**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 $?

Answered: Suppose that John's preferences over…

Suppose that John's preferences over meat (M) and vegetables (V) are represented by the following utility function where 0 < a < 1. U(M,V) = a ln(M) + (1 − a) ln(V)

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Concept explainers

Form Utility

The term utility in general refers to the satisfaction earned from usage of a particular product or service. Utility always plays an important role because it influences various important elements like demand & thereby the price of the good or service as well. Though utility is not easy to measure, various types of economic models can be used indirectly. There can be various types of utilities like economic utility, marginal utility, cardinal utility, form utility.

Question

**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 $?

Expert Solution

Step 1: Describe the given problem

Utility function : U = a ln (M ) + (1-a) ln (V)

V & M are two goods ,

0 <a <1

Price of M = Pm

Price of V = Pv

Income = I

Therefore ,

Budget Constraint : Pm (M ) + Pv (V ) = I

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