Question

I'm having a hard time understanding the Lagrange method in general, so any explanation is helpful. For example in part a), I know that $L_{M}$ = , , and . But I don't know what to do after that.

Answer 1

Solution:

a) For Lagrangian method: there is a single equation carrying the objective function (function to be maximized/minimized, eg utility function here) and all the constraints (here only one constraint: the budget constraint). Using this single Lagrangian equation, you solve for the optimizing parameters by taking partial derivatives with respect to them and Lagrangian multiplier(s), and solving for the first order conditions. After this, you reach to the three equations you have mentioned correctly as (taking Lagrangian multiplier as b, instead of lambda for ease of typing):

dL/dM = V² - b*Pm;

dL/dV = 2MV - b*Pv; and

dL/db = I - Pm*M - Pv*V

Next, we solve for first order conditions (FOC): dL/dx = 0 (x is denoted for any optimizer and Lagrangian multiplier; here x = M, V, b)

Then, FOC gives us following equations:

V² - b*Pm = 0 ... (i)

2MV - b*Pv = 0 ... (ii)

I - Pm*M - Pv*V = 0 ... (iii) (this is same as budget constraint; under Lagrangian multiplier, FOC corresponding to the multiplier always gives back the respective constraint (with equality))

Rewriting (i) and (ii) for multiplier value:

(i) gives b = V²/Pm, and (ii) gives 2MV/Pv

Using these two, we have V²/Pm = 2MV/Pv

**The above equation further rewritten gives us, V²/2MV = Pm/Pv. Note that this is how the optimal condition of tangency: marginal rate of substitution equal the price ratio (slope of indifference curve equals the slope of budget line) for utility maximization is derived.

So, V = 2M*Pm/Pv

Using this in (iii), we have Pm*M + Pv*(2M*Pm/Pv) = I

3Pm*M = I

M = I/(3*Pm)

Then V = 2*(I/(3*Pm))*Pm/Pv = 2*I/(3*Pv)

b) With I = 40, Pm = $1, and Pv = $1

Then, using part (a)

M = 40/3*1 = 40/3

V = 2*40/(3*1) = 80/3

Then utility level is: U = (40/3)*(80/3)² = 256000/27 = 9481.48 (approx)

c) With tax of $10 to each alien, income of an alien = 40 - 10 = $30

With this income, new consumption levels are:

M' = 30/3*1 = 10

V' = 2*30/(3*1) = 20

And new utility level = 10*20² = 4000

Clearly, utility level or happiness level has now decreased, as with tax, aliens are able to consume less.