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suppose when equilibrium has been achieved between supply and demand then suddenly internation oil rates decr. willsupply curve shift downwards

OCW Scholar - Principles of Microeconomics
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From what I can tell, if rates change, either supply or demand have changed. if demand has gone down, causing rates to drop, then the supply curve will stay the same, but less good will be produced.
Thumb rule : there is a shift in curve(demand or supply) only when the factors affecting the curve that are not represented in any of the axises(X and Y) change whereas three will be movement along the curve if the factors affecting the curve are there in the graph in either of the axises. Now when price changes there will be movement along the curve and not the shift in the curve ,because price being a factor represented along Y axis)
To answer precisely, we need clarity in the question: For what good is supply and demand in equilibrium? When you say "rate of oil decreases" do you mean the price of oil is falling, or the rate of production is falling? I assume you mean some good, like tennis shoes, for which oil is a factor of production (or at least a cost transportation). I also assume you mean the price of oil is falling. So what does that mean to the supply curve for tennis shoes? The easiest way to think about how the supply curve might shift in response to a change in a commodity price to produce a good is to put your finger on a point on the curve and ask yourself, "Given this starting quantity and price of production, will I as a producer be able to produce more or less stuff at this same price if the cost changes? If the cost of oil goes down, I can probably ship more tennis shoes at the old price, which moves my finger to the right at my starting price, or for any other price for that matter. The supply curve shifts to the right (or up) indicating an INCREASE IN SUPPLY due to the DROP IN COST of a FACTOR OF PRODUCTION.

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Other answers:

Shift in curve is not as same as movement along the curve. When curve shifts , one entirely gets a new curve placed differently in the graph, whereas movement happens with the placement of the curve unchanged but position. So one always needs to preserve the distinction between the two.
yes definitely. since oil is used in almost every process of production and distribution, it act as an input for every economic activity. thus when oil price decline, the input cost also declines and quantity supplied increases inth economy. so supply curve will shift upward. hope this will help
Since oil is one of the input factors for most firms. decrease in the price of the oil causes decrease in the cost of production.Producers will be ready to sell more product at that price per unit. This causes downward movement of supply curve.

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