Answer to Question #180977 in Microeconomics for Imran jamil

Question

Which optimal values can be calculated from following equation? How many techniques are used to solve these equations? Give answers after solution.Q =XY.......s.t...... M=PxX+PyY

Answer 1

Given the budget that he/she has, the maximum amount of utility that this person can possibly receive is computed using the function of utility. 

The mentioned equations are a part of the analysis of the "Utility". This set of equations will be of help in computing how much utility a person can possibly get, given a budget that is limited.

The given equations are as follows, 

$Q =XY$

$M=PxX+PyY$

In this, the first one is the equation of the utility, 

Utility function is $Q =XY$

Where Q=Utility

X and Y are commodities that constitute this Utility.

Budget constraint is $M=PxX+PyY$

where, 

M=Income

PX=Price  of the commodity X

PY=Price  of the commodity Y

Normally, in such sums, the values of the P_x

, P_y and M are already given.

The task here is to make sure that the utility is maximized and also to compute the quantities of these commodities X and Y. 

For instance, the values given for the same function are as follows, 

$100=1X+1Y$

Where M=100, P_x=1 and p_y=1

It is possible to use the technique of lagrangean, 

$L=XY−λ[100−2X−2Y]$

Solving this will give out the conditions of the first order

$Y−λ=0$

$X−λ=0$

$100-X-Y=0$

Solving these will give out, 

$Y=X=λ$

Using the method of substitution, we get

$100−X−Y=0$

$Because X=Y$

$100−X−X=0$

$100=2X$

$X=50$

This $Y=50$

Thus, we have found out the units(quantities of X and Y

X and Y) of both the commodities with which this person will get the maximum amount of utility, given the budget that he/she has.