The demand for a commodity is given by Q = β0 + β1P + u, whereQ denotes quantity, P denotes price, and u denotes factors other thanprice that determine demand. Supply for the commodity is given byQ = ϒ0 + ϒ1P + v, where v denotes factors other than price that determinesupply. Suppose that u and v both have a mean of zero, have variancesσ2u and σ2v, and are mutually uncorrelated.a. Solve the two simultaneous equations to show how Q and P dependon u and v.b. Derive the means of P and Q.c. Derive the variance of P, the variance of Q, and the covariancebetween Q and P.d. A random sample of observations of (Qi, Pi) is collected, and Qi isregressed on Pi. (That is, 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 regressioncoefficients. ii. A researcher uses the slope of this regression as an estimate of theslope of the demand function (β1). Is the estimated slope too largeor too small?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
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 = ϒ0 + ϒ1P + 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, 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
or too small?
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