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Consider a consumer with the utility function given by . The consumer has an income level Ip12 and faces prices and p for the two goods. The consumer makes consumption decisions to maximize her utility.
(a) What are the endogenous variables here? What are the exogenous variables/parameters?
(b) State the utility maximization problem and find the first order conditions.
(c) Solve for the utility maximizing consumption levels and the optimal value of the Lagrangian multiplier.
(d) Find an expression showing how a change in the price of good 1 would affect the consumption of good 1.
(e) You could have found the effect of a change in the price of good 1 directly from the first order conditions without solving explicitly in (c). Using the rule for finding derivatives implicitly, write out the expression for the effect of a change in the price of good 1 on consumption of good 1.
(f) Show that the approach in (e) gives the same result as you got by differentiating explicitly in (d).
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