**Problem 2**Suppose that John’s preferences over meat ( $ M $ ) and vegetables ( $ V $ ) are represented by the following utility function\[U(M, V) = \alpha \ln(M) + (1 - \alpha) \ln(V)\]where $ 0 < \alpha < 1 $.*No claim of realism is made for the numbers in this example.*---(a) Write down the Lagrangian for John’s optimization problem. (Recall that John maximizes utility given an income, $ I $, and prices $ p_M $ and $ p_V $ for the goods.)(b) Solve for John’s optimal consumption bundle $ (M^*, V^*) $ (as a function of income and prices) using the Lagrangian method.(c) Suppose $ \alpha = \frac{1}{5} $. Suppose also that John has income $ I = 200 $ and faces prices $ p_M = 1 $ and $ p_V = 2 $. What is the value of John’s optimal consumption bundle? What happens if John’s income doubles to $ I = 400 $?

The utility function for John is given as follows. The value of * lies between 0 and 1. The income…

Problem 2 Suppose that John's preferences over meat (M) and vegetables (V) are represented by the following utility function U(M,V) = a ln(M) + (1 − a) ln(V) where 0 < a < 1. ¹No claim of realism is made for the numbers in this example. 1 (a) Write down the Lagrangian for John's optimization problem. (Recall that John max- imizes utility given an income, I, and prices pm and py for the goods.) (b) Solve for John's optimal consumption bundle (M*, V*) (as a function of income and prices) using the Lagrangian method. 200 and faces prices PM = 1 and pv = 2. What is the value of John's optimal consumption bundle? What happens if John's income doubles to I = 400? (c) Suppose a = . Suppose also that John has income I =

Problem 2 Suppose that John's preferences over meat (M) and vegetables (V) are represented by the following utility function U(M,V) = a ln(M) + (1 − a) ln(V) where 0 < a < 1. ¹No claim of realism is made for the numbers in this example. 1 (a) Write down the Lagrangian for John's optimization problem. (Recall that John max- imizes utility given an income, I, and prices pm and py for the goods.) (b) Solve for John's optimal consumption bundle (M, V) (as a function of income and prices) using the Lagrangian method. 200 and faces prices PM = 1 and pv = 2. What is the value of John's optimal consumption bundle? What happens if John's income doubles to I = 400? (c) Suppose a = . Suppose also that John has income I =

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Chapter1: Making Economics Decisions

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