The first comparative static exercise is to derive so called income effects, i.e., how does the optimal bundle change as income changes, ceteris paribus? An increase in income is illustrated in figure shown below as a parallel outward shift of the budget constraint. The optimal quantities of both goods increase in this case, which means that both goods are so called normal goods. The dotted line, emanating from the origin, is called an income − offer curve, an income− expansion path, or, sometimes, an Engel curve, and shows how the optimal bundle changes as income changes. In the Cobb-Douglas case all these curves would be linear, which is easiest to see in the case of the Engel curve: Remember that the demand function for good 1 could be written as,
Q1 (Y, p1) = (a . Y)/p1,
Or, if we define α = a/p1, since p1 should be held constant all the time anyway,
Q1 (Y) = α . Y
Hence the Engel curve is linear with a slope equal to α, in this case. Again, the Cobb-Douglas is a special case where both goods must be normal goods.
Figure below shows an example where Q1 is a normal good for all income levels considered, but Q2 is first a normal good, but eventually becomes an inferior good (between Y2 and Y3).
The last figure shows that the income effect will normally vary along the income-offer curve, but we may be satisfied by getting a local estimate of the size and direction of the income effect. By this, it means a number which shows how the demand for each good will vary as we increase income by a small amount. To get such a measure we define the income elasticity of good 1 and 2 around the point (Q01, Q02, Y0) as,
Assume that Y0 = 10, and p01 = 1, if we assume Cobb-Douglas preferences (U(Q) = Q1 . Q2) the initial quantity demanded of good 1 is equal to Q01 = 5. If income increases to Y1 = 11 (ΔY0=1), the new quantity demanded is equal to Q01 = 5.5. Hence, we see that in this case the percentage increase in quantity demanded of good 1 is equal to the increase in income (+10%). The income elasticity is thus equal to η1 = 1. This is true for the second good as well. Remember that in our discussion of the Cobb-Douglas utility function, we could reduce the budget shares from the exponents of the utility function. Since these are constant, expressing tastes, they will not change as income and/or prices change. In order to keep the budget shares of all goods constant as income increases, the quantity demanded for all goods must increase to the same extent as income, i.e., all goods included in the Cobb-Douglas utility function have income elasticity equal to 1, and are thus normal goods.
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