The labour demand curve is a graph, indicating in a wage/employment diagram how much work (measured in work hours) firms demand at different wage rates. The curve is negatively sloping, meaning that firms want to cut down on employment if work becomes more expensive.
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The labour demand curve is derived from the partial production function (K fixed). Microeconomics teaches that utility-maximizing individuals buy another shirt if the utility derived from the shirt exceeds its price. In the same vein, a firm that maximizes profits hires another hour of work if the value of what will be produced during this hour exceeds the cost.
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We know that the slope of the partial production function (K fixed) measures the marginal product of labour (MPL), that is the output gained by employing one more hour of labour. The partial production function is steep when little labour is employed, it becomes successively flatter as firms employ more labour. Therefore, the MPL is high at low values of L and low at high values of L. Thus the MPL may be represented in a diagram with a marginal product of labour on the vertical axis and work hour on the horizontal axis as a line that falls from left to right.
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In a final step we need to show that this downwards sloping MPL curve is also the labour demand curve. Recall that the marginal product of labour indicates the value of one more work hour to the firm. If the hourly wage is, say, w₁ the MPL remains above this cost as long as less than L₁ work hours are being employed. If employment exceeds L₁, additional work costs more than the revenue it generates for the firm. Hence the profit-maximizing firm demands employment up to L₁, but not beyond.
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The same argument applies at other wage rates. Going to the right from a selected wage rate, the MPL curve always indicates how much labour firms may profitably employ. Hence the marginal product of labour curve is also the labour demand curve.

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Further reading on pp. 140-143.