We have examined the formation process of methanol by the reduction of formaldehyde under hydrothermal conditions. A formaldehyde absorbs a hydrogen molecule and turns to a methanol. Water molecules near a formaldehyde help to transfer protons to proceed the reduction process efficiently. The energy barrier for the reduction of a formaldehyde becomes 32.8 kcal/mol when a water cluster with five water molecules catalyzes the reduction. The ionic product becomes the largest under hydrothermal conditions. We introduce the acid-base catalytic effect due to hydronium and hydroxide on the reduction of formaldehyde. The energy barriers for the reduction of a formaldehyde are further reduced to 29.3 and 10.4 kcal/mol by the acid and base catalytic effects, respectively. The reduction of a formaldehyde is more effectively catalyzed by a hydroxide than a hydronium. The acid-base catalytic effect is not available at the high temperature of supercritical water due to the sudden decrease of the ionic product. It takes too long to form a reactant compound in supercritical water. The transition state theory is applied to calculate the reduction rate of a reactant compound, considering the tunneling effect of a proton. We confirmed that a metastable equilibrium state attains among single-carbon compounds except methanol and methane by reproducing the concentrations of the carbon compounds measured in a laboratory. We calculated the formation rate of methanol using the equilibrium concentration of formaldehyde. We compared the calculated formation rate with that determined by a laboratory experiment and confirmed that the present theoretical calculation is accurately able to describe the oxidative and reductive reaction network of single-carbon compounds under hydrothermal conditions. The present study can be applied to examine a reaction network of single-carbon compounds in hydrothermal vents on the Earth, Enceladus, and other solar system bodies such as Europa.