Hi D.,
I have 3 beautiful children ages 21, 20, and 16, so I've been down the "allowance" vs "chores" road, so here is my humble opinion.
An allowance is the money you ALLOW your children to have. An allowance is not earned money, as for a chore or job, and should not be given as a result of work performed, nor withheld or docked for work not performed. It is a tool you give them to help them learn to budget and save their money, and to make good decisions about money. It helps them understand that money is a limited resource, and we all have to make choices to live within those limits.
Deciding whether or not to give your child an allowance is up to you, and how much is also up to you and your family's available income and situation. My opinion would be somewhere around $10 a week for a 13yo, and somewhere around $7 or $8 for a 9 year old.
If you also require chores to be done, then they should be done simply because you are all members of a family, and family members are required to help each other out in order to meet the family goals. I explain to my children that I don't get paid for running the dishwasher, doing the laundry or cleaning the bathroom, and neither is any pay taken from me if I don't. The consequences of not doing my laundry is simply that I wear dirty clothes, which isn't a choice I want to make, so I do the laundry.
If my children ever wanted more money than they could save with their allowance (such as to buy Christmas presents or etc.), then that money needed to be earned. They could not earn them with their regular chores, but as you know, there is always work that needs to be done above and beyond the regular chores. I would allow them to clean out my car, a closet, or do something extra such as that. I would work out the compensation in advance. I usually paid by the job instead of by the hour, because working by the hour would give them the incentive to work slower, and by the job gives the incentive to finish one job in order to to another.
Lastly, I opened a savings account in their name at the bank. I required that a portion of their allowance go into the savings account each time they got paid. I required 10% go to savings, but my children usually chose to put more like 50% or more in savings. My name was also on the account, so they could not withdraw the money unless they had my permission. In this way, even my spender was able to see the value of saving her money, after she saw her savings account grow to over $300, she was thrilled and began to save even more of her money.
This system worked for us. I hope it helps,
L.