Figure 7: If F satisfies submodularity on the square on the diagonal, this implies
it satisfies submodularity on the rectangle.
situation depicted in figure 5 as we now show, say for F . Consider the case of
figure 7 with the four points (x, z), (z, z), (x, y) , (z, y) as shown. With z<x<y,
we know from A1 that, since F = U on ∆U ,
F(x, y) - F(x, x) ≥ F(z, y) - F(z, x).
From Lemma 8.2, submodularity holds for the configuration of the square (x, x),
(z, x), (z, z), (z, x), hence we have
F(x, x) - F (x, z) ≥ F(z, x) - F(z, z).
Adding the two inequalities yields
F(x, y) - F (x, z) ≥ F(z, y) - F(z, z),
which is just the definition of submodularity for the original points (x, z), (z, z),
(x, y) and (z, y).
It can be shown via analogous steps that the submodularity of F for any
other configuration of points can be reduced to showing submodularity for
squares with two vertices on the diagonal. The details are left out. ■
The next result allows us to conclude that the two reaction curves always
admit a discontinuity that skips over the diagonal, a key step for our endogenous
heterogeneity result.
37
More intriguing information
1. An Interview with Thomas J. Sargent2. Parallel and overlapping Human Immunodeficiency Virus, Hepatitis B and C virus Infections among pregnant women in the Federal Capital Territory, Abuja, Nigeria
3. ANTI-COMPETITIVE FINANCIAL CONTRACTING: THE DESIGN OF FINANCIAL CLAIMS.
4. Performance - Complexity Comparison of Receivers for a LTE MIMO–OFDM System
5. The name is absent
6. Putting Globalization and Concentration in the Agri-food Sector into Context
7. Assessing Economic Complexity with Input-Output Based Measures
8. The name is absent
9. Permanent and Transitory Policy Shocks in an Empirical Macro Model with Asymmetric Information
10. Income Mobility of Owners of Small Businesses when Boundaries between Occupations are Vague