Consumer Utility Assignment Discussion Paper

## Question

Suppose a manufacturer introduces a new good, X. If the product lives up to the claims made by the seller, a consumer’s utility is given by

U = X^{1/2}Y^{1/2}

Where Y is the composite commodity. There is, however, a 50% chance that the new good does not live up to the claims and delivers on ¼ of the expected flow of services, in which case the utility will be

U = (X/4)^{1/2}Y^{1/2}

Assuming both good cost $1 per unit, that it is impossible to test X before purchase and that no return is possible if it underperforms, how much X will be bought if income is $16?

If the individual believes that each unit of X bought yield a unit of service with certainty, how much will he buy? How much does the reduction in uncertain raise his welfare?

Suppose the probability of underperformance remains at 50%, but to compensate for this, the price is X is reduced to (1/2)(1) + (1/2)(1/4) = 5/8 so that $1 buys an expected quantity of one unit of X services. What will be the quantity of X purchased (I=$16) and will utility be as high as in B? Consumer Utility Assignment Discussion Paper

Step 1/2

To maximize utility, the customer will compare the negligible utility of X with its price. Assuming the likelihood of X living up to its claims is 50%, the anticipated utility of buying one unit of X is:

EU(X)=(12)(X12×Y12)+(12)((X4)12×Y12)

EU(X)=(12)(X12×Y12)+(12)X12×Y122

EU(X)=(58)X12Y12

The budget constraint is:

PXX+PYY=I

where PX=PY=$1andI=$16

Substituting P_X = $1, P_Y = $1 and I = $16 into the budget constraint:

X+Y=16

Solving for Y:

Y=16−X

Substituting Y into the expected utility equation:

EU(X)=(58)X12(16−X)12

Taking the derivative of EU(X) with respect to X and setting it equal to zero:

(516)(16−X)−12X−12=(516)(X−12)(16−X)−12

Simplifying:

(16−X)−12=X−12

Squaring both sides: Consumer Utility Assignment Discussion Paper

16−X=X

X=8

Step 2/2

Subsequently, on the off chance that the customer accepts that each unit of X bought yields a unit of benefit with certainty, he will purchase 8 units of X and 8 units of Y, investing the whole $16 budget.

The lessening in instability does not raise the consumer’s welfare in this case since the anticipated utility is the same as in portion (a).

If the price of X is reduced to (5/8) so that $1 buys an expected quantity of one unit of X services, the budget constraint becomes:

(58)X+Y=16

Solving for Y:

Y=(1285)−(85)X

Substituting Y into the expected utility equation:

EU(X)=X12((1285)−(85)X)12

Taking the derivative of EU(X) with respect to X and setting it equal to zero:

((1285)−(85)X)−12X−12=((1285)−(85)X)−12(12)X−32(−8)=0.

Explanation is given above.

Final answer

All answers is provided in the above steps1 and 2. Consumer Utility Assignment Discussion Paper