to a narrower class of subsets of , the single crossing differences condition is no longer necessary.

Definition (Interval):[4] Let . A set is an interval of if, whenever and are in , then any such that is also in .

For example, if , then is an interval of but not . Denote .

Definition (Interval Dominance Order):[5] The family obey the interval dominance order (IDO) if for any and , such that , for all , we have .

Like single crossing differences, the interval dominance order (IDO) is an ordinal property. An example of an IDO family is a family of quasiconcave functions where increasing in . Such a family need not obey single crossing differences.

A function is regular if is non-empty for any , where denotes the interval .

Theorem 2:[6] Denote . A family of regular functions obeys the interval dominance order if and only if is increasing in for all intervals .

Proof: To show the sufficiency of IDO, take any two , and assume that and . We only need to consider the case where . By definition , for all . Moreover, by IDO we have . Therefore, . Furthermore, it must be that . Otherwise, i.e., if , then by IDO we have , which contradicts that . To show the necessity of IDO, assume that there is an interval such that for all . This means that . There are two possible violations of IDO. One possibility is that . In this case, by the regularity of , the set is non-empty but does not contain which is impossible since increases in . Another possible violation of IDO occurs if but . In this case, the set either contains , which is not possible since increases in (note that in this case ) or it does not contain , which also violates monotonicity of . Q.E.D.

The next result gives useful sufficient conditions for single crossing differences and IDO.

Proposition 1:[7] Let be an interval of and be a family of continuously differentiable functions. (i) If, for any , there exists a number such that for all , then obey single crossing differences. (ii) If, for any , there exists a nondecreasing, strictly positive function such that for all , then obey IDO.

Application (Optimal stopping problem):[8] At each moment in time, agent gains profit of , which can be positive or negative. If agent decides to stop at time , the present value of his accumulated profit is

where is the discount rate. Since , the function has many turning points and they do not vary with the discount rate. We claim that the optimal stopping time is decreasing in , i.e., if then . Take any . Then, Since is positive and increasing, Proposition 1 says that obey IDO and, by Theorem 2, the set of optimal stopping times is decreasing.

Multi-dimensional optimization problems[edit]

The above results can be extended to a multi-dimensional setting. Let be a lattice. For any two , in , we denote their supremum (or least upper bound, or join) by and their infimum (or greatest lower bound, or meet) by .

Definition (Strong Set Order):[9] Let be a lattice and , be subsets of . We say that dominates in the strong set order ( ) if for any in and in , we have in and in .

Examples of the strong set order in higher dimensions.

  • Let and , be some closed intervals in . Clearly , where is the standard ordering on , is a lattice. Therefore, as it was shown in the previous section if and only if and ;
  • Let and , be some hyperrectangles. That is, there exist some vectors , , , in such that and , where is the natural, coordinate-wise ordering on . Note that is a lattice. Moreover, if and only if and ;
  • Let be a space of all probability distributions with support being a subset of , endowed with the first order stochastic dominance order . Note that is a lattice. Let , denote sets of probability distributions with support and respectively. Then, with respect to if and only if and .

Definition (Quasisupermodular function):[10] Let be a lattice. The function is quasisupermodular (QSM) if

The function is said to be a supermodular function if Every supermodular function is quasisupermodular. As in the case of single crossing differences, and unlike supermodularity, quasisupermodularity is an ordinal property. That is, if function is quasisupermodular, then so is function , where is some strictly increasing function.

Theorem 3:[11] Let is a lattice, a partially ordered set, and , subsets of . Given , we denote by . Then for any and

Proof: . Let , , and , . Since and , then . By quasisupermodularity, , and by the single crossing differences, . Hence . Now assume that . Then . By quasisupermodularity, , and by single crossing differences . But this contradicts that . Hence, .
. Set and . Then, and thus , which guarantees that, if , then . To show that single crossing differences also hold, set , where . Then for any guarantees that, if , then . Q.E.D.

Application (Production with multiple goods):[12] Let denote the vector of inputs (drawn from a sublattice of ) of a profit-maximizing firm, be the vector of input prices, and the revenue function mapping input vector to revenue (in ). The firm's profit is . For any , , , is increasing in . Hence, has increasing differences (and so it obeys single crossing differences). Moreover, if is supermodular, then so is . Therefore, it is quasisupermodular and by Theorem 3, for .

