In the short run, why do higher interest rate lower inflation? A common interpretation is "Higher interest rates put less borrowing power in the hands of consumers and businesses. And when they spend less, firms are not selling everything and prices naturally falls." In the perspective of AS-AD model, this is caused by a downward shift of the demand curve. 

However, I suppose it will cause a supply curve shift as well. Today, most production is funded by short-term borrowing and relied on people paying them for the goods and then pay back the loans. So when the interest rate goes up, the production costs go up, and hence causing a negative supply shock as well. 

But reality seems to confirm the fact that higher interest rate lower inflation. Why is that? Is that because the supply shock is not as significant as the demand shock? If so, why is that?
In the short run, why do higher interest rate lower inflation?


As you say:


  "Higher interest rates put less borrowing power in the hands of consumers and businesses. And when they spend less, firms are not selling everything and prices naturally falls."


There's also the effect on the velocity of money and the money supply that less borrowing brings to provide further downward pressure on prices.

I also think your concerns about supply falling in the short term are unfounded. Firms are slow to adjust their production/supply due to greater borrowing costs. Instead, they will attempt to keep up production and clear their inventories by lowering prices in the hopes that the next season will show them a success and bail them out. Any such effect from interest rates would be a much longer-run effect.