Solutions to Assignment #2

Transcription

1 ECON 20 (Fall 207) Department of Economics, SFU Prof. Christoph Lülfesmann exam). Solutions to Assignment #2 (My suggested solutions are usually more detailed than required in an I. Short Problems. The consumer s reservation price for a good, is the maximum price at which she - for given bundle (x, x 2 ) - would be willing to buy the next (marginal) unit of good. Formally, RP (x, x 2 ) is the price p for which - given x, x 2, and p 2 - the tangency condition MRS(x, x 2 ) = p /p 2 just holds: the consumer demands bundle (x, x 2 ) in this situation and price p is the reservation price of the last unit of good- consumption. Note that for any smaller price of good, the absolute slope if the indifference curve (i.e., the MRS) at (x, x 2 ) is bigger than the price ratio, which represents the slope of the budget line. In other words, for any such price the consumer would strictly prefer more consumption of good. The reverse argument applies for any price larger than p. Hence, p cannot be the reservation price of unit x in these situations. Taken together, the maximum price the consumer is willing to pay for the next unit is the price at which he is indifferent between trading x for x 2 or vice versa - and this is the price for which the tangency condition just holds. (Note also that because for strictly convex preferences the MRS falls in x, RP (x, x 2 ) falls in x as well.) 2. In May, Mei s income is 26 dollars. This means that the bundle (0, 5) which she consumed in April remains affordable, at a total expense of = 25. Accordingly, if Mei s preferences have not changed in the meantime and if they satisfy non-satiation, she will be better off in May because she can afford one additional tulip. Of course, she would have been better off in April if he had consumed a bundle on (and not beneath) her

2 budget line, but this does not change the argument because she actually did not consume the two additional roses. 3. This is false. The reservation price of a good is equal to the utility of the next unit of consumption, set in relation to the marginal utility - per dollar spending - the consumer would obtain from consuming the other good instead. Formally, this means that the reservation price of good, say, at a certain consumption level is RP = MU (x, x 2 ) MU 2 (x, x 2 )/p 2. The statement is correct only in the special case of quasilinear utility (i.e., MU 2 = ), and if p 2 =. 4. This is the demand function for Cobb-Douglas preferences. One specific representation of these preferences would be, e.g., the function U(x, x 2 ) = x /3 x 2/3 2. For this functional form, the consumer really spends /3 of this income on good (and 2/3 on good 2), as required by the given demand function. The price elasticity of demand is ϵ = x p = p x 3 m p 2 p (/3)m/p =. For the C-D utility function, a per cent raise in price reduces demand for a good by per cent. Why? The total spending on good, say, p x, is fixed for these preferences (recall the demand function), Accordingly, p x = c where c is some number. This means that p dx + x dp = 0, which translates into dx /dp = p /x, i.e., ϵ =. (Note: this formal argument was not required, of course). 5. This is the Law of Demand. Please consult our notes for an answer. 6. To graph the aggregate demand (which I will not do forgive me), notice that for prices in the range [30, 00] Tom is the only person to demand 2

3 ice cream. Hence, aggregate (inverse) demand in this price range is p = 00 Q. Since the slope of this demand function is, we have Q = 70 at the lower boundary p = 30 this is where a kink in demand occurs. Consider now prices lower than 30. At p = 0, total demand is Q(p = 0) = = 60. Accordingly, for prices below 30, aggregate demand is a straight line, starting at Q(30) = 70 and ending at Q(0) = 60. What is now the functional form for this aggregate demand? Answer: it is piecewise defined: aggregate demand reads Q = 00 p for p 30; and Q = q J + q T = 60 3p for p < 30. Translating into inverse demand, this means p(q) = 00 Q for Q 70, and p(q) = 60/3 Q/3 for Q > 70. II. Long Problems Problem a) Substituting the budget constraint x 2 = 00 4x into the utility function, Tom chooses his consumption of pop tarts (i.e., x ) in order to maximize U = 0x x x. () Setting the derivative with respect to x equal to zero (remember, his utility is maximized at a point where the utility function has zero slope), we immediately obtain 6 x = 0, i.e., x = 6. Inserting into the budget constraint then yields x 2 = 76. b) The consumer surplus from good is the utility which Tom derives from the consumption of pop tarts. Since in our example the utility function is quasilinear (there are no income effects) and the price of the other good is normalized to one, the CS is formally represented as the area underneath Tom s demand function for pop tarts. This demand function x (p ) can, e.g. be computed using the tangency condition for 3