Constrained optimization problems[edit]

In some important economic applications, the relevant change in the constraint set cannot be easily understood as an increase with respect to the strong set order and so Theorem 3 cannot be easily applied. For example, consider a consumer who maximizes a utility function subject to a budget constraint. At price in and wealth , his budget set is and his demand set at is (by definition) . A basic property of consumer demand is normality, which means (in the case where demand is unique) that the demand of each good is increasing in wealth. Theorem 3 cannot be straightforwardly applied to obtain conditions for normality, because if (when is derived from the Euclidean order). In this case, the following result holds.

Theorem 4:[13] Suppose is supermodular and concave. Then the demand correspondence is normal in the following sense: suppose , and ; then there is and such that and .

The supermodularity of alone guarantees that, for any and , . Note that the four points , , , and form a rectangle in Euclidean space (in the sense that , , and and are orthogonal). On the other hand, supermodularity and concavity together guarantee that for any , where . In this case, crucially, the four points , , , and form a backward-leaning parallelogram in Euclidean space.

Monotone comparative statics under uncertainty[edit]

Let , and be a family of real-valued functions defined on that obey single crossing differences or the interval dominance order. Theorem 1 and 3 tell us that is increasing in . Interpreting to be the state of the world, this says that the optimal action is increasing in the state if the state is known. Suppose, however, that the action is taken before is realized; then it seems reasonable that the optimal action should increase with the likelihood of higher states. To capture this notion formally, let be a family of density functions parameterized by in the poset , where higher is associated with a higher likelihood of higher states, either in the sense of first order stochastic dominance or the monotone likelihood ratio property. Choosing under uncertainty, the agent maximizes

For to be increasing in , it suffices (by Theorems 1 and 2) that family obey single crossing differences or the interval dominance order. The results in this section give condition under which this holds.

Theorem 5: Suppose obeys increasing differences. If is ordered with respect to first order stochastic dominance, then obeys increasing differences.

Proof: For any , define . Then, , or equivalntly . Since obeys increasing differences, is increasing in and first order stochastic dominance guarantees is increasing in . Q.E.D.

In the following theorem, X can be either ``single crossing differences" or ``the interval dominance order".

Theorem 6:[14] Suppose (for ) obeys X. Then the family obeys X if is ordered with respect to the monotone likelihood ratio property.

The monotone likelihood ratio condition in this theorem cannot be weakened, as the next result demonstrates.

Proposition 2: Let and be two probability mass functions defined on and suppose is does not dominate with respect to the monotone likelihood ratio property. Then there is a family of functions , defined on , that obey single crossing differences, such that , where (for ).

Application (Optimal portfolio problem): An agent maximizes expected utility with the strictly increasing Bernoulli utility function . (Concavity is not assumed, so we allow the agent to be risk loving.) The wealth of the agent, , can be invested in a safe or risky asset. The prices of the two assets are normalized at 1. The safe asset gives a constant return , while the return of the risky asset is governed by the probability distribution . Let denote the agent's investment in the risky asset. Then the wealth of the agent in state is . The agent chooses to maximize

Note that , where , obeys single crossing (though not necessarily increasing) differences. By Theorem 6, obeys single crossing differences, and hence is increasing in , if is ordered with respect to the monotone likelihood ratio property.

Aggregation of the single crossing property[edit]

While the sum of increasing functions is also increasing, it is clear that the single crossing property need not be preserved by aggregation. For the sum of single crossing functions to have the same property requires that the functions be related to each other in a particular manner.

Definition (monotone signed-ratio):[15] Let be a poset. Two functions obey signed{ -}ratio monotonicity if, for any , the following holds:

  • if and , then
  • if and , then

Proposition 3: Let and be two single crossing functions. Then is a single crossing function for any non{-}negative scalars and if and only if and obey signed-ratio monotonicity.

Proof: Suppose that and . Define , so that . Since is a single crossing function, it must be that , for any . Moreover, recall that since is a single crossing function, then . By rearranging the above inequality, we conclude that
To prove the converse, without loss of generality assume that . Suppose that
If both and , then and since both functions are single crossing. Hence, . Suppose that and . Since and obey signed{-}ratio monotonicity it must be that
Since is a single crossing function, , and so Q.E.D.

This result can be generalized to infinite sums in the following sense.

Theorem 7:[16] Let be a finite measure space and suppose that, for each , is a bounded and measurable function of . Then is a single crossing function if, for all , , the pair of functions and of satisfy signed-ratio monotonicity. This condition is also necessary if contains all singleton sets and is required to be a single crossing function for any finite measure .

