This **preview** has intentionally **blurred** parts. Sign up to view the full document

**Unformatted Document Excerpt**

401 ECON Section: Supply and Demand With Taxes Comparative Statics Sarah E. Taylor University of Michigan, Department of Economics January 10, 2011 In this set of notes I will pose to you some sample questions on taxes and some solution techniques for solving them. Then I will give you a detailed example. 1 How does the equilibrium price of consumers and suppliers change as a tax changes? Suppose that the price paid by consumers is pD and the price received by suppliers is pS . In addition, suppose we have a tax on the sale of a good such that the after-tax price received by suppliers is pS = pD .1 In lecture, we refer to the price paid by consumers as the equilibrium price. Suppose we want to nd dpD and dpS . d d 1.1 Explicitly 1. QS = QD and pD = pS Set supply equal to demand in equilibrium and the after-tax price paid by consumers equal to the price received by suppliers. These are your two equilibrium conditions. 2. Substitution 1 This solution technique can similarly used to solve for a tax on consumers; the equation relating the price paid by consumers and the price paid by suppliers will be the same. 1 Substitute in pS = pD into the supply side of the equation setting supply equal to demand. You will be left with an equation which has in it pD and , as well as any other exogenous variables. 3. Solve for pD in terms of , pD ( ) Isolate pD on one side of the equation setting supply equal to demand. 4. Solve for dpD d Take the derivative of the function you found in the last step. 5. Solve for dpS d Taking the fact that pD = pS and dierentiating both side we get that dpD 1 = dpS . Thus we can simply substitute the value we found in the last d d step into dpS = dpD 1 to get dpS . d d d 1.2 Implicitly 1. QS = QD and pD = pS Set supply equal to demand in equilibrium and the after-tax price paid by consumers equal to the price received by suppliers. These are your two equilibrium conditions. 2. dQS d = dQD d and dpD d 1= dpS d Take the derivative of both sides of the two equations with respect to . Remember that we have to apply the chain rule when we take the derivative of supply and demand with respect to . dQS d = QS dpS ( d ) p = QD dpD ( d ) p = dQD d 3. Substitution S D Substitute dpS = dpD 1 into Q ( dpS ) = Q ( dpD ). This will allow us to rst d d p d p d solve for the eect of the tax on the price of consumers. Now we have QS dpD ( d p 1) = QD dpD ( d ) p 2 4. Substitute in values The next step is to substitute in values for the partial derivatives using the original quantity supplied and demanded functions. Remember that now in the price the supply function is equal to pS and the price in the demand function is equal to pD . This will yield an equation of only dpD and constants. d 5. Solve for dpD d dpD d will enter into both sides of the equation. Solve for one side of the equation. 6. Solve for by isolating it on dpS d We can use the fact that 2 dpD d dpS d = dpD d 1 to then easily solve for dpS . d Example Consider the following demand and supply functions for cupcakes, where po is the price of cookies: QD = 100 2p + po and QS = 5p. Suppose we have placed a tax on the sale of cupcakes. What is the eect of a 1 dollar increase in the tax on the consumer price of cupcakes in equilibrium? What is the eect of a 1 dollar increase in the tax on the supplier price of cupcakes in equilibrium? As a further challenge, what is the eect on the equilibrium quantity of a 1 dollar increase in the tax? 2.1 Solution: dpD d and dpS d I will show you how to solve for this explicitly and then implicitly, using the solution techniques outlined above. Explicitly First we set supply equal to demand 100 2pD + po = 5pS . Then taking the fact that pS = pD and substituting, we have: 100 2pD + po = 5(pD ). Solving for pD in terms of we get: 100 7pD + po = 5 7pD = 100 5 po pD = 100 1 + 5 7 po . 7 7 Now we can dierentiate pD with respect to , yielding 3 dpD d = 5. 7 Now taking the fact that pS = pD and dierentiating we get that dpD 1 = 2 d 7 dpS d = Implicitly First we take our two equilibrium conditions: QS = QD and pD = pS . The next step is to dierentiate both equations with respect to : (1) dQS d = QS dpS ( d ) p = QD dpD ( d ) p = dQD d And, (2) dpD d 1= dpS d Now, substituting equation (2) into (1), we can transform equation (1) into an equation containing dpD and partial derivatives, which we can easily solve for: d QS dpD ( d p 1) = QD dpD ( d ) p Let us now solve for the partial derivatives: these values in, we get QS p D = 5, Q = 2. Substituting p 5( dpD 1) = 2( dpD ) 5( dpD ) 5 = 2( dpD ) d d d d Now taking equation (2), 2.2 Solution: dpS d = dpD d dpD d = 5 7 1 = 2 7 dQ d Again, we may solve for the change in the equilibrium quantity with the change in the tax using either quantity supplied or quantity demanded in equilibrium. Quantity Supplied 4 dQ d Now substituting in for QS p dQ d dQS d = = 5 and = dQS d dQ d = = dpS d QS dpS p d 2 = 7 , we have = 5( 2 ) = 10 . 7 7 Quantity Demanded Now substituting in for QD p dQ d dQD d = 2 and = dQD d = QD dpD p d dpD d = 5 , we have 7 = 2( 5 ) = 10 . 7 7 5 ... View Full Document