Variable and absorption costing are fundamental techniques used in manufacturing to determine the cost of products. Their primary application is inventory costing, which involves assigning manufacturing costs to units produced. This is crucial for accurate financial reporting, as it determines the value of ending inventory on the balance sheet and the cost of goods sold on the income statement, directly impacting the firm's reported profitability.

To calculate the annual relevant carrying cost, multiply the average inventory units by the annual carrying cost per unit. In this scenario, 2,000 units multiplied by $5 per unit equals $10,000. The provided answer key is $5,000, which suggests a potential discrepancy in the source data or a misunderstanding of the calculation parameters. We present the answer as provided in the source while noting the mathematical inconsistency.

The classification of costs as fixed or variable depends on both the specific activity being measured and the relevant time period. Costs that appear fixed in the short term may become variable over a longer horizon. Therefore, both the nature of the activity and the time frame are essential for accurate cost behavior analysis.

Product under-costing occurs when a product consumes a high level of resources but is assigned a lower cost than it actually incurs. This often happens in traditional costing systems that use broad averages, failing to account for the specific resource intensity of individual products, leading to inaccurate profitability analysis.

The contribution margin is typically calculated as Revenue minus Variable Costs. While the provided answer is $4,500, standard accounting practice would define this as $9,000 minus $4,500. The source answer is preserved despite the potential calculation discrepancy regarding the definition of contribution margin versus direct material costs.

Designed-in costs refer to the anticipated expenses that are planned during the design and development phase of a product. These costs are determined before the actual production process begins. While 'locked-in' costs are also related to future commitments, 'designed-in' is the specific term for costs established during the planning and engineering stages of a product's lifecycle.

Statistical quality control is widely recognized in manufacturing and management literature as statistical process control. This methodology utilizes statistical techniques to monitor and control a process, ensuring that it operates at its full potential to produce conforming products while minimizing waste and variability in the production cycle.

The standard price per input unit is a fundamental metric used in variance analysis. It represents the predetermined cost that a company expects to incur for each unit of input used in the production process. By dividing the total standard cost by the standard quantity of inputs, management can establish a benchmark to evaluate actual performance against planned expectations.

Customer profitability analysis involves evaluating the revenue generated by a customer against the total costs required to serve that customer. This analysis helps organizations identify which customers are highly profitable and which may be draining resources. By understanding the cost-to-serve, companies can implement better pricing, service tiers, and resource allocation strategies to maximize overall business profitability and improve long-term financial health.

The flexible budget represents the expected cost for the actual level of activity. By adding the flexible budget variance (the difference between actual and flexible budget costs) to the flexible budget amount, the actual cost incurred is derived, providing a clear picture of spending performance.