EconS Intermediate Microeconomics without Calculus Set 2 Homework Solutions

Transcription

1 Econ - Intermediate Microeconomics without Calculus et Homework olutions Assignment &. Consider the market for football tickets. It faces the following suly and demand functions = + = 8 + Y + B where is the rice for football tickets, Y is average income in ; s and B is the rice of basketball tickets. Let Y = and B =. (a) Plot both the suly and demand curve. First, we should substitute our values for Y = and B = = + = 8 + () + = 8 Next, solve both functions for to get the inverse suly and inverse demand functions = + = and we can lot these two functions to obtain * *

2 (b) In euilibrium, = and we can relace it with just. ince both inverse demand functions are solved for, we can just set them eual to each other. + = Multilying both sides by to get rid of the fractions + = = = and substituting = into the suly function gives our euilibrium. = + () = (c) Calculate the rice elasticity of demand, income elasticity, and cross-rice elasticity at the euilibrium rice and uantity. tarting with the rice elasticity of demand, let s look at the original demand curve. = 8 + Y + B We know that the rice elasticity of demand is eual to " = The coe cient of is, and we can substitute that value for. Furthermore, we solved for = and = from the last art. ubstituting these values in gives " = = which imlies that the demand for football tickets is uite elastic. Next, we look at the income elasticity which is eual to = Y The coe cient of Y is, and we can substitute that value for. Using the Y euilibrium values and Y = as given in the roblem, Y = = which imlies that football tickets are a normal good. Lastly, we look at the cross rice elasticity which is eual to " B = B B

3 The coe cient of B is, and we can substitute that value for B. Using the euilibrium values and B = as given in the roblem, " B = = which imlies that football tickets and basketball tickets are substitutes. (d) I exlained the elasticities in art (c), but yes, they do make sense. Peole should demand more football tickets as their incomes rise, and substitute them for basketball tickets. Furthermore, football games are not a necessity, and can be relaced uickly if the rice rises too much, which exlains the rice elasticity of demand. Assignment. Consider the following suly and demand functions we used in assignment from the last roblem set (You can refer to that for some values you might need later) = + = (a) The government decides to charge a er unit tax of t = to the buyer. What is the new euilibrium rice and uantity? Adding the tax to the buyer, our e ective demand function becomes = ( + ) = = 8 olving for inverse suly and demand curves, = + = = E which gives us the following gure * E *

4 etting uantity sulied eual the the uantity e ectively demanded, = + = 8 = 9 = which gives a rice received by the seller of and a rice aid by the buyer of + =. olving for euilibrium uantity, = + = + = (b) How much tax revenue does the government bring in? How much of that revenue is aid by the buyer? The seller? Udating our gure, Buyer eller We can calculate the total tax revenue by alying our formula, T R = t = () = and thus the government brings in $ with this tax. To gure out the burden on both the buyer and the seller, we need the original euilibrium rice from the undistorted market,. Then, using the gure, we just calculate the area of each of the above rectangles. Buyer = ( )() = eller = ( )() = and thus the buyer bears $ of the tax while the seller bears $ of the tax. (c) What are the Consumer urlus, Producer urlus, Welfare, and ead Weight Loss levels due to this tax?

5 Udating our gure, C TR WL P We can use triangle formulas to nd the consumer and roducer surluses, Adding u total welfare, C = (MAX ) = ( )() = P = ( MIN ) = ( )() = W = C + P + T R = + + = 9 And using the triangle formula for dead weight loss, W L = ( )( UN I ) = ( )( ) =

6 Assignment. Consider the gure given below. Indi erence curve I asses through oints A, B, and C. Ales I A B C Oranges (a) Judging by the curvature of the indi erence curve, what tye of references does this erson have? ince the indi erence curve is a erfectly straight line (no curvature), this erson views ales and oranges as erfect substitutes. (b) What is the marginal rate of substitution from oint A to oint B? Calculating the marginal rate of substitution, MR = Change in Oranges Change in Ales = = = which imles that this erson would be willing to give u one ale to gain another orange. (c) What is the marginal rate of substitution from oint A to oint B? Calculating the marginal rate of substitution, MR = Change in Oranges Change in Ales = = = which imles that this erson would be willing to give u one ale to gain another orange. (d) What do the results from arts (b) and (c) imly about the marginal rate of substitution for this kind of references? With erfect substitutes, the marginal rate of substitution will always be the same no matter which bundle the erson selects.

7 Assignment. Consider the following utility function where x and z are both goods. U = x : z : (a) What are the marginal utilities with resect to x and z? Taking the ower rule for both x and z MU x = :x : z : MU z = :x : z : (b) What is the marginal rate of substitution? (Remember to combine the exonents!) Using the formula, MR = MU x = :x : z : z = MU z :x : z : x