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A numerical study is made of the spectrum of broadband electrostatic noise (BEN) generated by ion beams in the plasma sheet boundary layer (PSBL), as introduced by Grabbe and Eastman (1984). Two cases are analyzed. The first case has a single hot electrons species present. In that case, two modes are unstable. One is a beam-acoustic mode, which drives instabilities over a broad frequency and angular range. This is a negative energy resistive medium instability in which inverse Landau damping causes growth and which occurs in a broad-angle cone centered about the magnetic field. Another is an ion-ion two-stream instability which has a much narrower frequency and angle range and considerably greater growth rates. The angle of maximum growth of this instability is typically around 75° (as in the work by Grabbe (1985a, b)) and is found to vary from the high 60s to the low 80s. It is only unstable in the low-frequency part of the BEN spectrum. The polarization, as measured in the lower part of the spectrum, is predicted to occur at an angle near 75° with respect to the magnetic field, in very good agreement with previous polarization measurements. The high-frequency part of the spectrum is predicted to be more closely field aligned. A second case analyzed is that in which a second population of cold electrons (arising from diffusion from the lobe into the plasma sheet boundary layer) is present. This cold electron population significantly increases the upper frequency range of the unstable spectrum, and the associated growth rates are enhanced. Its primary influence is found to be on the field-aligned component at small angles, but if the concentration of cold electrons is sufficiently large, it decreases the growth rates of the oblique ion-ion instability, thus stabilizing the larger angles. The field-aligned spectrum undergoes a transition from a negative energy to a positive energy two-stream instability. This second electron population generates the very high frequency part of the observed BEN spectrum that is just below the plasma frequency and creates wave amplitudes that are as large as the one produced by the large-angle instability. These results explain the gradual rise in the upper frequency range of BEN as the spacecraft approaches the plasma sheet from the lobe, and the sudden jump in that frequency up to near the plasma frequency upon crossing the PSBL. Also, it is predicted that the frequencies of field-aligned waves close to the plasma frequency are produced in a region where the cold electrons diffuse in from the lobe, and the large angles of growth are produced outside this region in the PSBL.