Application (Monopoly problem under uncertainty):[17] A firm faces uncertainty over the demand for its output and the profit at state is given by , where is the marginal cost and is the inverse demand function in state . The firm maximizes

where is the probability of state and is the Bernoulli utility function representing the firm’s attitude towards uncertainty. By Theorem 1, is increasing in (i.e., output falls with marginal cost) if the family obeys single crossing differences. By definition, the latter says that, for any , the function

is a single crossing function. For each , is s single crossing function of . However, unless is linear, will not, in general, be increasing in . Applying Theorem 6, is a single crossing function if, for any , the functions and (of ) obey signed-ratio monotonicity. This is guaranteed when (i) is decreasing in and increasing in and obeys increasing differences; and (ii) is twice differentiable, with , and obeys decreasing absolute risk aversion (DARA).

See also[edit]

Selected literature on monotone comparative statics and its applications[edit]

  • Basic techniques – Milgrom and Shannon (1994).,[18] Milgrom (1994),[19] Shannon (1995),[20] Topkis (1998),[21] Edlin and Shannon (1998),[22] Athey (2002),[23] Quah (2007),[24] Quah and Strulovici (2009, 2012),[25] Kukushkin (2013);[26]
  • Production complementarities and their implications – Milgrom and Roberts (1990a, 1995);[27] Topkis (1995);[28]
  • Games with strategic complementarities – Milgrom and Roberts (1990b);[29] Topkis (1979);[30] Vives (1990);[31]
  • Comparative statics of the consumer optimization problem – Antoniadou (2007);[32] Quah (2007);[33] Shirai (2013);[34]
  • Monotone comparative statics under uncertainty – Athey (2002);[35] Quah and Strulovici (2009, 2012);[36]
  • Monotone comparative statics for models of politics – Gans and Smart (1996),[37] Ashworth and Bueno de Mesquita (2006);[38]
  • Comparative statics of optimal stopping problems – Quah and Strulovici (2009, 2013);[39]
  • Monotone Bayesian games – Athey (2001);[40] McAdams (2003);[41] Quah and Strulovici (2012);[42]
  • Bayesian games with strategic complementarities – Van Zandt (2010);[43] Vives and Van Zandt (2007);[44]
  • Auction theory – Athey (2001);[45] McAdams (2007a,b);[46] Reny and Zamir (2004);[47]
  • Comparing information structures – Quah and Strulovici (2009);[48]
  • Comparative statics in Industrial Organisation – Amir and Grilo (1999);[49] Amir and Lambson (2003);[50] Vives (2001);[51]
  • Neoclassical optimal growth – Amir (1996b);[52] Datta, Mirman, and Reffett (2002);[53]
  • Multi-stage games – Vives (2009);[54]
  • Dynamic stochastic games with infinite horizon – Amir (1996a, 2003);[55] Balbus, Reffett, and Woźny (2013, 2014)[56]

References[edit]