4 utility maximization, which in the present case is 0 x = p Recall that the price which solves this equation for a given x is Tom s reservation price for the x -th unit of good. Solving for p, we obtain the inverse demand function p = 0 x which can easily be plotted. Gross consumer surplus is the area beneath this demand function, summed up over all quantities x 6. We obtain CS gross = [0 4]6(/2) + 24 = 42, while the net consumer surplus is obviously CS net = CS gross p x = = 8. Notice that Tom s total utility is thus equal to 8, the sum of his gross CS from pop tarts, and his consumption = 76 of fitness classes. We can check that this number is correct by substituting the numbers for x, x 2 into Tom s utility function. c) If p 2 = 4, Tom s demand function (again using the tangency condition) becomes 0 x = p 4 Again solving for x yields the demand function x = 0 p /4 and, for p = 4, x = 9 and x 2 = (00 36)/4 = 6. Substituting these numbers into Tom s utility function, we find that his CS from pop tarts is /2 = 99/2, which is higher than before because the increase in p 2 led Tom to consume more pop tarts. His utility from consumption of fitness classes is 6, and his overall utility is as expected, the price increase reduced Tom s utility. Importantly, we cannot anymore compute Tom s CS from pop tarts by calculating the area underneath his inverse demand curve. To see this, solve the demand function for p to obtain p = 40 4x : while it is true that his reservation price for the say, st (small) unit of pop tarts has increased to 40 now, it is not true 4

5 that this increase in the reservation price raises his actual utility from consuming the first (or any other) unit. Rather, the inverse demand curve now shows values that are four times the marginal utility from good. This finding confirms that the area underneath the demand function is a measure for the CS only if the price of the other good (here good 2) is normalized to unity. d) Note that Tom s demand function for pop tarts (as given by the tangency condition) is x = 0 p /p 2. It follows that his optimal consumption does (for an interior solution which we have here) not dependent on his income m. Accordingly, the income change does not affect his optimal consumption of pop tarts, which remains x = 6. e) You should explain here that because of quasilinear preferences (and since we have an interior solution for x ), the substitution effect of the price increase is negative, whereas the income effect is zero - see also our arguments in class. Problem 2 a) In an equilibrium, the demand price must be equal to the supply price. Accordingly, 2q = 3q and solving yields the equilibrium quantity q = /5. Finally, Substituting into P D (or P S ) yields the equilibrium price p = 2/5 = 3/5. b) (Net) Consumer surplus is the area below the (inverse) demand curve, and above the equilibrium price. In words, it is here the aggregated difference between a consumer s reservation price for a unit of the good, and the price he is paying for it. Similarly, producer surplus is the area between price and supply curve, i.e., the aggregate difference between 5

6 sales price and costs per unit. (Note: these verbal explanations were not really required here). Now on to the computations (you ll find all graphical results on a separate solution sheet). size Producer surplus is Consumer surplus is a triangle with an area CS = [ 3 5 ] 5 2 = 2 5 P S = = = c) If the government imposes a quantity tax of t = /0, this drives a wedge between demand and supply price. Specifically, we have P D = P S + /0 here. The new equilibrium is given by the condition P D = 2q = P S + t = 3q + /0, and solving yields the new equilibrium quantity q = 9/50. Substituting, we obtain P S = 27/50 and P D = 32/50. d) We have CS = [ ] = = Producer surplus is P S = = Note that in equilibrium 40 per cent (or 32/50 30/50 = 2/50 in absolute terms) of the tax burden of /0 per unit is borne by consumers, and 60 per cent of the tax burden (30/50 27/50 = 3/50) is borne by producers. Geometrically, the deadweight loss (DW) is of triangular shape. Specifically, it encompasses the areas between the old and new quantity, below the original demand and above the original supply curve. Formally, it is given as (I find it easiest to divide the total triangle into two small 6

7 triangles, with the original equilibrium price as dividing line. Then, I compute the area of each small triangle and sum up) DW = = 000. What is the meaning of the DW? Suppose the government were to spend the entire tax revenue on projects from which each citizen gets a utility exactly identical to the amount of her tax payment (or, alternatively, the government simply returns the money). Then, DW represents the social loss of taxation to society. 7