1The graph was created using the utility function U(x) = x11/3x22/3. The budget line d-e isgiven by 9x1 + 18x2 = 135. The demand functions that maximize utility are D1(I,P1,P2) = I/3P1and D2(I,P1,P2) = 2I/3P2. The indirect utility function is V(I,P1,P2) = I(3P1)-1/3(3P2/2)-2/3.10 5 10 15 20 good 20 5 10 15 20 25 good 1aedConsumer Theory: The Mathematical Core Dan McFadden, 100ASuppose an individual has a utility function U(x) which is a function of non-negativecommodity vectors x = (x1,x2), and seeks to maximize this utility function subject to the budgetconstraint P1x1 + P2x2 # I, where I is income and P = (P1,P2) is the vector of commodity prices.Define u = V(I,P) to be the value of utility attained by solving this problem; V is called the indirectutility function. Let x = D(I,P) = (D1(I,P),D2(I,P)) be the commodity vector that achieves the utilitymaximum subject to this budget constraint. The functions D(I,P) are called this consumer’s marketdemand functions. Since these functions maximize utility subject to the budget constraint, V(I,P)/ U(D(I,P)) / U(D1(I,P),D2(I,P)). The figure below shows the budget line d-e, and the point a thatmaximizes utility.1 Note that the point a is on the highest indifference (constant utility) curve thattouches the budget line, and that at a the indifference curve is tangent to the budget line, so that itsslope, the marginal rate of substitution (MRS) is equal to the slope of the budget line, -P1/P2. Note that the indirect utility function and the market demand functions all depend on income and onthe price vector. For example, the demand for good 1, written out, is x1 = D1(I,P1,P2). When demandfor good 1 is graphed against its own price P1, then changes in own price P1 correspond to movementalong the graphed demand curve, while changes in income I or in the cross-price P2 correspond toshifts in the graphed demand curve. If all income is spent, then P1D1(I,P) + p2D2(I,P) / I. If incomeand all prices are scaled by a constant, the budget set and the maximum attainable utility remain the2same. This implies that V(I,P) and D(I,P) are unchanged by rescaling of income and prices.Functions with this property are said to be homogeneous of degree zero in income and prices.The consumer is said to be locally non-satiated if near any commodity vector there is alwaysanother that has strictly higher utility. When the consumer is locally non-satiated, the utility-maximizing consumer will spend all income, no commodity vector that costs less can give themaximum utility level, and the indirect utility function V(I,P) is strictly increasing in I. Local non-satiation rules out “fat” indifference curves. Given utility level u, let I = M(u,P) be the minimum income needed to buy a commodityvector that gives utility u; M is called the expenditure function. Let x = H(u,P) = (H1(u,P),H2(u,P))be the commodity vector that achieves the minimum expenditure subject to the constraint that utilitylevel u be attained. Then, M(u,P) / P"H(u,P) / P1H1(u,P) + P2H2(u,P). The demand functionsH(u,P) are called the Hicksian, constant utility, or compensated demand functions. If all prices arerescaled by a constant, then the commodity vector solving the minimum income problem isunchanged, so that H(u,P) is homogeneous of degree zero in P. The expenditure is scaled up by thesame constant as the prices, and is said to be linear homogeneous in P. If in the figure above, insteadof fixing the budget line and locating the point a where the highest indifference curve touches thisline, we could have fixed the indifference curve and varied the level of income for given prices tofind the minimum income necessary to touch this indifference curve. This minimum again occursat a. Then, solving the utility maximization problem for a fixed I and attaining a maximum utilitylevel u, and solving the expenditure minimization problem for this u and attaining the income I, leadto the same point a. The mathematical condition for this graphical result to hold is that the consumerbe locally non-satiated, so that all income is spent. In this case, if the consumer utility level is u =V(I,P), then I is the minimum income at which utility level u can be achieved. Then, the result thatthe utility maximization problem and the expenditure minimization problem pick out the same pointa implies that I / M(V(I,P),P) and u / V(M(u,P),P); i.e., V and M are inverses of each other forfixed P. Further, D(I,P) / H(V(I,P),P) and D(M(u,P),P) / H(u,P).The demand for a commodity is said to be normal if demand does not fall when income rises,and inferior if demand falls when income rises. A commodity is a luxury good if its budget sharerises when income rises, and is otherwise a necessary good. The budget share of the first commodityis s1 = P1D1(I,P1,P2)/I. Define the income elasticity of demand,η = (I/D1(I,P1,P2))@MD1(I,P1,P2)/MI.This is the percentage by which demand for good 1 increases when income goes up by one percent.Show as an exercise that the income elasticity of the budget share satisfies(I/ s1)@Ms1/MI = η - 1.30 0.1 0.2 0.3 Budget Share of Good 10 100 200 300 IncomeEngle CurvesRegressiveLuxuryNecessaryThe income elasticitiesof demand at I = 250 are-0.33 for Regressive,0.8 for Necessary,1.2 for Luxury.0 1 2 3 4 5 Good 20 0.5 1 1.5 2 2.5 3 Good 1Locus of Demand PointsIncome-ExpansionPathThen, a luxury good has an income elasticity of its budget share greater than zero, a necessary goodhas an income elasticity of the budget share less than zero, and an inferior good has an incomeelasticity of the budget share less than -1. A graph of the demand for a good against income is calledan Engle curve. The figure below shows the Engle curves for three cases. It is possible to trace out the locus of demand points in an indifference curve map as income changeswith prices fixed; this locus is called an income-offer curve or income-expansion path. Points on anincome-expansion path correspond to points on Engle curves for each of the commodities. Thefigure below shows an income-expansion path when good 1 is a necessary good.40 1 2 3 4 5 Good 20 0.5 1 1.5 2 2.5 3 Good 1Locus of Demand PointsOffer Curve0 2 4 Price of Good 10 0.5 1 1.5 2 2.5 3 Quantity of Good 1Demand FunctionIn an indifference curve map, it is also possible to trace out the locus of demand points as theprice of a good changes; this
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