DSC3707 Sol Ass01 S2-2022_e969c1389271e9024ecd3a45edaf7ddb

DSC3707: Solutions for Assignment 01, Semester 2, 2022 Question 1 (a) Suppose that the market for a commodity is governed by supply and demand sets defined as follows. The supply set S is the set of pairs ( q, p ) for which 5 p - 8 q = 60 and the demand set D is the set of pairs ( q, p ) for which p + 2 q = 30 , where the price p is in rands per unit quantity q . Sketch S and D and determine the equilibrium set E = S ∩ D , the supply and demand functions q S , q D , and the inverse supply and demand functions p S , p D . (Answer without wxMaxima, but you can check your answer with wMaxima: Choose [Equations], then [Solve algebraic system]. Solution: Supply Function: q S ( p ) = 5 p - 60 8 = 5 8 p - 15 2 . Demand Function: q D ( p ) = 30 - p 2 = 15 - 1 2 p. Inverse Supply Function: p S ( q ) = 60+8 q 5 = 12 + 8 5 q. Inverse Demand Function: p D ( q ) = 30 - 2 q. Equilibrium occurs when supply is matched by demand. Hence q S ( p ) = q D ( p ) , and we solve the equation: - 15 2 + 5 8 p = 15 - p 2 . This gives p = 20 , and q = 15 - 20 2 = 5. ± 2 2 0 1 5 5 0 3 0 2 G S u p p l y : G = F - 5 8 1 5 2 D e m a n d : G = 1 5 - F 1 2 e = S ∩ D = { ( p ; q ) } = { (20; 5) } (b) Suppose that the government decides to impose an excise tax of T rands on each unit of the commodity in (a). What price will the consumers end up paying for each unit of the commodity?

Solution: Supplier sees the price as p - T . Hence, to get new equilibrium price we solve 5 8 ( p - T ) - 15 2 = 15 - p 2 . This gives p = 20 + 5 9 T , which is the price the consumer pays. (c) Find a formula for the amount of money the government obtains from taxing the commodity in the manner described in (b). Determine this quantity explicitly when T = 7 , 20. Solution: The new equilibrium value for q is bracketleftBigg 15 - ( 20 + 5 9 T ) 2 bracketrightBigg = 5 - 5 18 T. So the government revenue is q T = parenleftbigg 5 - 5 18 T parenrightbigg T = 5 T - 5 18 T 2 In T = R7 , 20 = 7 1 5 = 36 5 , the revenue is 5 ( 36 5 ) - 5 18 ( 36 5 ) 2 = R21 , 60 . Question 2 Suppose that the supply and demand sets for a particular market are S = { ( q,p ) | p - 8 q = 24 } , D = { ( q,p ) | p + q 2 + 2 q = 99 } , where q is the quantity of units sold, and p is the price in rands per unit. (a) Sketch S and D and determine the equilibrium set E = S ∩ D . (wxMaxima: Choose [ Plot2D] . Choose [Equations] , then [Solve algebraic system] .) Solution: We have p D ( q ) = 99 - q 2 - 2 q and p S ( q ) = 24 + 8 q 2 4 6 4 9 5 0 9 9 2 G

Solution: After 15 years the amount in the savings account is bracketleftbig 50 000(1 , 08) 8 bracketrightbig [exp(7 × 0 , 05)] = 50 000(1 , 8509302)(1 , 4190675) = R131 329 , 75 Question 4 Suppose that the demand function for a good is q D ( p ) = 9 000 p 2 + 2 , where q is the quantity and p is the price in rands. If the price is decreased from R9 to R8,50, determine, using a calculus method, the approximate increase in the quantity sold. Compare your answer with the answer obtained by using the obvious method of simple substitution. (Answer without using wxMaxima, but you can then check your answer using wxMaxima.) Solution: Using the chain rule: dq D dp = 9 000 parenleftbigg - 1 ( p 2 + 2) parenrightbigg (2 p ) = - 18 000 p ( p 2 + 2) 2 Hence Δ q D ≈ dq D dp | (8 , 50 - 9 , 00) p =9 NB: The symbyl A | p =9 means that A is evaulated at p = 9 = - (18 000)(9) (9 2 + 2) 2 ( - 0 , 5) ≈ 11 , 76 or 12 . Hence q D increases by approximately 12. “Exact” Calculation: q D (9) = 9 000 83 = 108 , 43373 q D (8 , 50) = 9 000 8 , 5 2 + 2 = 121 , 21212 The increase is approx = 12 , 778391 or 12 , 78 ≈ 13. The first calculation underestmates the increase by a small amount. Question 5 Suppose that the supply and demand sets for a certain good are S = { ( q,p ) | 2 p - 3 q = 19 } , D = { ( q,p ) | 2 p + q = 23 } , and suppliers operate according to the cobweb model. That is, if p t and q t are (respectively) the price and quantity in year t , then p t = p D ( q t ) and q t = q S ( p t - 1 ). Suppose also that the initial price is p 0 = 9. (a) Derive a recurrence equation for p t , and solve the recurrence equation.

Solution: First, p D ( q ) = 23 - q 2 , and q 6 = q 6 p = - 19+2 p 3 . Hence P t = p D ( q t ) = 23 - q t 2 = 23 - q S ( p t - 1 ) 2 = 23 - bracketleftBig 2 p t - 1 - 19 3 bracketrightBig 2 = 1 2 parenleftbigg 69 3 + 19 3 - 2 3 p t - 1 parenrightbigg = 88 6 - 2 6 p t - 1 Hence p t = 88 6 - 1 3 p t - 1 = 44 3 - 1 3 p t - 1 The time independent quantity p * satisfies p * = 44 3 - 1 3 p * . Hence p * = 11 . p t = p * + ( p 0 - p * ) parenleftbigg - 1 3 parenrightbigg t = 11 + (9 - 11) parenleftbigg - 1 3 parenrightbigg t = 11 - 2 parenleftbigg - 1 3 parenrightbigg t (b) How does p t behave as t tends to infinity? Solution: Since ( - 1 3 ) t → 0 as t → ∞ , we get p t → 11 as t → ∞ . (c) How does q t behave as t tends to infinity? Solution: p t = 23 - q t 2 Hence q t = 23 - 2 p t Hence, as t → ∞ , q t → 23 - 2(11) = 1 . (d) Use your results to determine S ∩ D , and then check the correctness of your answers for (b) and (c). Solution: The equilibrium price p is 11, and equilibrium quantity q is 1.