# Quiz 3: Preferences and Utility

(a) For a utility function to have strictly diminishing marginal rate of substitution, it should be strictly quasi-concave. A function is strictly quasi-concave, if the following condition holds: ...... (1) The partial differentials for the utility function are given below: Putting in the values in the following equation ...... (2) We get, As the inequality (1), does not hold, the function is not strictly quasi-concave and hence does not have strictly diminishing rate of substitution. Both the marginal utilities U x and U y are positive and are constant as reproduced below. As both the marginal utilities are constant, value of equation (2), will be zero. Hence, quasi-concavity condition shown in inequality (1) will not hold. This is true in general for all cases in which marginal utilities of both the goods are constant and are not influenced by the amount of consumption of the other good. (b) Let us set utility equal to 10. Then the equation for the curve is To simplify the function, while leaving the ordering unchanged, we can square it so the indifference curve is represent by Using the above equation, at utility equal to 10, we get the following value of the partial differentials: Putting in the above values in equation (2), We get, Except for when x and/or y are equal to 0, resulting in utility equal to 0, x and y will always be positive quantities. Thus, Hence, strictly quasi-concavity inequality (1), holds and the utility function has strictly diminishing marginal rate of substitution. The marginal utilities for both the goods are reproduced below. Since these are partial differentials (i.e. holding the quantity of the good fixed), the marginal utility of each good is constant in terms of the other good and does not exhibit diminishing marginal utility. However, as was shown above, the function is strictly quasi-concave. This is because diminishing marginal utility for each good is not a necessary condition for diminishing MRS and consequently strict quasi-concavity. This can be seen below. From the above equation, it can be seen that, as x increases, the marginal rate of substitution will decrease. Hence, it is necessary that the ratio of the marginal utilities decrease for strict quasi-concavity, irrespective of whether each good has diminish marginal utility or not. (c) For this utility function, we get the following values: Substituting the above values in equation (2), We get, As x is always positive, Hence, inequality (1) holds and the utility function has strictly diminishing MRS. The marginal utilities of both goods are reproduced below. It can be seen from the above equations that while good x has diminish marginal utility, marginal utility for good y is constant at 1.As has been shown above, the function is strictly quasi-concave. This is because; diminishing marginal utility for each good is not a necessary condition for diminishing MRS and consequently, strict quasi-concavity and only the marginal rate of substitution should decrease. This can be seen below. From the above equation, it can be seen that, as x increases, the marginal rate of substitution will decrease. Hence, it is necessary that the ratio of the marginal utilities decrease for strict quasi-concavity, irrespective of whether each good has diminish marginal utility or not. (d) Let us arbitrarily set utility equal to 4. Then the equation for the curve is To simplify the function, while leaving the ordering unchanged, we can square both sides so the indifference curve is represent by The values for the partial differential of the above function are as follows: Putting in the above values in equation (2), We get, For the function to be strictly quasi-concave, The above inequality holds only when: As x and y will always be positive quantities, the above inequality will hold true only when However, looking at the original utility function reproduced below: We see that for utility to be defined, As the quantities are consumed in positive amounts, this also means that This is exact opposite to the condition for quasi-concavity i.e. x y Hence, for all the defined values of utility, the utility function is not strictly quasi-concave. The marginal utilities of both the goods are reproduced below. As can be seen from the above equations, while good x exhibits increasing marginal utility, good y exhibits decreasing marginal utility. Above, we found that the function is not strictly quasi-concave in the defined values of utility in which y x. This is because when y x, the increasing marginal utility of good x overpowers the diminishing marginal utility of good y. The net result is that the curve does not exhibit diminishing MRS (i.e. it is not strictly quasi-concave). (e) The values for the differential of the utility function are as follows: Substituting the above values in equation (2), We get, The above equation would always be negative. Hence, inequality (1) holds and the function is strictly quasi-concave. The marginal utilities of the two goods are reproduced below. They both exhibit diminishing marginal utility. As shown above, the function is strictly quasi-concave and thus had diminishing marginal rate of substitution. However, this result does not hold in general. Diminishing marginal utility of both goods is not a sufficient condition for diminishing marginal rate of substitution. This is because it is their ratio that matters, and not the absolute values.

a. Marginal utility is addition in total utility by consuming one more unit of a commodity. The utility function is given below: The marginal utilities of good x and y for the utility function are given below: As marginal utility of x is y and marginal utility of y is x and they both need to be kept constant at the derivative is calculated of other good. Therefore, it shows the constancy of marginal utilities of x and y. The marginal rate of substitution (MRS) is the units of one good an individual likes to give up for one additional unit of other good. The marginal rate of substitution is the ratio of marginal utility of one good to marginal utility of other good. The formula for the MRS is given below: Substituting in the values for the marginal utilities the value of MRS is calculated below: As x and y are the commodities and their consumption can't be negative and there is a negative sign with second derivative therefore, marginal rate of substitution must be decreasing with increase in x b. The utility function is given below: The marginal utilities of good y and x for the utility function are given below: As both x and y is the consumption and consumption can't be negative. Therefore, both of these marginal utilities are increasing with y and x. Hence, this function shows increasing marginal utilities. Substituting in the values of the marginal utilities, the value of MRS can be calculated as shown below: As x and y are quantities of goods and this can't be negative, MRS decreases with increase in x. Hence, this utility function shows decreasing MRS. c. The utility function is given below: The marginal utilities of good x and y for the utility function are given below: As the value of x increases, there will be a fall in marginal utility of x. Hence, the goods x shows decreasing marginal utility. As the value of y increases, there will be a fall in marginal utility of y. Hence, the goods y shows decreasing marginal utility. Substituting the values for the marginal utilities, the value of MRS can be calculated as shown below: x and y both are quantities of good and they can't be negative. Therefore, increase in x result in fall in MRS. Hence, this utility function has diminishing MRS. All the three utility functions of commodity x and y show decreasing MRS.