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I was studying the decomposition of price effect into substitution and income effects. I am finding it a bit complicated.

This is what I have understood:

(0) Let us assume that there are two commodities. We have a budget line and an indifference map. The budget line is tangent to one of the indifference curves. This point of tangency is the initial equilibrium point.

(1) Due to the change in price of a commodity, the budget line rotates to a new position and we obtain a new equilibrium point. This change in position of the equilibrium point is known as the price effect.

(2) Due to price change, the relative price of the commodity as well as the real income of the consumer changes. Due to change in relative price of the commodity, the consumer will substitute the relatively less expensive commodity for the relatively more expensive commodity and we get a new equilibrium point. This change in equilibrium point is the substitution effect. Due to change in real income, the consumer will change his consumption of the two commodities depending on whether they are normal or inferior commodities and we get a new equilibrium point. This change in equilibrium point is the income effect. Hence we can say that the price effect is the resultant of the substitution effect and the income effect.

(3) To segregate the two effects we draw a hypothetical budget line by shifting the rotated budget line to compensate for the change in real income and the magnitude of the shift varies according to the approach of compensation followed (Hick's approach or Slutsky's approach).

(4) The budget line thus obtained as a result of rotation and shift starting from the original budget line shows the substitution effect.

(5) The remaining part shows the income effect.

(6) Thus we get the substitution effect by rotating and shifting the budget line and then the price effect can be viewed as the combination of substitution and income effects.

Have it grasped it correctly? I am finding all this a little twisted and kind of a reversed logic. First we rotate the budget line to get the price effect and then shift the budget line to obtain the substitution effect or segregate the substitution and income effects. Finally to get to the price effect, we first get to the substitution effect and then the income effect.

I do not understand the rationale behind the decomposition of price effect. Why are we decomposing the effect of a change in price of one commodity into all these effects which are (as I have understood), all hypothetical. Is there any particular advantage of this decomposition? Also I don't exactly understand the meaning of real income. Does it mean the quantity that we can buy given the income and price? If someone could explain it with an example I would be very grateful. Also in substitution effect, why do we substitute the relatively less expensive commodity for the relatively more expensive commodity? Why not the reverse? Also what is the significance of the compensated demand curve? Why do we need two demand curves.

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Why are we decomposing the effect of a change in price of one commodity into all these effects which are (as I have understood), all hypothetical.

They're hypothetical, but they are nevertheless meaningful. We can easily imagine what it would be like if an agent had a larger budget while prices remain unchanged; we can also imagine what it would be like if, under diminishing marginal utility assumptions, the relative prices of goods were to change.

One reason we do this is that it helps develop intuition and show why naive assumptions don't necessarily hold. For example, if we have goods $x$ and $y$, prices $p_x$ and $p_y$ with budget $B$ in an initial equilibrium, we might naively think that setting $p_x^{'} = 0.5p_x$ means that we consume twice as much of $x$, when this is not the case for most utility functions.

Also in substitution effect, why do we substitute the relatively less expensive commodity for the relatively more expensive commodity? Is there any particular advantage of this decomposition?

The answer to the second question is that the Slutsky decomposition allows us to tell a story that answers the first question - when the price of $x$ falls, we may buy more of $x$ (an income effect), but we will also buy more of $y$ because diminishing marginal utilities implies that after using the price-savings to buy some number of additional units of $x$, we get more "bang for the buck" by purchasing an additional unit of $y$ instead. This is the substitution effect.

Also I don't exactly understand the meaning of real income.

Real income is income adjusted for inflation from a base year, so that it reflects real purchasing power. In the context of these Econ 101 budget-line models, it's implied that the budget is in real units for pedagogical reasons. There's no base year, but the point is to ensure that any price-changes contemplated are "real", and to get you to think of consumption as the basic economic unit, rather than money per se. Often you'll see simplifications where one of the goods is priced at $1/unit, and referred to as the "numeraire".

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