Under perfect competition, the wage rate equals the value of the marginal product.

Perfect competition is a market structure characterized by a large number of buyers and sellers, homogeneous products with no differentiation, free entry and exit of firms, and perfect information. In such a market structure, no single firm can influence the market price, and all firms are price takers.

The wage rate, usually denoted (w), is the price paid per unit of labor, usually expressed as an hourly or daily rate. The value of the marginal product (VMP) is the additional revenue that a firm can generate by employing one more unit of input, in this case labor, keeping other inputs constant. It is calculated by multiplying the marginal product of labor (MPL) by the market price (P) of the output. Mathematically, VMP = P x MPL

In a perfectly competitive labor market, the equilibrium wage rate is determined by the interaction of demand and supply factors. On the demand side, firms will hire labor until the cost of employing an additional unit of labor (wage rate) equals the value of the additional output (VMP) produced by that unit of labor. On the supply side, workers will supply their labor at a wage rate equal to their marginal cost of labor (which can be interpreted as the opportunity cost of not working). Under perfect competition, both labor demand and labor supply will be perfectly elastic, leading to an equilibrium wage rate that equals the VMP.

To show that the wage rate equals the value of the marginal product under perfect competition, consider the profit-maximizing behavior of a firm hiring labor. A firm hires labor until the wage rate equals the VMP, as shown by the following condition: w = P x MPL This condition implies that the wage rate equals the additional revenue that a firm can generate by employing one more unit of labor, keeping other inputs constant. Under perfect competition, this condition will hold true for all firms in the labor market. Thus, the equilibrium wage rate will be determined by the interaction of labor demand and supply factors. At this point, both firms and workers act as price takers and accept the market-determined wage rate that equals the value of the marginal product.

Under perfect competition, the wage rate equals the value of the marginal product. This outcome is due to the interaction of labor demand and supply factors in a perfectly competitive labor market, where firms maximize profits by hiring labor until the cost of employing an additional unit of labor (wage rate) equals the value of the additional output (VMP) produced by that unit of labor.