  1. ^ See Veinott (1992): Lattice programming: qualitative optimization and equilibria. MS Stanford.
  2. ^ See Milgrom, P., and C. Shannon (1994): “Monotone Comparative Statics,” Econometrica, 62(1), 157–180; or Quah, J. K.-H., and B. Strulovici (2012): “Aggregating the Single Crossing Property,” Econometrica, 80(5), 2333–2348.
  3. ^ Milgrom, P., and C. Shannon (1994): “Monotone Comparative Statics,” Econometrica, 62(1), 157–180.
  4. ^ Quah, J. K.-H., and B. Strulovici (2009): “Comparative Statics, Informativeness, and the Interval Dominance Order,” Econometrica, 77(6), 1949–1992.
  5. ^ Quah, J. K.-H., and B. Strulovici (2009): “Comparative Statics, Informativeness, and the Interval Dominance Order,” Econometrica, 77(6), 1949–1992.
  6. ^ Quah, J. K.-H., and B. Strulovici (2009): “Comparative Statics, Informativeness, and the Interval Dominance Order,” Econometrica, 77(6), 1949–1992.
  7. ^ Quah, J. K.-H., and B. Strulovici (2009): “Comparative Statics, Informativeness, and the Interval Dominance Order,” Econometrica, 77(6), 1949–1992.
  8. ^ Quah, J. K.-H., and B. Strulovici (2009): “Comparative Statics, Informativeness, and the Interval Dominance Order,” Econometrica, 77(6), 1949–1992; and Quah, J. K.-H., and B. Strulovici (2013): “Discounting, Values, and Decisions,” Journal of Political Economy, 121(5), 896-939.
  9. ^ See Veinott (1992): Lattice programming: qualitative optimization and equilibria. MS Stanford.
  10. ^ Milgrom, P., and C. Shannon (1994): “Monotone Comparative Statics,” Econometrica, 62(1), 157–180.
  11. ^ Milgrom, P., and C. Shannon (1994): “Monotone Comparative Statics,” Econometrica, 62(1), 157–180.
  12. ^ See Milgrom, P., and J. Roberts (1990a): “The Economics of Modern Manufacturing: Technology, Strategy, and Organization,” American Economic Review, 80(3), 511–528; or Topkis, D. M. (1979): “Equilibrium Points in Nonzero-Sum n-Person Submodular Games,” SIAM Journal of Control and Optimization, 17, 773–787.
  13. ^ Quah, J. K.-H. (2007): “The Comparative Statics of Constrained Optimization Problems,” Econometrica, 75(2), 401–431.
  14. ^ See Athey, S. (2002): “Monotone Comparative Statics Under Uncertainty,” Quarterly Journal of Economics, 117(1), 187–223; for the case of single crossing differences and Quah, J. K.-H., and B. Strulovici (2009): “Comparative Statics, Informativeness, and the Interval Dominance Order,” Econometrica, 77(6), 1949–1992; for the case of IDO.
  15. ^ Quah, J. K.-H., and B. Strulovici (2012): “Aggregating the Single Crossing Property,” Econometrica, 80(5), 2333–2348.
  16. ^ Quah, J. K.-H., and B. Strulovici (2012): “Aggregating the Single Crossing Property,” Econometrica, 80(5), 2333–2348.
  17. ^ Quah, J. K.-H., and B. Strulovici (2012): “Aggregating the Single Crossing Property,” Econometrica, 80(5), 2333–2348.
  18. ^ Milgrom, P., and C. Shannon (1994): “Monotone Comparative Statics,” Econometrica, 62(1), 157–180.
  19. ^ Milgrom, P. (1994): “Comparing Optima: Do Simplifying Assumptions Affect Conclusions?,” Journal of Political Economy, 102(3), 607–15.
  20. ^ Shannon, C. (1995): “Weak and Strong Monotone Comparative Statics,” Economic Theory, 5(2), 209–27.
  21. ^ Topkis, D. M. (1998): Supermodularity and Complementarity, Frontiers of economic research, Princeton University Press, ISBN 9780691032443.
  22. ^ Edlin, A. S., and C. Shannon (1998): “Strict Monotonicity in Comparative Statics,” Journal of Economic Theory, 81(1), 201–219.
  23. ^ Athey, S. (2002): “Monotone Comparative Statics Under Uncertainty,” Quarterly Journal of Economics, 117(1), 187–223.
  24. ^ Quah, J. K.-H. (2007): “The Comparative Statics of Constrained Optimization Problems,” Econometrica, 75(2), 401–431.
  25. ^ Quah, J. K.-H., and B. Strulovici (2009): “Comparative Statics, Informativeness, and the Interval Dominance Order,” Econometrica, 77(6), 1949–1992; Quah, J. K.-H., and B. Strulovici (2012): “Aggregating the Single Crossing Property,” Econometrica, 80(5), 2333–2348.
  26. ^ Kukushkin, N. (2013): “Monotone comparative statics: changes in preferences versus changes in the feasible set,” Economic Theory, 52(3), 1039–1060.
  27. ^ Milgrom, P., and J. Roberts (1990a): “The Economics of Modern Manufacturing: Technology, Strategy, and Organization,” American Economic Review, 80(3), 511–528; Milgrom, P., and J. Roberts (1995): “Complementaries and fit. Strategy, structure and organizational change in manufacturing,” Journal of Acconuting and Economics, 19, 179–208.
