The demand for a commodity is given by Qt = βo + β1Pt + Ut, where Q denotes quantity, P denotes price, and U denotes factors other than price that determine demand. Supply for the commodity is given by Qt = γo + γ1Pt+ Vt, where V denotes factors other than price that determine supply. Suppose that U and V both have a mean of zero, have variances σ2u, σ2v, respectively and are mutually uncorrelated.

a. Solve the two equations for Q and P 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 (Q, P) is collected, and Q is regressed on P. (That is, Q 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 (β). Is the estimated slope too large or too small?