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The demand for oil is inelastic in the short term because oil is a necessity, and there's not yet a direct (or perfect) substitute goods.
Generally speaking, the market price is influenced greatly by the the oil supply that OPEC controls. When the OPEC (or decided to) decrease supply, the market faces an increase in oil price. However, because demand for oil is relatively inelastic, the quantity demanded only decreases very slightly - if any.
On the other hand, the demand characteristic for oil is slightly different from normal goods. The Law of Demand states that as the price of a goods increases, the quantity demanded for the goods decrease, vice versa and ceteris paribus. However, with oil, there is a perception that an oil price increase is a result eventual extinction of oil. Hence, countries may increase its oil reserve despite rising oil prices.