In my post "more on the law of demand, buy why?", I outlined the difficulty of explaining the law of demand by not using the substitution effect and income effect. In particular, some people may think that the demand curve is downward sloping simply because the marginal benefit or marginal utility (MU) of a good is diminishing with the quantity consumed of the good. In that past post, I said that MU is not price or money value while the vertical axis of the demand curve diagram is price or money value. Hence, one can't jump to conclude that diminishing MU implies downward sloping demand curve. We need something more than that and the extra factors involve (eventually also) substitution and income effects. To relate MU to demand curve is not straightforward, actually quite complicated. In that post, I stop after stating these and urge my readers to simply use substitution and income effects to explain the law of demand.
In this post, I want to fill this gap, showing you the "complicated" explanation that relates MU to demand curve and the substitution-income effect. I write this post because readers may be curious after reading my past post, wondering how all these can be related. My presentation below borrows heavily from a discussion in the Chapter II of John Hicks' A Revision of Demand Theory. In fact, only after I read this chapter, I learn how to explain the relation between MU and substitution-income effect.
First, demand curve is a price-quantity relation while MU is a (marginal) utility-quantity relation. To relate the two curves, we need to convert utility into price and so an equation below:
The price P one is willing to pay for a unit of a good = (MU of the good)/(MU of money).
The left side is the price involved in the demand curve of a good. The right side is related to the quantity of the good. If the right side is decreasing with quantity, we have a downward sloping demand curve. That's what we need to demonstrate.
Why dividing MU of a good by the MU of money? The unity of MU (of a good) is simply utility, it is not money or price. So, you need to divide it by MU of money to get a price. For example, suppose that MU of a good is 6 units (consuming one more unit of a good gives you 6 extra units of utility), and that MU of money is 2 units (giving you one more dollar, your utility increases by 2 units). So, you are willing pay $3 for getting one more unit of the good (you get 6 units of utility, which is what $3 can generate).
Now, although MU of the good is diminishing with the quantity of this good, the MU ratio of the good and of the money need not be diminishing with the quantity of this good. We can't jump directly from MU to price. Viewed from another perspective, if the price of a good falls, the left side of the equation is lower. To maintain the equality, the right side should also fall. But does it necessarily imply a higher quantity of the good? If MU of money is unchanged but MU of the good is diminishing with its quantity, the answer is yes. But is MU of money unchanged when the price of a good falls? Furthermore, there are extra complications involved as explained below.
Revisit the equation above. This is for only one good. In fact, a consumer buys more than one good. For each of these goods bought, we have an equation as above for the good. Suppose there are only three goods, X, Y and Z. We also use Px, Py, MUx, MUy, etc to denote the price of the respective good and MU of the respective good. We use MUM to stand for MU of money. Then, for X, Y, and Z, we have the equation above to be expressed respectively as
Px = MUx/MUM,
Py = MUy/MUM,
Pz = MUz/MUM.
What interests us is the case where the price of one good changes, other things being equal. Let's consider that Px falls but Py and Pz are unchanged. Further, income is fixed. Given the price change, a person will change its quantity demanded to maintain the equalities above. If not, for example, Px < MUx/MUM, the person would find buying more good X brings about a benefit valued in money term higher than the price of X, he or she should buy more X (until the equality holds again).
In fact, if Px falls to Px' but all quantities in X, Y and Z are unchanged (such that the right sides are unchanged), exactly Px' < MUx/MUM will happen. As said, the person will be induced to buy more X. Since MUx is diminishing with the quantity of X (denoted as Qx), MUx/MUM should decline, given the same MUM, and the equality will then be resumed. This is exactly the naïve explanation of the law of demand (via only MUx). But this ignores the MUM. This assumes MUM is unchanged.
Is MUM constant when Px falls? Even if this is assumed, there is still certain further effect that cannot be ignored.
First, when one buy more X, MUy or MUz may change. For instance, if X is coffee powder and Y is tea bags, one may find MUy lower when more coffees are consumed. If Z is milk, one may find MUz higher when more powders are bought. If MUy and MUz change as described, given unchanged Py and Pz, at the same quantity of Y and Z, MUy/MUM < Py and MUz/MUM > Pz if MUM is unchanged. Adjustment in Y and Z should take place. The quantity of Y (Qy) should decrease and so MUy increases and the quantity of Z (Qz) should increase and so MUz decreases. Yet if Qy and Qz change, MUx may also change again and further adjustments take place.
Further, don't forget these changes in quantities can't involve a total expenditure higher than the income. The quantity changes are constrained by no change in the total expenditure (given fixed income). The quantity changes that can keep the total expenditure unchanged are unlikely to be the same as the quantities that can restore the equalities between prices and MU-ratios for the three goods if MUM keeps unchanged.
Hence, for most cases, it is only reasonable to assume that MUM will change when Px changes. Well, if MUM changes, all the three equations above will be affected. It is not that only MUx matters (via diminishing MUx) but MUM also matters (as it will change) to the slope of the demand curve.
Although the whole process involves changes in MUx and MUM, we may still divide it into two steps.
In the first step, when Px falls, we analyze as if MUM is fixed. Though MUM generally will change when the price and the quantities consumed of goods change, we can keep it fix by assuming that there is a hypothetical reduction in income such that, the price and income change combined generate an unchanged MUM. If MUM is fixed, we can concentrate on how MUx (together with MUy and MUz) changes is associated with the Qx change. The effect of Px on Qx in this step is indeed about substitution effect: the price of X triggers rearrangement of quantities in X,Y,Z while the income is varied to keep MUM fixed.
In the second step, the hypothetical reduction income to keep MUM fixed as Px falls is (hypothetically) returned. Then, there is an effect on the quantities from this income rise and MUM change. This is about the income effect indeed.
Now, we return to the familiar substitution effect and income effect though we present them in a rather different (and more complicated) manner so as to relate them to MU of goods and MUM. Perhaps, as a regular reader of hi,economics, you are interested in this (complicated) relation. But the point I want to make is: Sometimes we are tempted to use simple explanation. Using MU to explain the law of demand is exactly one such occasion. However, opting for simplicity may sometimes generate more confusions. What looks to be a simple explanation may be just a naive explanation.
Hence, as concluded earlier, for explaining the law of demand, I think one should use substitution and income effects, instead of using diminishing MU. Don't think the latter is an easier explanation.
