age. If he just undertakes this and project A, then he can consume $850,000 now and S340.000 later on Project c and so on There are a number of other projects he thinks of. We can trace out a curve to approximate these, e.g., see above With projects alone he can consume anywhere along the curve Proiects and the bank We now consider what he can do if we take account of both his projects and the ossibility of borrowing and lending at the bank Project A Suppose he just undertakes the first project, A. He has $950,000 now and $200,000 later on--what can he do? (i He could simply consume $950,000 now and $200,000 later on without going to the bank (i) Alternatively, he can use all the money he receives to have a marvelous time He has $950,000 now and $200,000 later on, which he can use to repay a loan that he spends now. How much can he borrow to repay with the $200,000 later on; i.e., what is the present value of $200,000 at 20%? PV(200,0000at20%)=200.000=166666 Total possible in consumption youth= 950,000+ 166, 666 $1.1167M 6
6 age. If he just undertakes this and project A, then he can consume $850,000 now and $340,000 later on. Project C and so on There are a number of other projects he thinks of. We can trace out a curve to approximate these, e.g., see above. With projects alone he can consume anywhere along the curve. Projects and the Bank We now consider what he can do if we take account of both his projects and the possibility of borrowing and lending at the bank. Project A Suppose he just undertakes the first project, A. He has $950,000 now and $200,000 later on--what can he do? (i) He could simply consume $950,000 now and $200,000 later on without going to the bank. (ii) Alternatively, he can use all the money he receives to have a marvelous time. He has $950,000 now and $200,000 later on, which he can use to repay a loan that he spends now. How much can he borrow to repay with the $200,000 later on; i.e., what is the present value of $200,000 at 20%? PV(200,000OA at 20%) = 200,000 = 166,666 1.20 Hence, Total possible in consumption youth = 950,000 + 166,666 = $1.1167M
OA 1.34M 2M () 1.1167M (iii) Alternatively he can plan to spend it all next period Total in old age= 1.2x.000+200.000 $1,340,000 As before, we can go on doing this until we trace out a straight-line budget constraint as before. The equation for this line is COA=1.34M-1.2C￥ The slope is again.2 since the interest rate is 20%. You can see that this follows from the structure of the problem since all values in Old Age are multiplied by 1.20 compared to their values in youth Project B Now suppose he undertakes the first and second projects, A and B, so that he produces at A+B. We can go through the same calculations again and get another line representing his consumption possibilities
7 (iii) Alternatively he can plan to spend it all next period. Total in old age = 1.2 x 950,000 + 200,000 = $1,340,000 As before, we can go on doing this until we trace out a straight-line budget constraint as before. The equation for this line is COA = 1.34M - 1.2 CY The slope is again - 1.2 since the interest rate is 20%. You can see that this follows from the structure of the problem since all values in Old Age are multiplied by 1.20 compared to their values in Youth. Project B Now suppose he undertakes the first and second projects, A and B, so that he produces at A + B. We can go through the same calculations again and get another line representing his consumption possibilities
OA 136M 134M A+B A Y In this case, Intercept on COA axis=1.2 X 850,000+ 340,000=136 M Hence, analytically the budget constraint is given by CoA=1.36M-1.2C We can see that by undertaking the first project he can push out the line representing his possible consumption, similarly when he undertakes the second project, and so on. If he prefers more money to less, he is better off if his budget constraint is pushed out further since this allows him to consume more in both periods. Hence no matter what his preferences are, he is better off with a budget constraint that is farther out To see this we can represent preferences in this diagram by an indifference curve This is the locus of combinations of Coa and Cy such that he is indifferent 8
8 In this case, Intercept on COA axis = 1.2 x 850,000 + 340,000 = 1.36 M. Hence, analytically the budget constraint is given by COA = 1.36 M - 1.2 CY We can see that by undertaking the first project he can push out the line representing his possible consumption, similarly when he undertakes the second project, and so on. If he prefers more money to less, he is better off if his budget constraint is pushed out further since this allows him to consume more in both periods. Hence no matter what his preferences are, he is better off with a budget constraint that is farther out. To see this we can represent preferences in this diagram by an indifference curve. This is the locus of combinations of COA and CY such that he is indifferent
OA utility increasing We can represent differences in preferences for consumption in old age and youth by differences in the shape of the indifference curves. Suppose, for example, somebody has a strong preference for consumption in old age Then her indifference curves will look something like this OA Miser Her curves are flat for the following reason. Since she is a miser a small reduction in consumption in old age must be compensated for by a large increase in youth to make her indifferent
9 We can represent differences in preferences for consumption in old age and youth by differences in the shape of the indifference curves. Suppose, for example, somebody has a strong preference for consumption in old age. Then her indifference curves will look something like this: Her curves are flat for the following reason. Since she is a miser a small reduction in consumption in old age must be compensated for by a large increase in youth to make her indifferent
Somebody who prefers consumption in his youth, who we shall call a spender, will have the type of indifference curves below. They are steep because a large reduction in consumption in old age can be compensated for by a small increase in youth to make him indifferent OA Spender Now if we put budget constraints on these diagrams we can see that pushing a budget constraint out makes both better off OA 10
10 Somebody who prefers consumption in his youth, who we shall call a spender, will have the type of indifference curves below. They are steep because a large reduction in consumption in old age can be compensated for by a small increase in youth to make him indifferent. Now if we put budget constraints on these diagrams we can see that pushing a budget constraint out makes both better off