We prove here that own price effect is the sum of own substitution effect and income effect for a price change, which is known as Slutsky's theorem. This is shown in the figure given bellow:
At initial prices and money income, budget line is AB and according to the condition of the equilibrium e0 is the initial equilibrium point. The consumer gets U0 level of utility. Suppose at constant income and p2, p1 decreases (say by one unit). Consequently, the intercept of the budget line (M/p2) remains unchanged but absolute slope of the budget line (p1/p2) decreases. The new budget line becomes flatter with the same intercept. It is denoted by AC line. New equilibrium can be achieved at any point on the new budget line AC (and therefore own price effect can take any algebraic sign). Suppose the equilibrium takes place at point e1. Hence, as p1 decreases, for given p2 and M, demand for good I increases from x10 to x11. This is the own price effect for x1 and here it is negative. A part of this change is due to change in real income (since for given p2 and M as p1 decreases, real income increases) and another part is originated at constant real income. To decompose these effects, we reduce money income (M) of the consumer in such a way that real income in terms of utility remains unchanged. After such reduction of M, intercept of the new budget line AC, i.e., (M/p2) decreases with the same slope (p1/p2) for given p1and p2. Hence the new budget line shifts parallely downwards subject to the fact that after the shift, it is tangent to the previous indifference curve. The consumer can attain the same level of utility and the real income remains constant in terms of utility after adjusting money income and utility is also maximised. After adjustment of money income, budget line is A'C' along which real income in terms of utility remains constant after change in p1 for given p2. This budget line is known as compensated budget line. Under such budget line equilibrium will necessarily take place at point e1'. Hence under constant real income in terms of utility, as p1decreases for given p2, x1 increases (from x10 to x11') by substituting x2 (from x10 to x21). This is known as own price substitution effect for x1 which is negative and indifference curve is downward sloping strictly convex to the origin. But as x1 increases from x10 to x1 and real income also increases, the demand for good I increases from x10 to x1' through a rise in real income. This would indicate that by income effect for a price change, x1 is a normal good. Clearly, we have own price effect consists of own substitution effect and income effect for a price change, where own substitution effect in negative but income effect for a price change can take any algebrical sign depending on the good is normal, superior or inferior.