Consider a consumer whose preferences are defined over bundles of non-negative consumption quantities for each of two commodities: Widgets anda composite consumption commodity known as “All Other Goods”. Thisconsumer’s preferences are both rational and locally non-satiated. Denote2the price of Widgets by p1, the consumption of Widgets by q1, the price of“All Other Goods” by p2, the consumption of “All Other Goods” by q2, andthe consumer’s income by y. Use the composite consumption commodity(that is, “All Other Goods”) as the numeraire throughout this question. (Inother words, assume that p2 = $1 throughout this question. Note that thisassumption allows you to interpret the amount of “All Other Goods” con-sumption as expenditure on “All Other Goods”, because q2 = 1q2 = p2q2.)Suppose that this consumer’s uncompensated demand function for Wid-gets is given byq1 (p1, y; p2 = 1) = (0.02) y − (2) p1.(Uncompensated demands are also known as Marshallian demands, or Wal-rasian demands, or ordinary demands.) Initially, this consumer has an in-come of $6, 500 and faces a Widget price of $50 per Widget. Following aneconomic shock, the Widget price rises to $60 per widget. This economicshock has no impact whatsoever on either this consumer’s income or the“All Other Goods” price.1. Illustrate the total impact of the economic shock on this consumer’soptimal consumption decision using an “indifference curve / budgetline” diagram. (Assume that the consumer’s indifference curves havethe conventional “downward sloping and bowed-in towards the ori-gin” shape.) Make sure that you indicate the various numerical valuesof the intercepts of the budget lines with the axes, along with the nu-merical values of the pre-shock and post-shock optimal consumptionbundles.2. Illustrate the revealed preference approximation of the Slutsky de-composition of the total effect of the economic shock on this con-sumer in an “indifference curve / budget line” diagram, using the“pre-shock consumption bundle, post-shock prices” approach to thisdecomposition. (Assume that the consumer’s indifference curves havethe conventional “downward sloping and bowed-in towards the origin”shape.) Be sure to identify the numerical values of each commodityin all of the relevant consumption bundles for this decomposition, themovement between bundles that constitutes the substitution effect,the movement between bundles that constitutes the income effect, thesize (including both magnitude and sign) of the substitution effect interms of widgets, and the size (including both magnitude and sign)of the income effect in terms of widgets.3. Does the revealed preference approximation to the substitution effectthat you found in part 3 of this question overstate, exactly equal, or3understate the true (or Hicksian) substitution effect that it is approx-imating? Does the revealed preference approximation to the incomeeffect that you found in part 3 of this question overstate, exactlyequal, or understate the true (or Hicksian) income effect that it isapproximating? Justify your answers.

Consider a consumer whose preferences are defined over bundles of non-negative consumption quantities for each of two commodities: Widgets anda composite consumption commodity known as “All Other Goods”. Thisconsumer’s preferences are both rational and locally non-satiated. Denote2the price of Widgets by p1, the consumption of Widgets by q1, the price of“All Other Goods” by p2, the consumption of “All Other Goods” by q2, andthe consumer’s income by y. Use the composite consumption commodity(that is, “All Other Goods”) as the numeraire throughout this question. (Inother words, assume that p2 = $1 throughout this question. Note that thisassumption allows you to interpret the amount of “All Other Goods” con-sumption as expenditure on “All Other Goods”, because q2 = 1q2 = p2q2.)Suppose that this consumer’s uncompensated demand function for Wid-gets is given byq1 (p1, y; p2 = 1) = (0.02) y − (2) p1.(Uncompensated demands are also known as Marshallian demands, or Wal-rasian demands, or ordinary demands.) Initially, this consumer has an in-come of$6, 500 and faces a Widget price of $50 per Widget. Following aneconomic shock, the Widget price rises to$60 per widget. This economicshock has no impact whatsoever on either this consumer’s income or the“All Other Goods” price.1. Illustrate the total impact of the economic shock on this consumer’soptimal consumption decision using an “indifference curve / budgetline” diagram. (Assume that the consumer’s indifference curves havethe conventional “downward sloping and bowed-in towards the ori-gin” shape.) Make sure that you indicate the various numerical valuesof the intercepts of the budget lines with the axes, along with the nu-merical values of the pre-shock and post-shock optimal consumptionbundles.2. Illustrate the revealed preference approximation of the Slutsky de-composition of the total effect of the economic shock on this con-sumer in an “indifference curve / budget line” diagram, using the“pre-shock consumption bundle, post-shock prices” approach to thisdecomposition. (Assume that the consumer’s indifference curves havethe conventional “downward sloping and bowed-in towards the origin”shape.) Be sure to identify the numerical values of each commodityin all of the relevant consumption bundles for this decomposition, themovement between bundles that constitutes the substitution effect,the movement between bundles that constitutes the income effect, thesize (including both magnitude and sign) of the substitution effect interms of widgets, and the size (including both magnitude and sign)of the income effect in terms of widgets.3. Does the revealed preference approximation to the substitution effectthat you found in part 3 of this question overstate, exactly equal, or3understate the true (or Hicksian) substitution effect that it is approx-imating? Does the revealed preference approximation to the incomeeffect that you found in part 3 of this question overstate, exactlyequal, or understate the true (or Hicksian) income effect that it isapproximating? Justify your answers.