9.5 The demand for a commodity is given by Q = b0 + b1P + u, where Q denotes quantity, P denotes price, and u denotes factors other than price that determine demand. Supply for the commodity is given by Q = g0 + g1P + v, where v denotes factors other than price that determine supply. Suppose u and v both have a mean of 0, have variances s2 u and s2v

, and are mutually uncorrelated.

a. Solve the two simultaneous equations to show how Q and P depend on u and v.

b. Derive the means of P and Q.

c. Derive the variance of P, the variance of Q, and the covariance between Q and P.

d. A random sample of observations of (Qi, Pi) is collected, and Qi is regressed on Pi. (That is, Qi is the regressand, and Pi is the regressor.)

Suppose the sample is very large.

i. Use your answers to

(b) and

(c) to derive values of the regression coefficients. [Hint: Use Equations (4.7) and (4.8).]

ii. A researcher uses the slope of this regression as an estimate of the slope of the demand function 1b12. Is the estimated slope too large or too small? (Hint: Remember that demand curves slope down and supply curves slope up.)