• the Marginal Rate of  Substitution (MRS = MU_y/MU_x) equals the slope of the Budget Line = Δy/Δx;

• the MRS also equals - (/P_x/P_y); and,

•  MU_x/P_x = MU_y/P_y.

2. Changed Assumptions

    In calculating the constrained maximization of utility it was assumed that income (I), the price of x (Px) and the price of y (Py) where constant.  If these assumptions are changed the point of maximization will change. 

If income increases the budget line moves up parallel to the original budget line assuming Px & Py remain fixed (M&Y10th Fig 3.7; M&Y 11th Fig. 2.8; B&Z Fig. 3.14; B&B Fig. 5.2).  If I and Py remains constant but Px decreases then the price ratio changes and the budget line rotates from the initial y-intercept as a larger quantity of x can be purchased (M&Y 10th Fig. 3.8; M&Y 11th Fig. 2.7; B&Z Fig. 3.10; B&B Fig. 5.1).  If Px increases the opposite will occur as a smaller quantity of x can be purchased with fixed I.

A change in I, assuming fixed prices and no change in taste, leads to a straight forward change in the position of the best affordable point, e.g. an increase in I results in a higher maximum point but with identical criteria as the initial tangency, i.e. MRS = -(Px/Py) and MUx/Px = MUy/Py.  A change in Px (or Py), however, results in two distinct changes - an income and a substitute effect - that collectively is called the price effect (M&Y 10th Fig. 4.6; M&Y 11th Fig. 3.8; B&Z Fig. 4.3; B&B Fig. 5.6).

a) Price Effect for Normal Goods

An increase in the price of a given commodity (x) causes the slope of the budget line to increase lowering the level of consumer utility, i.e. a new equilibrium on a lower indifference curve - assuming constant income and constant prices for the other good (y).  As noted above the overall effect is called the 'price effect' (M&Y 10th Fig. 4.6; M&Y 11th Fig. 3.8; B&Z Fig. 4.3; B&B Fig. 5.6), i.e. from U to V or, with respect to the quantity of x consumed, from x₁ to x₂, or with respect to utility from U₁ to U₂.

b) Substitution Effect

If, however, I is increased to maintain the initial level of utility then the quantity of the x consumed will still decrease as the slope of the budget curve increases in response to the price rise, i.e. from U to W or, with respect to the quantity of x consumed from x₁ to x₃. This decrease in consumption due only to the price increase, while maintaining I to stay on the initial indifference curve, is called the substitution effect. It measures how much less of a now more expensive commodity (x) will be consumed simply because of a price increase. It  follows from the basic economic principle of substitution, i.e. consumers will tend to substitute a less expensive for a more expensive good or service.  Notice that the initial conditions for utility maximization changes because the price ratio changes as does the MRS.

c) Income Effect

If, however, I is not increased to maintain the initial level of utility (U₁), then consumption of x will decline further- from x₃ to x₂ (M&Y 10th Fig. 4.6; M&Y 11th Fig. 3.8; B&Z Fig. 4.3; B&B Fig. 5.6), because, in effect, the purchasing power of fixed I is reduced. Put another way, a higher Px reduces the total amount of both x and y that can be purchased shifting the consumer from U₁ to U₂. This is called the income effect.  The income effect thus reflects the change in the price of a commodity upon the demand for that commodity by changing the purchasing power of income available for expenditure.

d) Inferior Goods

    The substitution effect is always negative, that is if the price of a commodity (x) goes up, the quantity consumed goes down. The income effect, however, can be positive or negative. For 'normal' goods, an decrease in income results in decreased consumption - assuming constant prices. If consumption increases when income decreases then the commodity is an 'inferior' good (M&Y 10th  Fig. 4.7; M&Y 11th Fig. 3.9; B&Z Fig. 4.5; B&B Fig. 5.8) and the income effect is negative with consumption increasing from x₃ to x₂. In most cases, if the price of an inferior good decreases consumption will still increase even if income rises.