1. For a market, assume that the supply function is given by QS = 0.5P – 10 and the demand function is given by QD = 80 – 0.5P.
a. Write out the inverse supply and demand functions.
b. Find the equilibrium market price and quantity.
c. A sales tax is levied on buyers with a tax rate of t, i.e., the real price paid by buyers is (1+t)P. Find the new equilibrium market price and quantity.
d. (5, extra credits) Write out the tax revenue function and find the maximum revenue (just a mathematical application; in reality, no government should maximize tax revenue).
2. A market is modeled by the following demand and supply functions: QS = 0.5P – 10 and QD = 80 – 0.5P. For a market shock (P0, not the equilibrium price), assume that price adjustment is given by Pt = (1.05)Pt-1 + 0.4[QD(Pt-1) – QS(Pt-1)]. Solve this equation and tell if the price has a limit (also indicate the limit if it exists).
3. Find the first-order derivatives for the following functions.
a. y = 4(5e-2x + e3x)
b. y = xx(1+lnx)
4. Find the critical point(s) x2e-x, and determine if it (they) is (are) maximum, minimum, or inflection point(s).
5. A competitive firm has a cost function C=q3-40q2+500q+3000.
a. Find the startup point (q, P) and write out the supply function.
b. Find the break-even point (q, P). Try to keep two decimal points in your final answer.
c. Find the optimal output at price of 192.
d. At price of 192, find the optimal profit.
6. The demand function faced by a monopolist is given by: P = 54-Q. The firm’s cost function is given by: TC = 100 + 6Q + Q2.
a. Find the monopoly’s profit maximizing output level.
b. Find the monopoly’s profit maximizing price.
c. Find the monopoly’s maximum profit.
7. Suppose that the total cost function TC(q) is differentiable, which is the sum of variable cost VC and fixed cost FC. Let MC and AVC be marginal cost and average variable cost, respectively. (Use no specific function to) Prove:
a. if MC>AVC, AVC increases; if MC < AVC, AVC decreases;
b. at the minimum of AVC, AVC=MC.
