Quasi-linear Utility Functions:

At some point we should step back and reflect on what we have been doing here. Is it really correct to add these areas together and use them as indicators of we change in the desirability of implementing a project or not? Or, to assess the cost of a tax with the size of the dead weight loss? Well, we can start by going back to our discussion of the marginal rate of substitution between two goods (1 and 2). Where we said that MRS₁₂ shows the marginal willingness to pay for one additional unit of good 1, in terms of good 2, and we also characterized the consumer’s optimum by the condition that MRS₁₂ = −p₁/p₂. However, in the analysis above there have been no mentioning of the price of good 2, and should we not take into account that the demand for good 2 also will change as the price of good 1 is increased? After all, we’ve learned that it is the relative price that matters! OK! We must admit it. The analysis above is very partial in nature; we simply neglect all these other repercussion effects and assume (implicitly) that all cross-price effects are negligible. However, we could remember that I suggested that good 2 could be interpreted as “all other goods”, or simply “money”. In this case the analysis is all right, at least then it comes to what we are expressing the marginal willingness to pay in terms of (namely money), and it is also OK to add these areas together. Another problem is that we saw in the section about the Slutsky equation that a price increase in general has an income effect. This means that the consumer’s budget is affected by the price increase and he will suffer a decline in utility due to this effect.

There is one form for the utility function which makes all these problems disappear, and that is the so called quasi-linear utility function. This utility function can be written (in our usual two good case) as,

U(Q₁,Q₂) = ν (Q₁) + Q₂.

Note that this function is linear in Q₂, but the sub-utility function ν (Q₁) may be nonlinear, so the whole utility function is not quite linear (but quasi- linear). If we, for example, assume that ν (Q₁) = lnQ₁, the graph of this function is concave, i.e., its slope becomes flatter and flatter as Q₁ increases. This is the same as to say that the marginal utility of good 1, ν′ (Q₁), is diminishing. Note that in the ordinal utility approach this is not a necessary assumption, in fact the marginal utilities of both goods can be increasing, as long as their ratio, the MRS, is diminishing. For the quasi-linear utility function the MRS will be constant for each level of Q₂, as Q₁ is increased. What this means is that if we shift the budget lines in a parallel fashion (as we do when we illustrate the income effect), the consumption of good 2 will be unaffected. Hence, by using the quasi-linear utility function we’ve effectively eliminated the income effect from consideration, what this means is that the total effect on quantity demanded of a price change of good 1 can be attributed completely to the substitution effect. It is in fact only in this case that the usual welfare analysis, by using the consumers’ surplus measure, is valid, and that is not very encouraging since we have to admit that the quasi-linear utility function is probably only applicable for very insignificant goods (like paper-clips), and for cost-benefit analysis, therefore uninteresting.

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