The demand for a commodity is given by Q = 0 + 1P + u, where O

The demand for a commodity is given by Q = β0 + β1P + u, where O denotes quantity, P denotes price, and u denotes factors other than price that determine demand. Supply for the commodity is given by Q = γ0 + γ1}P + v, where v denotes factors other than price that determine supply. Suppose that u and v both have a mean of zero, have variances σ2u, and σ2v, 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 the Qi is the regressand and Pi is the regressor.) Suppose that the sample is very large.
i. Use your answers to (b) and (c) to derive values of the regression coefficients.
ii. A researcher uses the slope of this regression as an estimate of the slope of the demand function (β1). Is the estimated slope too large or too small?