Econ. 410TauchenMathematics Practice ProblemsBe sure that you can solve these practice problems. Review calculus and algebra texts ifnecessary. (If you do not have a text, go to the Mathematics Library in Phillips Hall.) We want tobe sure that you are familiar with the mathematics so that when it is time to apply these techniques,you can concentrate on the economic principles rather than the mathematical calculations. In theseproblems, you will be concentrating only on the mathematics.1. (Calculus and graphing review) A firm’s inverse demand function is D(q) = 100− 2q. [Reminder:For each price, the demand function gives the quantity demanded at that price. For each quantity,the inverse demand function gives the price at which that quantity would be demanded.] Thus ifa firm sells q units of the good, it receives a price 100 − 2q for each unit sold. The firm’s totalrevenue is the quantity times the price orR(q) = 100q − 2q2.a. The firm’s marginal revenue is the derivative of the revenue function. Compute the marginalrevenue function.b. Graph the firm’s demand function and the firm’s marginal revenue function. The quantity ismeasured on the horizontal axis; the price and marginal revenue are measured on the vertical axis.2. (Calculus review) A firm’s inverse demand function is D(q) = a − bq where a and b are bothpositive numbers. The firm’s revenue is R(q) = aq − bq2.a. Compute the marginal revenue function.b. Compare the intercept of the demand and marginal revenue functions. [The intercept is thevalue of the function for q = 0.] Also compare the slopes of the demand and the marginal revenuefunctions.3. (Calculus and algebra review) A firm has costs C(q) = 10q2.a. Determine the value of q for which 20 = C0(q). Denote this value of q as q∗. Compute20q∗− C(q∗).b. Let p be a positive price. Find the value of q for which p = C0(q). [Hint: Compute C0(q).Equate p to C0(q). Solve for q as dependent on p.] Let q∗denote the value of q for which p = C0(q).Is q∗increasing or decreasing in p?4. (Algebra review) Solve the following two equalities for K and L in terms of w, r, and q. Yourexpression for K should involve w, r, and q but not L. Similarly, your expression for L shouldinvolve w, r, and q but not K.KL=wrq = KL.Hint: Solve the first equation above for K in terms of w, r, and L. Substitute this expression forK into the second equality. Finally, solve for L.5. (Algebra review) Solve the following two equalities for K and L in terms of w, r, and q. Yourexpression for K should involve w, r, and q but not L. Similarly, your expression for L shouldinvolve w, r, and q but not K..5K−.5.5L−.5=wrq = K.5+ L.5.6. (Algebra review) Solve the following two equalities for x and y. Your expression for x maycontain the variables α and I but not y. Similarly, your expression for y may involve α and I butnot x. [Hint: Solve the first equality for y, substitute this expression for y into the second equality,and then solve for x.]yx + a= 2 and I = 4x + 2y.7. (Algebra review) Solve the following two equalities for x and y in terms of px, py, and I. Yourexpression for x may involve px, py, and I but not y. Similarly, your expression for y may involvepx, py, and I but not x. Note that your expressions for x and/or y may not necessarily include allthree of the variables px, py, and I.yx + a=pxpyI = pxx + pyy.8. (Optimization) A firm’s profit depends upon the quantity that it produces. Specifically, thefirms profit isΠ(q) = (200 − q)q − 100q.Find the profit maximizing quantity. [Hint: Find the quantity for which the derivative of the profitfunction is zero. Show that the second derivative of the profit function is negative.] Determine thefirm’s profit at the profit-maximizing quantity. [Hint: Substitute this quantity into the above profitfunction.]9. (Optimization) Suppose now that the firm’s profit function isΠ(q) = (200 − q)q − cqwhere c is a positive number. (In the problem above, c has the value 100.) Find the profitmaximizing quantity. [Hint: Find the quantity for which the derivative of the profit function iszero. Note that the quantity will depend upon c. Show that the second derivative of the profitfunction is negative.] Determine the firm’s profit at the profit-maximizing