Monotone comparative statics

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

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Motivation

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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

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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

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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

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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

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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

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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 equivalently  . 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

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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

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Selected literature on monotone comparative statics and its applications

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  • 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

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  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 Accounting 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 Complementarities,” 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.
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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.
  51. ^ Vives, X. (2001): Oligopoly Pricing: Old Ideas and New Tools. MIT Press, ISBN 9780262720403.
  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.
  56. ^ Balbus, Ł., K. Reffett, and Ł. Woźny (2013): “Markov Stationary Equilibria in Stochastic Supermodular Games with Imperfect Private and Public Information,” Dynamic Games and Applications, 3(2), 187–206; Balbus, Ł., K. Reffett, and Ł. Woźny (2014): “A Constructive Study of Markov Equilibria in Stochastic Games with Strategic Complementaries,” Journal of Economic Theory, 150, p. 815–840.