The isocost line represents all combinations of inputs (typically labor and capital) that result in the same total cost for a firm. Imagine a scenario where a company must decide between employing additional workers or acquiring more machinery, all while adhering to a strict budget. The slope of the isocost line captures the tradeoff between different affordable combinations of inputs.

The slope of the isocost line is mathematically defined as the negative ratio of the wage rate to the rental rate of capital, indicating how one input can be substituted for another.

Example: In a manufacturing setting, if w = $20/hour and r = $40/hour, the slope of the isocost line is -1/2. This means that to keep costs constant, reducing capital by 1 unit allows for an increase of 2 units of labor.

Impact of Input Price Changes:

• If w increases to $40/hour (with r constant), the new slope becomes -1, indicating labor has become relatively more expensive.
• If w decreases to $10/hour (with r constant), the new slope becomes -1/4, making labor relatively cheaper.

Budget Changes:

• An increase in the firm's budget shifts the isocost line outward, parallel to the original line.
• A decrease in the budget shifts the line inward.
• The slope remains unchanged as long as input prices are constant.