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The consumption function is a mathematical relationship between consumption (C) and income (Y). It is represented graphically as the slope of the consumption curve. The consumption function is generally written as C = a + bY, where 'a' is autonomous consumption (consumption when income is zero) and 'b' is the marginal propensity to consume. This relationship helps us understand how changes in income affect the level of consumption in an economy.

Mathematically, the MPC is the derivative of the consumption function with respect to income (Y). It is the rate at which the consumption changes with respect to a small change in income. Taking the derivative of the consumption function C = a + bY with respect to Y, we get: \[MPC = \frac{dC}{dY} = b \] The MPC is constant and equal to the coefficient 'b' in the linear consumption function.

The geometric meaning of the MPC can be understood in the context of the graph of the consumption function. In a graph where the horizontal axis represents income (Y) and the vertical axis represents consumption (C), the consumption function appears as a straight line with a slope equal to the MPC. Therefore, the MPC is the slope of the consumption function. In economic terms, the geometric meaning of the MPC represents the extent to which consumption changes when there is a change in income. A higher MPC indicates a steeper consumption function, which means that a higher proportion of additional income is spent on consumption. Conversely, a lower MPC indicates a flatter consumption function, which means that a smaller proportion of additional income is spent on consumption. So, the geometric meaning of the marginal propensity to consume is the slope of the consumption function, representing how consumption changes as income changes.