Utility Maximization Assignment

Surname 1

Name of the Student

Name of the Professor

Name of the Course

Date of Submission

Utility Maximization Assignment

Q1: Let I=6,000, U(X, Y) =X

Y, P

2, P

=3. Find the optimal combination of X and Y that

maximizes utility.

Since the expenditure cannot exceed income I, the proportions of good X and Y

consumed must be equal to 6000.

Amount spent on good X= Px* X = 2X

Amount spent on good Y= Py*Y = 3Y

Therefore, 2X + 3Y≤ 6000

For maximum utility, it is assumed that the consumer will spend all the income on all the goods.

Therefore, the budget constraint becomes

2X+3Y = 6000

U(X, Y) = X

For maximization, we follow four key steps

Max U(X, Y) = X

Subject to

2X + 3Y= 6000 => 2x+3y-6000=0

Step 1: Express the two functions into a Langrage equations that take the following form

( , , ) u(x,y) + (I )

z x y P x P y



  

Where



is the Langrage multiplier.

Surname 2

Therefore, the above equation becomes

Z(x,Y,



) =x

y -



(2x + 3y – 6000) => x

y - 2



x - 3



y+6000



Step 2: Find the partial derivatives of all the variables that form Z

= 3x

y-2



=0 =>





---- (i)

= x

-3





----- (ii)

-2 3F x y





+ 6000=0------ (iii)

However, maximum utility occurs when

mu 3 ,

2 9 4.5

mu p

x y x

mu p x dy

Therefore

x y x y

mu x



    



     

Step 3: plugging in equation i and ii into equation iii and substituting for



x = $2250, and y = $500

Q2: In theory, what is the optimality condition for a consumer, i.e. what condition must hold

when the consumer has maximized production? Please express your answer using relevant

ratios and not specific numbers.

Optimality condition for a consumer is the maximum satisfaction that a consumer could

derive from consuming different goods and services constrained by domestic income and product

prices. Ideally, the goal of the consumer is to derive the maximum utility from consuming given

goods and services limited by budget. Though most people are unaware of the economics of

consumption, everyone tries daily to solve the problem of getting the best of income they have.

Mathematically, the problem of maximizing utility hence reaching the consumer optimum can be

illustrated as follows:

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Max U= f(X, Y)

Subject to P

(X) + P

(Y) ≤ I

Where Px and Py are the proportion of good X and Y respectively, I is the income and f(X, Y) is

the consumer utility function.

From the objective function above it becomes possible to draw indifference curves

representing the different preference to consumption the consumer might have. From the

indifference curve, it is possible to obtain the marginal rate of substitution (MRS) =MUx/MUy.

The marginal rate of substation gives the rate at which a consumer is ready to give up a given

good or service in exchange for another. However, the MRS is constrained by the budget

function, P

(X) + P

(Y) ≤ I which can be rewritten as Y = I/Py - (Px/Py)X.

The indifference curve for a consumer with the above objective function can be represented as

follows,

The shaded region shows bundles of good x and good y that the consumer can attain.

However, such values within the shaded region represent an inefficient use of income. As such,

the consumer must choose bundles of good x and good y at the points that lie on the intersection

of the budget constraint curve and any utility curve IC

, IC

, or IC

. However, the optimum point

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is only attainable when the gradient of the budget constraint curve surpasses or is surpassed by

the gradient of the indifference curve. Under the latter conditions, the consumer can do better by

substituting the amount of good y or good x purchased.

Therefore, for a consumer to reach the optimum point in which the income is used optimally to

satisfy consumption occurs when

MUx/MUy or MRS

= Px/Py

As such, optimality condition of a consumer will be achieved when the marginal utility of

consuming good X relative to the marginal utility of consuming good y equals the ratio of the

proportion of good x to the proportion of good y. Marginal utility is the change in customer

satisfaction or utility that would occur if a consumer changes the quantity of a given product

when the other the product remains constant.

Q3: Suppose the consumer chooses a combination of X and Y where the marginal rate of

substitution of X for Y is equal to 2/3. Should the consumer adjust her choice of

consumption? Explain. If so, should the consumer choose more or less of X and/or Y?

If the consumer chooses a combination of X and Y such that:

MUx/MUy=2/3, then the consumer should not adjust her choice of consumption as he or she

would have already reached the consumer optimum.

As illustrated in question Q2, optimum consumption is achieved when MRS

=Px/Py

In line with question Q1,

Px =2

Py =3

Consequently, Px/Py = 2/3

On the other hand, the marginal rate of substitution (MRS

) =MUx/MUy.

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In this case, the customer MRS = 2/3

Therefore,

MRS

= 2/3 = Px/Py

As such, the consumer is already at the equilibrium and should not change the choice of

combination of goods X and Y.

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