{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Macroeconomics Exam Review 60

# Macroeconomics Exam Review 60 - c 2001 Michael Carter All...

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

For analysis, it is convenient to represent the technology 𝑉 ( 𝑦 ) by a production function (Example 2.24). The firm’s optimization can then be expressed as max x ∈ℜ 𝑛 + 𝑝𝑓 ( x ) 𝑛 𝑖 =1 𝑤 𝑖 𝑥 𝑖 and the profit function as Π( 𝑝, w ) = max x ∈ℜ 𝑛 + 𝑝𝑓 ( x ) 𝑛 𝑖 =1 𝑤 𝑖 𝑥 𝑖 2.14 1. Assume that production is profitable at p . That is, there exists some y 𝑌 such that 𝑓 ( y , p ) > 0. Since the technology exhibits constant returns to scale, 𝑌 is a cone (Example 1.101). Therefore 𝛼 y 𝑌 for every 𝛼 > 0 and 𝑓 ( 𝛼 y , p ) = 𝑖 𝑝 𝑖 ( 𝛼𝑦 𝑖 ) = 𝛼 𝑖 𝑝 𝑖 𝑦 𝑖 = 𝛼𝑓 ( y , p ) Therefore { 𝑓 ( 𝛼 y , p ) : 𝛼 > 0 } is unbounded and Π( p ) = sup y 𝑌 𝑓 ( y , p ) sup 𝛼> 0 𝑓 ( 𝛼 y , p ) = + 2. Assume to the contrary that there exists p ∈ ℜ 𝑛 + with Π( p ) = 𝜋 / ∈ { 0 , + , −∞ } . There are two possible cases. (a) 0 < 𝜋 < + . Since 𝜋 = sup 𝑦 𝑌 𝑓 ( y , p ) > 0, there exists y 𝑌 such that 𝑓 ( y , p ) >
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}