The Lefschetz-Hopf theorem and axioms for the Lefschetz number

The reduced Lefschetz number, that is, the Lefschetz number minus 1, is proved to be the unique integer-valued function L on selfmaps of compact polyhedra which is constant on homotopy classes such that (1) L(fg) = L(gf), for f:X -->Y and g:Y -->X; (2) if (f_1, f_2, f_3) is a map of a cofiber sequence into itself, then L(f_2) = L(f_1) + L(f_3); (3) L(f) = - (degree(p_1 f e_1) + ... + degree(p_k f e_k)), where f is a map of a wedge of k circles, e_r is the inclusion of a circle into the rth summand and p_r is the projection onto the rth summand. If f:X -->X is a selfmap of a polyhedron and I(f) is the fixed point index of f on all of X, then we show that I minus 1 satisfies the above axioms. This gives a new proof of the Normalization Theorem: If f:X -->X is a selfmap of a polyhedron, then I(f) equals the Lefschetz number of f. This result is equivalent to the Lefschetz-Hopf Theorem: If f: X -->X is a selfmap of a finite simplicial complex with a finite number of fixed points, each lying in a maximal simplex, then the Lefschetz number of f is the sum of the indices of all the fixed points of f.Comment: 9 page

Does Housing Wealth Make Us Less Equal? The Role of Durable Goods in the Distribution of Wealth

We study the role an illiquid durable consumption good plays in determining the level of precautionary savings and the distribution of wealth in a standard Aiyagari model (i.e. a model with heterogeneous agents, idiosyncratic uncertainty, and borrowing constraints). Transactions costs induce an inaction region over which the durable stock and the associated user cost are not adjusted in response to changes in income, increasing, on average, the volatility of non-durable consumption. The volatility of total consumption is then a function of the share of the durable good in the utility function and the width of the inaction region. We are particularly interested in parameterizations which increase the precautionary motive for saving through an increase in "committed expenditure risk". We find, for an empirically relevant share of durable consumption and for all transaction costs below an upper threshold, that the level of precautionary savings is increasing in the transaction costs. Transaction costs have only a modest impact on the degree of wealth dispersion, as measured by the Gini index, as the associated increase in savings is close to linear in wealth. While we are unable to match the dispersion of wealth in the data, we increase the dispersion over a single asset model (Gini index of .71 for financial assets and .37 for total wealth) and we are able to match the relative dispersion of financial to durable assets, i.e. we find financial assets much more unequal than durable assets. We also match the ratio of housing wealth to total wealth for the median agent. We calibrate the model to data from the PSID, the CES, and the SCFPrecautionary Savings, Wealth Distribution, Durable Goods

Trace formula for a dielectric microdisk with a point scatterer

Two-dimensional dielectric microcavities are of widespread use in microoptics applications. Recently, a trace formula has been established for dielectric cavities which relates their resonance spectrum to the periodic rays inside the cavity. In the present paper we extend this trace formula to a dielectric disk with a small scatterer. This system has been introduced for microlaser applications, because it has long-lived resonances with strongly directional far field. We show that its resonance spectrum contains signatures not only of periodic rays, but also of diffractive rays that occur in Keller's geometrical theory of diffraction. We compare our results with those for a closed cavity with Dirichlet boundary conditions.Comment: 39 pages, 18 figures, pdflate

The demand for homeowners insurance with bundled catastrophe coverages : Wharton project on managing catastrophic risks

In this paper, we estimate the demand for homeowner insurance in Florida. Since we are interested in a number of factors influencing demand, we approach the problem from two directions. We first estimate two hedonic equations representing the premium per contract and the price mark-up. We analyze how the contracts are bundled and how contract provisions, insurer characteristics and insured risk characteristics and demographics influence the premium per contract and the price mark-up. Second, we estimate the demand for homeowners insurance using two-stage least squares regression. We employ ISO's indicated loss costs as our proxy for real insurance services demanded. We assume that the demand for coverage is essentially a joint demand and thus we can estimate the demand for catastrophe coverage separately from the demand for noncatastrophe coverage. We determine that price elasticities are less elastic for catastrophic coverage than for non-catastrophic coverage. Further estimated income elasticities suggest that homeowners insurance is an inferior good. Finally, we conclude based on the results of a selection model that our sample of ISO reporting companies well represents the demand for insurance in the Florida market as a whole

The Rent-Price Ratio for the Aggregate Stock of Owner-Occupied Housing

We construct a time series of the rent-price ratio for the owner- occupied stock of housing, starting in 1960:1, by merging micro data from the last five Decennial Censuses of Housing with price indexes for house prices and rents.House Prices, Housing, Rents, CMHPI, Capitalization Rates