We use the Dirac equation with a ``(asymptotically free) Coulomb + (Lorentz scalar) linear '' potential to estimate the light quark wavefunction for \( q\bar Q\) mesons in the limit \(m_Q\to ...\infty\). We use these wavefunctions to calculate the Isgur-Wise function \(\xi (v.v^\prime )\) for orbital and radial ground states in the phenomenologically interesting range \(1\leq v.v^ \prime \leq 4\). We find a simple expression for the zero-recoil slope, \(\xi^ \prime (1) =-1/2- \epsilon^2 <{r_q}^2>/3\), where \(\epsilon\) is the energy eigenvalue of the light quark, which can be identified with the \(\bar\Lambda \) parameter of the Heavy Quark Effective Theory. This result implies an upper bound of \(-1/2\) for the slope \(\xi^\prime (1)\). Also, because for a very light quark \(q (q=u, d)\) the size \(\sqrt {<{r_q}^2>}\) of the meson is determined mainly by the ``confining'' term in the potential \((\gamma_\circ \sigma r)\), the shape of \(\xi_{u,d}(v.v^\prime )\) is seen to be mostly sensitive to the dimensionless ratio \(\bar \Lambda_{u,d}^2/\sigma\). We present results for the ranges of parameters \(150 MeV <\bar \Lambda_{u,d} <600 MeV\) \((\bar\Lambda_s \approx \bar\Lambda_{u,d}+100 MeV)\), \(0.14 {GeV}^2 \leq \sigma \leq 0.25 {GeV}^2\) and light quark masses \(m_u, m_d \approx 0, m_s=175 MeV\) and compare to existing experimental data and other theoretical estimates. Fits to the data give: \({\bar\Lambda_{u,d}}^2/\sigma =4.8\pm 1.7 \), \(-\xi^\prime_{u,d}(1)=2.4\pm 0.7\) and \(\vert V_{cb} \vert \sqrt {\frac {\tau_B}{1.48 ps}}=0.050\pm 0.008\) ARGUS '93; \({\bar\Lambda_{u,d}}^2/\sigma = 3.4\pm 1.8\), \(-\xi^\prime_{u,d}(1)=1.8\pm 0.7\) and \(\vert V_{cb} \vert \sqrt { \frac {\tau_B}{1.48 ps}}=0.043\pm 0.008\) CLEO '93; ${\bar\Lambda_{u,d}}^2/
We consider an exchange economy in which price rigidities are present. In the short run the non-numeraire commodities have a exible price level with respect to the numeraire commodity but their ...relative prices are mutually fixed. In the long run prices are assumed to be completely exible. For a given price level and fixed relative prices, markets can be equilibrated by means of quantity rationing on demand and supply. Keeping markets in equilibrium through rationing, we provide an adjustment process in prices and quantities converging from a trivial equilibrium with complete demand rationing on all non-numeraire markets to a Walrasian equilibrium. Along the path initially all relative prices are kept fixed and the price level is increased. Rationing schemes are adjusted to keep markets in equilibrium. Doing so the process reaches a short run equilibrium with only demand rationing and no rationing on the numeraire and at least one of the other commodities. The process allows for a downward price adjustment of non-rationed non-numeraire commodities and reaches a Walrasian equilibrium in the long run.
