{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Macroeconomics Exam Review 66

# Macroeconomics Exam Review 66 - Solutions for Foundations...

This preview shows page 1. Sign up to view the full content.

2.33 By Exercise 2.31 and Example 2.53, each 𝑔 𝑛 is strictly increasing on + . That is 𝑥 1 < 𝑥 2 = 𝑔 𝑛 ( 𝑥 1 ) < 𝑔 𝑛 ( 𝑥 2 ) for every 𝑛 (2.32) and therefore lim 𝑛 →∞ 𝑔 𝑛 ( 𝑥 1 ) lim 𝑛 →∞ 𝑔 𝑛 ( 𝑥 2 ) This suﬃces to show that 𝑔 ( 𝑥 ) = lim 𝑛 →∞ 𝑔 𝑛 ( 𝑥 ) is increasing (not strictly increasing). However, 1 + 𝑥 is strictly increasing, and therefore by Exercise 2.31 𝑒 𝑥 = 1 + 𝑥 + 𝑔 ( 𝑥 ) is strictly increasing on + . While it is the case that 𝑔 = lim 𝑔 𝑛 is strictly increasing on + , (2.32) does not suﬃce to show this. 2.34 For every 𝑎 > 0, 𝑎 log 𝑥 is strictly increasing (Exercise 2.32) and therefore 𝑒 𝑎 log 𝑥 is strictly increasing (Exercise 2.28). For every 𝑎 < 0, 𝑎 log 𝑥 is strictly increasing and therefore (Exercise 2.30 𝑎 log 𝑥 is strictly decreasing. Therefore 𝑒 𝑎 log 𝑥 is strictly decreasing (Exercise 2.28). 2.35 Apply Exercises 2.31 and 2.28 to Example 2.56.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Ask a homework question - tutors are online