Unformatted text preview: Middle East Technical University Department of Economics ECON 201 Halis Akder (01), Erol akmak (02) TA: zlem Tongu Answers to Problem Set 2 1 Fall 2008 a) No, because if the consumer consumes only x1 or only x2, he will have zero utility. Consuming from each such that , 0 results in a utility greater than zero, so the consumer has no reason to consume only one of the goods. b) Our problem is UMP, i.e. max . , . We set up the Lagrangian: , , , , The first order conditions are: (1) 0 (2) (3) 0 0 , and plug this into (3): 3 2 . Hence the consumer spends 40% of his income on From (1) and (2) we derive Then, we have good 2. 2 5 c) The Marshallian demand for good 2 is demand for good 1 is d) From the f.o.c., we found
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. . The Marshallian , , . Thus = . . . represents the marginal utility of income in UMP, i.e. the marginal . change in utility when income is changed by one unit. e) The income expansion path is derived already from the f.o.c. and is f) The Engel curve is derived from the Marshallian demands of each good, by taking the variables other than income as constants. So the Engel curve for good 1 is , and for good 2 it is . g) The priceconsumption curve is derived from the Marshallian demands of each good, by taking the variables other than the good's own price as constants. So the PC curve for good 1 is . 2) The consumer considers coffee and sugar as complements, so we have a utility function in the form , {c, /2}. Thus, we cannot derive the demands using the Lagrange method. Graphical analysis is required. We see that the optimum consumption bundle is along the line 2 . Thus we may rewrite the budget constraint as 2 So, c* = and s* = . 3) We need to do graphical analysis again. It can be seen from the graph that the consumer will want to consume at a corner point, i.e. at one of the intercepts. Therefore she will consume only one of the goods. Whether she consumes good 1 or good 2 depends on the relative price of the goods. So we have a partially defined demand function for the goods. From the demand correspondence, we can see that both of the goods are normal. ,0 , , , 0, 3 3 3 , , The income expansion path is partially defined, too: ,0 , , , 0, , 3 3 3 , 4) max , . , , The first order conditions are: (1) 0 (2) (3) 0 . We set up the Lagrangian: , , 0 0.5/ 0.5/ By substitution, we find the Marshallian demands as 1 , , , So the expenditure function at is , 1 This result is not surprising, of course, since we have set our constraint as . What if we have set the constraint as ? In order to answer this question, let's consider the utility function of the consumer: 0.5 0.5 0 0 0 0 which implies that the utility of the consumer increases as her consumption of goods increases (albeit at a decreasing rate, but that's not important). So, even if we set the budget constraint as , we know that the consumer will spend all of her . Knowing this, we income, thus consuming at a point that satisfies can say that if we increase the income of the consumer, she will be better off. Conversely, if her income is decreased, her well being will be reduced. 5) Our problem is EMP, i.e. min , . Lagrangian: , , 8 The first order conditions are: (1) 0 (2) (3) 8 0 0 , , . We set up the From (1) and (2) we derive 8 Then, we have Similarly, we have If we plug in the values 149.9. , and plugging this into (3), we get: 8 . . 20, 15, we find 200.1 and , where , where 1385, The minimum expenditure is E* = 6250.5 represents the marginal cost of utility, i.e. the change in the expenditure function, when the given utility changes by one unit. From the f.o.c. we already found that 6) a) b) max , (1) (2) (3) , 1 1 0 0 0 . Marshallian (uncompensated) demands: 1 , , 2 1 , , 2 c) min , (4) (5) (6) , 1 1 0 0 0 . 1 1 By substitution into (6) we find the following expression for the Hicksian (compensated) demand of good 1: 2 0 0 Then, , , , assuming that , , 0. 7) If one of the goods is inferior, say good 1, then it means that if the income of the consumer increases, the amount of good 1 consumed decreases (we know this from the first derivative of Engel curve being negative for inferior goods, i.e. a negative relationship between consumption and income). However, assuming that the consumer has a monotonic utility function (i.e. utility increases as the consumption increases), and he consumes at the optimal consumption bundle after the increase in income, then a higher utility is now attained by the consumer. But this means the consumer must indeed be consuming more of good 2, because he is consuming less of good 1 as his income increased. Hence, we have a positive relationship between consumption of good 2 and income. Therefore, good 2 cannot be inferior given good 1 is inferior. 8) Consider the UMP max , . . , , and From the f.o.c., we find the Marshallian demands as , , . For homogeneity, we check the value of , , , , , , , , , for the following equation: , which implies that the Marshallian demands are homogenous of degree 0. Graphically, the changes in income and prices correspond to a change in the budget set via the budget line. Since both the prices and income double, the budget line remains as before. Given the same constraint, the consumer's demand functions remain as before, too. 9) From the figure we can see that unlike the perfect substitutes case, the consumer prefers to consume both of the goods (interior solution) instead of consuming only one of them (corner solution). Similar to the perfect complements case, we cannot use the Lagrangian conditions to determine an interior solution, since at the kink, the utility function is not differentiable. Notice that these type of utility functions are characterized by , Actually, it can be said that perfect complements case (where special case of these type of utility functions. 0 , . 0 is a ...
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 Spring '10
 çAKMAK
 Microeconomics, Utility, Marshallian, income expansion path, Marshallian demands

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