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We establish some exact asymptotic results for a matching problem with respect to a family of beta distributions. Let X1,…,Xn be independent random variables with common distribution the symmetric Jacobi measure $d\mu (x) = C_{d} (1-x^{2})^{\frac d2 -1} dx$ with dimension d ≥ 1 on [− 1,1], and let $\mu _{n} = \frac {1}{n} {\sum }_{i = 1}^{n} \delta _{X_{i}}$ be the associated empirical measure. We show that $$ \lim\limits_{n \to \infty} n{\mathbb{E}} \left[ {W_{2}^{2}}(\mu^{n}, \mu ) \right] = \sum\limits_{k = 1}^{\infty} \frac{1}{k(k+d-1)}, $$ where W2 is the quadratic Kantorovich distance with respect to the intrinsic cost $\rho (x, y) = |\arccos (x) - \arccos (y)|$ , (x,y) ∈ [− 1,1]2, associated to the model. When μ is the product of two Jacobi measures with dimensions d and $d^{\prime }$ respectively, then $$ {\mathbb{E}} \left[ {W_{2}^{2}}(\mu^{n}, \mu ) \right] \approx \frac{\log n}{n} . $$ In the particular case $d = d^{\prime } = 1$ (corresponding to the product of arcsine laws), $$ \lim_{n \to \infty} \frac{n}{\log n} {\mathbb{E}} \left[ {W_{2}^{2}}(\mu^{n}, \mu ) \right] = \frac{\pi}{4}. $$ Similar results do hold for non-symmetric Jacobi distributions. The proofs are based on the recent PDE and mass transportation approach developed by L. Ambrosio, F. Stra and D. Trevisan.