An economist hired by a company that makes a popular line of indoor grills predicts that consumers in one market will buy $x$ units per week if the price is $p=-0.12 x+62$ dollars. The profit is given by the equation
\[
P=-0.16 x^{2}+45 x-557
\]

Step 1 ：Given the profit function \(P=-0.16 x^{2}+45 x-557\) and the price function \(p=-0.12 x+62\), we are to find the number of units and the price at which maximum profit is achieved.

Step 2 ：To find the maximum profit, we need to find the derivative of the profit function and set it equal to zero. This will give us the value of x (number of units) at which the profit is maximized.

Step 3 ：The derivative of the profit function is \(P' = 45 - 0.32x\). Setting this equal to zero, we find that \(x = 140.625\).

Step 4 ：Substituting this value of x into the price function, we find that the price at which maximum profit is achieved is \(p = 62 - 0.12*140.625 = 45.125\).

Step 5 ：Final Answer: The maximum profit is achieved when 140.625 units are sold at a price of \$45.125 per unit. So, \(\boxed{140.625, 45.125}\).