Assignment:
Assume that the demand for welding services is QD = a - bP where a > 0 and b > 0 are exogenous parameters. The baseline case is b = 0.5 and a = 34. For parts (a-g), assume the baseline case.
(a) What is the elasticity of demand at the point where P=48?
(b) At what point on the demand curve is demand most elastic? (Hint: use the formula for elasticity to determine how the elasticity of demand changes as QD changes)
(c) Calculate revenue if Q=24.
(d) Calculate revenue as a function of Q (the revenue function).
(e) Calculate, using calculus, marginal revenue as a function of Q (the marginal revenue function).
(f) Calculate revenue if Q=25.
(g) Using the marginal revenue function, calculate the price that maximizes revenue.
For parts (h-n), you should leave a and b as exogenous variables with unspecified values.
This means that most of your answers will be functions of a and b. (Treat a and b like constants and solve for everything in terms of a and b)
(h) Calculate the inverse demand function.
(i) Calculate the two intercepts and the slope of the demand curve.
(j) Calculate revenue as a function of Q.
(k) Graph the MR curve and the demand curve to show that the marginal revenue curve is a line having the same P-intercept as the demand curve but twice the slope.
(l) Calculate the elasticity of demand as a function of Q.
(m) For what values of Q is the elasticity of demand less than -1?
(n) For what values of Q is marginal revenue positive?
Assume that the inverse supply function in this market is PS(Q) = 4 + 10Q + Q2. Parts (o,p) use only the supply function, not the demand function.
(o) Draw the supply curve, including the numerical coordinates of at least three points on the curve.
(p) Calculate elasticity of supply at the point where Q=1.
Parts (q) and (r) combines supply and demand.
(q) Derive the equilibrium price and quantity, using algebra, in the baseline case.
(r) Calculate revenue at the equilibrium point.
3. Assume a seller faces the following demand function is QD = 3 - ln(P)
(a) Compute the inverse demand function.
(b) Compute the seller's marginal revenue as a function of Q.
(c) What price maximizes seller's total revenue?
(d) Compute elasticity of demand as a function of the quantity demanded.