Quiz 2: Mathematics for Microeconomics


Utility function: It refers to a function use to measure the consumer's level of satisfaction derives from the consumption of goods and services. The utility function is given by, img a. Given the utility function, the partial differentiation with respect to a single variable, in a function of several variables is calculated as follows; img b. Partial differentiation with respect to x is img and with respect to y is img . Compute partial derivatives at x= 1 and y = 2 by substituting the values of x and y in the above partial derivates computed in previous part. It is computed below: img img c. Total differentiation is written with the help of partial differentiation. img d. Total differentiation is written with the help of partial differentiation. img So, img It means trade-off between x and y is not linear. The negative sign shows that getting more x required forgoing some y and vice-versa. e. To prove that value of U is 16, when img , substitute the value of x and y in the utility function and solve as follows: img Hence, the value of utility is 16. f. To determine the change in ratio of x and y while holding utility constant at 16, substitute the value of x and y in the img computed in part d. It is computed below: img Hence, (-2/3) ratio of x and y needed to hold U constant at 16. g. To plot the utility function, compute the horizontal and the vertical corner points as follows: img Vertical points: When x is 0, then y will be, img Thus, vertical point is (0,2.30) Horizontal points: When y is 0, then x will be, img Thus, horizontal point is (2,0). The following figure shows the shape of utility function when utility is 16: img As shown in the above figure, the shape of U is an ellipse centered at the origin. Slope of this equation is img .

Total revenue is the amount earned by the firm by its operations. Total cost is the production cost incurred by the firm to produce the commodity. Profit is the difference between the total revenue and total cost. a) Profit maximizing firm produces the output at that level where difference between total revenue and total cost is maximum. img Necessary condition is as follows: img Sufficient condition is as follows: img Calculate the profit as follows: img Thus, firm produces img of output and earns img profits. b) Second-order condition of profit maximization is img . It is also called sufficient condition. img Thus, second order condition is satisfied at 10 unit of output. c. Profit maximizes where marginal revenue is equal to marginal cost. Marginal revenue is additional revenue in total revenue while selling one more unit of product. Marginal cost is additional cost in total cost while producing one more unit of product. Calculate as follows: img img Yes, the solution obeys the rule 'marginal revenue equals to marginal cost'.

Substitute goods are those good that can be used in place of other good and satisfaction level of both products is almost the same. Substitution method Sum of x and y is 1. Therefore, we can write it as:- img Condition of consumer equilibrium (which seeks to solve the maximization and minimization problems subject to some constraints) can be easily understood with the help of Lagrange multiplier. Lagrange multiplier method:- Sum of x and y is 1. Budget constraint can be written as:- img Taking first order condition:- img img img