  28. ^ Topkis, D. M. (1995): “Comparative statics of the firm,” Journal of Economic Theory, 67, 370–401.
  29. ^ Milgrom, P., and J. Roberts (1990b): “Rationalizability, Learning and Equilibrium in Games with Strategic Complementaries,” Econometrica, 58(6), 1255–1277.
  30. ^ Topkis, D. M. (1979): “Equilibrium Points in Nonzero-Sum n-Person Submodular Games,” SIAM Journal of Control and Optimization, 17, 773–787.
  31. ^ Vives, X. (1990): “Nash Equilibrium with Strategic Complementarites,” Journal of Mathematical Economics, 19, 305–321.
  32. ^ Antoniadou, E. (2007): “Comparative Statics for the Consumer Problem,” Economic Theory, 31, 189–203, Exposita Note.
  33. ^ Quah, J. K.-H. (2007): “The Comparative Statics of Constrained Optimization Problems,” Econometrica, 75(2), 401–431.
  34. ^ Shirai, K. (2013): “Welfare variations and the comparative statics of demand,” Economic Theory, 53(2)Volume 53, 315-333.
  35. ^ Athey, S. (2002): “Monotone Comparative Statics Under Uncertainty,” Quarterly Journal of Economics, 117(1), 187–223.
  36. ^ Quah, J. K.-H., and B. Strulovici (2009): “Comparative Statics, Informativeness, and the Interval Dominance Order,” Econometrica, 77(6), 1949–1992; Quah, J. K.-H., and B. Strulovici (2012): “Aggregating the Single Crossing Property,” Econometrica, 80(5), 2333–2348.
  37. ^ Gans, J. S., and M. Smart (1996): “Majority voting with single-crossing preferences,” Journal of Public Economics, 59(2), 219–237.
  38. ^ Ashworth, S., and E. Bueno de Mesquita (2006): “Monotone Comparative Statics for Models of Politics,” American Journal of Political Science, 50(1), 214–231.
  39. ^ Quah, J. K.-H., and B. Strulovici (2009): “Comparative Statics, Informativeness, and the Interval Dominance Order,” Econometrica, 77(6), 1949–1992; Quah, J. K.-H., and B. Strulovici (2013): “Discounting, Values, and Decisions,” Journal of Political Economy, 121(5), 896-939.
  40. ^ Athey, S. (2001): “Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information,” Econometrica, 69(4), 861–889.
  41. ^ McAdams, D. (2003): “Isotone Equilibrium in Games of Incomplete Information,” Econometrica, 71(4), 1191–1214.
  42. ^ Quah, J. K.-H., and B. Strulovici (2012): “Aggregating the Single Crossing Property,” Econometrica, 80(5), 2333–2348.
  43. ^ Van Zandt, T. (2010): “Interim Bayesian-Nash Equilibrium on Universal Type Spaces for Supermodular Games,” Journal of Economic Theory, 145(1), 249–263.
  44. ^ Vives, X., and T. Van Zandt (2007): “Monotone Equilibria in Bayesian Games with Strategic Complementaries,” Journal of Economic Theory, 134(1), 339–360.
  45. ^ Athey, S. (2001): “Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information,” Econometrica, 69(4), 861–889.
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  47. ^ Reny, P. J., and S. Zamir (2004): “On the Existence of Pure Strategy Monotone Equilibria in Asymmetric First-Price Auctions,” Econometrica, 72(4), 1105–1125.
  48. ^ Quah, J. K.-H., and B. Strulovici (2009): “Comparative Statics, Informativeness, and the Interval Dominance Order,” Econometrica, 77(6), 1949–1992.
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  50. ^ Amir, R., and V. E. Lambson (2003): “Entry, Exit, and Imperfect Competition in the Long Run,” Journal of Economic Theory, 110(1), 191–203.
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  52. ^ Amir, R. (1996b): “Sensitivity Analysis of Multisector Optimal Economic Dynamics,” Journal of Mathematical Economics, 25, 123–141.
  53. ^ Datta, M., L. J. Mirman, and K. L. Reffett (2002): “Existence and Uniqueness of Equilibrium in Distorted Dynamic Economies with Capital and Labor,” Journal of Economic Theory, 103(2), 377–410.
  54. ^ Vives, X. (2009): “Strategic Complementarity in Multi-Stage Games,” Economic Theory, 40(1), 151–171.
  55. ^ Amir, R. (1996a): “Continuous Stochastic Games of Capital Accumulation with Convex Transitions,” Games and Economic Behavior, 15(2), 111-131; Amir, R. (2003): “Stochastic Games in Economics and Related Fields: An Overview,” in Stochastic games and applications, ed. by A. Neyman, and S. Sorin, NATO Advanced Science Institutes Series D: Behavioural and Social Sciences. Kluwer Academin Press, Boston, ISBN 978-94-010-0189-2.
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