What can cause an increase in equilibrium price and an increase in equilibrium supply?

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I’m going to assume you meant to say “equilibrium quantity” because supply is a curve that dictates equilibrium. The changes in supply and demand will disrupt and shift equilibrium price and quantity, but there is no “equilibrium supply” to increase.

When supply goes down, the equilibrium will change. Price will go up because of scarcity of a good. However, quantity will go down, because with less available, there is a lower quantity purchased. Conversely, if supply increases, price will decrease, but supply will increase.

If demand decreases, the price and quantity will both decrease. If both price and quantity are to increase, there would need to be an increase in demand. This can be driven by better marketing, a change in the market itself, a change in the technology used, or something else—it is a very dynamic construct. However, the underlying economic notion is simple: if you want quantity and price to increase, demand must increase.

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It seems to me that there must be some mistake in this question.  The reason for this is that there is no such thing as “equilibrium supply.”  Supply is a curve, as is demand.  At the point where the supply and demand curves intersect, we have equilibrium.  At that point, there is an equilibrium price and an equilibrium quantity that is both supplied and demanded.  However, there is no equilibrium supply.  So, I believe that this question must actually be concerning what could cause both the equilibrium price and the equilibrium quantity to rise.  In that case, the answer would be that only an increase in demand can cause such a combination of effects.

When supply rises, the equilibrium quantity rises, but the equilibrium price drops.  When supply declines, the equilibrium price rises, but the equilibrium quantity drops.  When demand declines, both equilibrium price and equilibrium quantity drop as well.  However, when demand rises, equilibrium quantity goes up (because people are willing and able to buy more things) and equilibrium prices rise (because it costs more to make more things and because people who demand more are willing to pay more). 

In order to have both price and quantity rise, we need to have an increase in demand.

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