Question

Problem 1: Utility maximization

There two goods, X and Y , available in arbitrary non-negative quantities (so the consumption set is R2+). A consumer has preferences over consump- tion bundles that are strongly monotone, strictly convex, and represented by the following (differentiable) utility function:

u(x, y) = x + xy + y,
where x is the quantity of good X and y is the quantity of good Y .

The consumer has wealth w > 0. The price for good X is p > 0, and the price for goodY is q>0.

(A) In an appropriate diagram, illustrate the consumers map of indifference curves. Make sure you label the diagram clearly, and include as part of your answer any calculations about the slopes of the indifference curves and where the indifference curves intersect either of the axes.

(B) Formulate and solve the consumer’s utility maximization problem. Your final answer should describe the consumer’s demand for goods X and Y as a function of w, p, and q, as well as the consumer’s indi- rect utility function.

(C) Now assume that w=100 and p=10.
(i) Illustrate the consumer’s demand function for good Y in an appropriate diagram.

(ii) Illustrate the consumer’s cross-price demand function for good X in an appropriate diagram.

Answer 1

The Budget Constraint of the consumer could be given as

W = P_X. X + P_Y. Y

where W is the wealth of the consumer, P_x is the price of good X, X is the quantity of good X, P_Y is the price of good Y and Y is the quantity of good Y.

Utility function is given as

U(X, Y) = X + XY + Y

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(A) An indifference curve shows all the possible bundles that a consumer can consume with having same level of satisfaction (utility) throughout. Indifference curve is constructed in the ascpect of the utility function provided.

The budget constraint is constructed using the budget equarion of the consumer. It is a straight line with inverse relation with the demand for good X and good Y. If consumption of one good is increased at he given prices, the units of other good will have to be given up. The slope of budget constraint is given as -Px/Py.

Slope of Indifferene curve is given as MUx/Muy which is also known as Marginal Rate of Substitution.

In this problem,

Slope of IC = MRS = (1 + Y) / ( X + 1)

The equilibrium consdition or the choice of bundle of the consumer will be at the point where IC is tangent to the budget constraint. i.e. Slope of IC = Slope of Budget Constraint

i.e. (1 + Y)/ (X + 1) = - p / q

The consumer will choose the bundles that are on the IC2 as IC2 is tangent to the budget constraint and gives higher utility as compared to IC1.

The consumer will not choose bundles on the IC1 as they are not affordable by the consumer. The bundles on IC1 are higher in cost than what the consumer's budget allows and hence out of reach for him.

Similarily, IC3 gives the lowest utility to the consumer. Even though it is affordable, the consumers demand is not fully satisfied on this IC and leaves him with the excess budget. The available resources are not fully satisfied at this point.

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(B) Solving the maximisation problem using lagrange function:

First Order Conditions

From the above equations, we get

Y = p/q (X + 1) - 1

Substituting the value of Y in third equation,

----------------------------------- eq 1

------------------------------------eq 2

eq1 and eq 2 are the demand functions of X and Y.

Indirect Utility function

The values of X and Y will be substituted in the utility function U(X, Y) = X + XY + Y

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(C)

as calculated above.

Now, W = 100 and p = 10

Y = 45/q - 1.5

CROSS PRICE DEMAND OF GOOD X

(as derived above)

Now, W = 100, p = 10

X = (100 - 10 - q)/2q

X = 45/ q - 0.5

Cross price demand curve of good X is given as

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