This paper estimates the cost of the lockdown of some sectors of the world economy in the wake of COVID-19. We develop a multi sector disequilibrium model with buyer-seller relations between agents located in different countries. The production network model allows us to study not only the direct cost of the lockdown but also indirect costs which emerge from the reductions in the availability of intermediate inputs. Agents determine the quantity of output and the proportions in which to combine inputs using prices that emerge from local interactions. The model is calibrated to the world economy using input-output data on 56 industries in 44 countries including all major economies. Within our model, the lockdowns are implemented as partial reductions in the output of some sectors using data on sectoral decomposition of capacity reductions. We use computational experiments to replicate the temporal sequence of the lockdowns implemented in different countries. World output falls by 7% at the early stage of the crisis when only China is under lockdown and by 23% at the peak of the crisis when many countries are under a lockdown. These direct impacts are amplified as the shock propagates through the world economy because of the buyer-seller relations. Supply-chain spillovers are capable of amplifying the direct impact by more than two folds. Naturally, the substitutability between intermediate inputs is a major determinant of the amplification. We also study the process of economic recovery following the end of the lockdowns. Price flexibility and minor technological adaptations help in reducing the time it takes for the economy to recover. The world economy takes about one quarter to move towards the new equilibrium in the optimistic and unlikely scenario of the end of all lockdowns. Recovery time is likely to be significantly greater if partial lockdowns persist.