The demand for a commodity is given by Q = Bo + ß₁P + 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 = Yo + Y₁P + v, where v denotes factors other than price that deter- mine supply. Suppose that u and v both have a mean of zero, have variances o and o, 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 (Q₁, P;) is collected, and Q; is regressed on P₁. (That is, Q; is the regressand, and P; is the regressor.) Suppose that the sample is very large. (i) (ii) Use your answers to (b) and (c) to derive values of the regression coefficients. A researcher uses the slope of this regression as an estimate of the slope of the demand function (B1). Is the estimated slope too large or too small? (Hint: Remember that demand curves slope down and supply curves slope up.)