. The demand for a commodity is given by Q = β0 + β1P + 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 standard deviations su and sv, respectively, 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.
Question asked in two parts. Please help with the bolded questions. The demand for a commodity is given by Q = β0 + β1P + 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 standard deviations su and sv, respectively, 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 Q 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 coefficient.
[Hint: Use Equations (4.7) and (4.8) in Stock and Watson’s book.]
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? (Hint: Remember that demand
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