lagrangian equation help( urgent)

Question

A consumer’s demands x, y for two different goods are chosen to maximize the
utility function
𝑈(𝑥, 𝑦) = √𝑥 + √𝑦 (x≥ 0, y ≥ 0)
subject to the budget constraint 𝑝𝑥 + 𝑞𝑦 = 𝑚 (where p, q, m > 0)

 

(d) Find the utility-maximizing demands for both goods, as well as the Lagrange
multiplier λ, all as functions of the three variables (p, q, m).

 

 

 

Answer 1

We need ∇U = λ∇(px+qy) so the system is:

 

1/(2√𝑥) =  pλ

1/(2√y) = qλ

px+qy=m

 

So we have:

 

q/√𝑥 = p/√y ⇒ q²y = p²x

 

(p+p²/q)x=m ⇒ p(q+p)x = mq ⇒ x=mq/[p(p+q)]

 

y = mp/[q(p+q)]

 

The corresponding Lagrange multiplier is:

 

λ = √[(1/p + 1/q) / m] / 2.

 

To test that this is where the maximum is one option is that by the Extreme Value Theorem, either this point or one of the two endpoints of the constraint must be the site of the maximum.

 

At the critical point found: U = √[m(1/p+1/q)]

 

At the endpoint (0,m/q): U = √(m/q)

 

At the endpoint (m/p,0): U = √(m/p)

 

Clearly, the critical point wins out.

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