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Econ 20002_2022_SM1. Tutorial 4. Consumer & Producer Theory T4sol. Page 1 of 5 ECON 20002 Intermediate Microeconomics Tutorial 4 solutions. Consumer & Producer Theory Problem 1. Labour – leisure choice and taxation. Problem 2. Elmo reveals his preferences (adapted from Varian). a) The budget lines and points chosen. b) Visit A: A is revealed preferred to C, D, E – as they all were available during visit A, when A was chosen over any of them. We don’t know about B – B was not available when A was chosen. Visit B: B is revealed preferred to C, D, E – as they all were available during visit B, when B was chosen over any of them. We don’t know about A – A was not available when B was chosen. QTT Qoreos 20 0 10 20 30 40 10 30 40 A E B C D Econ 20002_2022_SM1. Tutorial 4. Consumer & Producer Theory T4sol. Page 2 of 5 Visit C: C is revealed preferred to D and E – as they were available during visit C, when C was chosen over any of them. We don’t know about A and B – A and B were not available when C was chosen. Visit D: can’t say anything as A, B, C and E were not available when D was chosen. Visit E: can’t say anything as A, B, C and D were not available when E was chosen. There are no contradictions – A is revealed preferred to C, D, E; B is revealed preferred to C, D, E; and C is revealed preferred to D and E. c) Worse than C. C is RP to all the bundles that were available during visit C (budget set triangle for this visit). D is RP to all the bundles that were available during visit D (corresponding budget set triangle for this visit); C is RP to D, so due to transitivity, C is better than all the bundles in budget set during visit D. Same for E. d) Better than C. “More is better”: all bundles that have at least as much of both goods and more of at least one good than bundle C, are better than C. All bundles that have at least as much of both goods and more of at least one good than bundle A, are better than A and therefore, are better than C (transitivity). Same for B. What about convexity Consider bundles on a straight line between C and A. If Elmo had been indifferent between A and C, than all these mixtures of A and C would have been better QTT Qoreos 20 0 10 20 30 40 10 30 40 A E B C D QTT Qoreos 20 0 10 20 30 40 10 30 40 A E B C D Econ 20002_2022_SM1. Tutorial 4. Consumer & Producer Theory T4sol. Page 3 of 5 for him than either A or C (that’s what convexity means – mixtures are better than corners/extremes). But A is better for Elmo than C, so when Elmo “mixes up” C with A (i.e. chooses bundle on a straight line between them), he will end up with a better bundle than C. Same for B. Therefore, all the bundles on the straight line between A and C are better than C; all the bundles that are better than these mixtures due to “more is better” are also better than C. Same for B. e) Where are all the bundles such that Elmo is indifferent between them and the bundle C They must be located in the uncoloured area – you can draw an indifference curve there. Problem 3. Production Technology a) You can produce 10 units by using any one technology – that gives you L-shaped isoquant for each particular technology. Let’s call the kink points A, B and C. If you can choose technology, but have to use only one, you will choose the best one given how much labour/capital you have and will end up with step isoquant. If you combine A and B; and B and C, then connecting lines between them become part of an isoquant. If you combine A and C, the straight line lies above AB and BC segments – so it is not part of an isoquant – given that much labour and capital, you can produce more than 10 units either using A&B or B&C. Or, in other words, you can use less labour and capital to produce 10 units using technologies A&B or technologies B&C, and use the rest to produce more. Combining all three is not efficient either – you can always do at least as well by using A&B or B&C. Even though, for each plant, labour and capital are perfect complements (there is no substitution between the two), if we have several plants that allow us to combine different fixed-proportion technologies, we end up with downward sloping isoquant – labour and capital are substitutes. When number of technologies increases, you end up with a smooth convex isoquant. QTT Qoreos 20 0 10 20 30 40 10 30 40 A E B C D Econ 20002_2022_SM1. Tutorial 4. Consumer & Producer Theory T4sol. Page 4 of 5 b) From graph you can see that your best choice is to combine technologies B and C. Assume you produce proportion x of 10 units using Tech. B and (1-x) using Tech. C. Then you need for Tech. B: L = 8x, K = 5x to produce x*10 units. For Tech. C: L = 15(1-x), K = 2(1-x) to produce (1-x)*10 units. For example, if you want to produce 30% of q = 10 using Tech. B, then you need to use 30% of labour and capital used to produce 10 units as it is a fixed-proportions technology (i.e. use 0.3*8 = 2.4 labour and 0.3*5 = 1.5 capital; x = 0.3). Total labour available = 8x + 15(1-x); we have 11.5, so 8x + 15(1-x) = 11.5 gives us x = 0.5, i.e. we want to produce 5 units with Tech.B and 5 units with Tech.C. Let’s check if we have enough capital to do it: we need 5x + 2(1-x) = 5*0.5 + 2*0.5 = 3.5, so we have exactly correct amount of capital. The input bundle is on the isoquant q=10. Produce half (5 units) using each of technologies B and C. So, we can produce 5 units using Tech. B with L = 4, K = 2.5 and 5 units using Tech. C with L = 7.5, K = 1. We have exactly enough labour and capital to be able to do it. The input point on the graph is exactly in the middle of BC. Econ 20002_2022_SM1. Tutorial 4. Consumer & Producer Theory T4sol. Page 5 of 5 For curious math lovers: if you are changing proportion x from 0 to 1, you will get all the points on the straight line connecting B and C. x = 0 corresponds to point C; x = 1 to B. Labour = 8x + 15(1-x) = 15-7x and capital 5x + 2(1-x) = 2 + 3x with give you horizontal and vertical coordinates. c) Same as b), but x*10 using Tech. A and (1-x)*10 using Tech. B. For Tech. A: L = 2x, K = 10x For Tech. B: L = 8(1-x), K = 5(1-x) labour: 2x + 8(1-x) = 7.8. That gives us x = 0.033, i.e. we want to produce about 0.33 units with Tech.A and 9.67 units with Tech.B. Let’s check if we have enough capital to do it: we need 10x + 5(1-x) = 10*0.033 + 5*0.967 = 5.165, but we have much more capital – 7.5. The input bundle is above the isoquant q=10, it is on a higher isoquant. With these inputs, we will be able to produce more than 10 units. Extra ( for curious future Honours): how much can we produce Assume we can produce y units of output by producing proportion x of them with Tech.A and proportion (1-x) with Tech.B. Then we need: For Tech. A: L = 0.2xy, K = xy For Tech. B: L = 0.8(1-x)y, K = 0.5(1-x)y. We have L=7.8, K=7.5. Therefore, using all labour and capital: 0.2xy + 0.8(1-x)y = 7.8 xy + 0.5(1-x)y = 7.5 we obtain x = 0.25 and y = 12. (to solve the above system by hand, you can, for example, write it as: xy + 4(1-x)y = 39 or 4y – 3xy = 39 2xy + (1-x)y = 15 y + xy = 15 and then get rid of xy by multiplying the second equation by 3 and adding to the first one to obtain (4+3)y=39+15*3 => 7y = 84 => y = 12. Substituting it into the second equation, we obtain 12(1+x) = 15 => x = 0.25. We can produce 12 units of output by producing 25%, or 3 units, with Tech.A and the rest, 9 units, with Tech.B. We will employ L = 0.6, K = 3 in Tech.A and L = 7.2, K = 4.5 in Tech.B.