To determine what the price needs to be in order to sell at least 1600 hats next year, we can use the concept of elasticity of demand. Elasticity of demand is a measure of how much the quantity demanded changes in response to a change in price. If demand is elastic, then a small change in price will result in a large change in quantity demanded. If demand is inelastic, then a change in price will result in a small change in quantity demanded.
To calculate elasticity of demand, we can use the formula:
Elasticity of demand = % change in quantity demanded / % change in price
Let's use the data provided to calculate the elasticity of demand for the hat shop:
% change in quantity demanded = (2000 - 1500) / 1500 = 0.333 or 33.3%
% change in price = (12 - 10) / 12 = 0.167 or 16.7%
Elasticity of demand = 33.3% / 16.7% = 2
This means that demand for hats at the shop is elastic, and a decrease in price will result in a larger increase in quantity demanded.
Now we can use the following formula to determine the new price needed to sell at least 1600 hats:
New price = Old price x (New quantity demanded / Old quantity demanded) ^ (-1 / Elasticity of demand)
Plugging in the values we have:
New price = 10 x (1600 / 2000) ^ (-1 / 2) = 9.05 dollars
Therefore, the hat shop owner would need to sell hats at 9.05 dollars in order to sell at least 1600 hats next year, assuming demand remains elastic.