5. Consider the example in Section 3.3 but now assume that the market rate of interest is 10%. Graphically show the operating plan for the following two transformation curves: a) 3K+2K2 30 b) (K + 1)2 + K3 = 221 Using the transformation curve in part (b), find the firm's optimal financing decision if the firm's initial resources at time 1 is: c) 5 d) 0 e) Find the optimal consumption decision assuming the transformation curve in part (b) and further assuming that the only owner of the firm has the following utility function: U(K1, K2) = K K22 3.3 EXAMPLE: AGING WINE In this section we present a mathematical example as supporting detail for the graphical reasoning presented above. Consider the problem of a businessperson trying to decide whether to sell a product now or refine it further and sell it later at a higher price. For example, assume a vintner who is trying to maximize the market of initially available resources in the form of new wine cither by selling the current vintage of wine at time l or aging it for sale at time 2. Let the market rate of interest be 6% The objective function of the vintner will be K max Ki + (3.4) 1.06 where K, = value of new wine sold at time 1 K = value of aged wine sold at time 2 Assume the maximization is subject to K+K 212.36, a transformation curve chosen for its mathematical tractability rather than for its interpretive realism. The transformation curve implies that the time I cash value of all the raw wine is (5212.36), or approximately $14.60. 42 Creating Wealth by Investing in Productive Opportunities Proceeding by substitution, we can rewrite equation (5.4)*: max (212.36-K) K 1.06 (3.5) Taking the derivative of equation (3.5) with respect to K and setting the result equal to zero, we obtain: + 1 1.06 = 0 1/2 4-2K) (3.6) (212.36-K?) This step is equivalent to saying "choose a production plan such that the marginal rate of return equals the market rate of interest." Rewriting equation (3.6), we obtain the following sequence of calculations: 1 K: 1.06 (212.36-K) K:(1.06) = (212.36-K)" K}(1.00 - $212.36-K} K${1+(1.061*1 = $212.36 K($2.1236) = $212.36 K3 = $10.00 therefore KT = (S212.36 - $100.00)" = $10.60 and 10.00 MVI = S10.60+ = $20.03 1.06 This situation is shown graphically in Figure 3.7. Note the following: (1) The vintner has increased initial wealth (MV.) to $20.03 from the original $14.60 available by selling all the raw wine at time 1; (2) the vintner will employ wine in the aging process up to the point where the marginal rate of return on aging wine just equals the market rate of interest. 5. Consider the example in Section 3.3 but now assume that the market rate of interest is 10%. Graphically show the operating plan for the following two transformation curves: a) 3K+2K2 30 b) (K + 1)2 + K3 = 221 Using the transformation curve in part (b), find the firm's optimal financing decision if the firm's initial resources at time 1 is: c) 5 d) 0 e) Find the optimal consumption decision assuming the transformation curve in part (b) and further assuming that the only owner of the firm has the following utility function: U(K1, K2) = K K22 3.3 EXAMPLE: AGING WINE In this section we present a mathematical example as supporting detail for the graphical reasoning presented above. Consider the problem of a businessperson trying to decide whether to sell a product now or refine it further and sell it later at a higher price. For example, assume a vintner who is trying to maximize the market of initially available resources in the form of new wine cither by selling the current vintage of wine at time l or aging it for sale at time 2. Let the market rate of interest be 6% The objective function of the vintner will be K max Ki + (3.4) 1.06 where K, = value of new wine sold at time 1 K = value of aged wine sold at time 2 Assume the maximization is subject to K+K 212.36, a transformation curve chosen for its mathematical tractability rather than for its interpretive realism. The transformation curve implies that the time I cash value of all the raw wine is (5212.36), or approximately $14.60. 42 Creating Wealth by Investing in Productive Opportunities Proceeding by substitution, we can rewrite equation (5.4)*: max (212.36-K) K 1.06 (3.5) Taking the derivative of equation (3.5) with respect to K and setting the result equal to zero, we obtain: + 1 1.06 = 0 1/2 4-2K) (3.6) (212.36-K?) This step is equivalent to saying "choose a production plan such that the marginal rate of return equals the market rate of interest." Rewriting equation (3.6), we obtain the following sequence of calculations: 1 K: 1.06 (212.36-K) K:(1.06) = (212.36-K)" K}(1.00 - $212.36-K} K${1+(1.061*1 = $212.36 K($2.1236) = $212.36 K3 = $10.00 therefore KT = (S212.36 - $100.00)" = $10.60 and 10.00 MVI = S10.60+ = $20.03 1.06 This situation is shown graphically in Figure 3.7. Note the following: (1) The vintner has increased initial wealth (MV.) to $20.03 from the original $14.60 available by selling all the raw wine at time 1; (2) the vintner will employ wine in the aging process up to the point where the marginal rate of return on aging wine just equals the market rate of interest