The embodied carbon emissions includes the direct and indirect carbon emitted for the entire production process. These values are needed to estimate impacts on carbon emissions due to changes in consumption patterns. We have developed a model of the economic system to account for every production process. This model illustrates interrelations among production processes by incorporating sectors that produce multiple products. At each multiple product-producing sector, inputs differ in accordance with the material content of the output product. The model is consistent with available input-output coefficients and the physical law of material flow in each process. By exploring our model, we derived carbon embodiment functions that evaluate the embodied carbon emissions of an arbitrary product in terms of material content. Empirical analysis has been fulfilled using the available 405-sector input-output table of Japan.