upply, Demand and the Identification Problem In this problem we will understand why economists began to think about instrumental variables in the first place. We will explore simultaneity bias. Suppose that you are an economist trying to predict the effect that raising tariffs will have on the price and output of coffee. To do this, you need to understand the supply and demand functions of coffee. For now, suppose we just want to understand demand. Suppose that demand has the following simple form: This says that demand is linear in price but there are occasionally "demand shocks" that make everyone prefer more of a good or less of a good regardless of price. We suspect that B1 < 0, as people demand less of a good when price is high Suppose that the market supply function for coffee has the following form: Qs = 00 + α1P4η We suspect now that α1 > 0 since as price is higher, the market is more willing to supply coffee since more producers will enter Finally, we know that in equilibrium, supply must equal demand. Now, you have collected data on many markets indexed by m = 1, , M. Each market has the same supply and demand functions but differ in their random shocks. We will work through the problem of estimating supply and demand. For simplicity, we will assume that Var(η)-σ , Var(e)-σ? and Cov(η, e)-0 a) First, think of at least one component of e- something that will differ across markets that might affect demand for coffee that isn't the price of coffee. Do you think this would also affect supply? b) Second, think of at least one component of η something that will differ across coffee producers that might affect supply at any given price. Do you think this will also affect demand? c) Now, use the fact that QS Q in equilibrium to solve for price and quantity in a market m as a function of ε, η and the model parameters d) Solve for Cov(PQ) in terms of σ| and σ, e) Solve for Var(P) in terms of σ and σ; f) A data scientist who hasn't thought too hard about the economics runs a regression of Q on P across markets and tells you that he has estimated demand. In other words, he believes that he has run this regression From parts (d) and (e), solve for population version of the OLS estimator in this regression (i.e., Cov(X, Y)/Var(X)). Under what conditions, if any, will this be a consistent estimator of g) Now suppose that we have data on whatever you suggested in part (b). Call this Z and assume that it is uncorrelated with e, regardless of what you said. I.e., Coe(Z. η)メ0 but Cov(Ze)-0 Denote Cov(Z,n) by ơz'.. Prove that if one uses Z as an instrument for price then ß{v is a consistent estimator of βί. To do this, calculate Cov(Z.Q), Cov(Z,P) and plug it into the IV estimator formula, Cov(Z, Q)/Cov(Z, P h) The same logic as above allows us to use shocks to demand to estimate the supply function. Your friend the data scientist, wowed by your insights, suggests using the price of tea as a shock to demand for coffee. After all, if tea is expensive people will demand more coffee no matter the price. Do you think your friend has chosen a good instrument? Hnt: If demand for coffee depends on the demand for tea, then the demand for tea is likely to depend on the demand for coffee. So what does that tell us about the price of tea and shocks to the coffee market?
