Suppose that in a given production process

Suppose that in a given production process a blueprint (B) can be produced using either an hour of computer time (C) or 4 hours of a manual draftsman’s time (D). (You may assume C and D are perfect substitutes. Thus, for example, the firm could also produce a blueprint using 0.5 hour of C and 2 hours of D.)

1. a) Write down the production function corresponding to this process (i.e., express B as a function of C and D).
2. b) Suppose the price of computer time (p_c) is 10 and the wage rate for a manual draftsman (p_D) is 5. The firm has to produce 15 blueprints. What are the cost minimizing choices of C and D? On a graph with C on the horizontal axis and D on the vertical axis, illustrate your answer showing the 15-blueprint isoquant and isocost lines.

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1. a) Computers are four times as productive as draftsmen; an alternative way of saying this is that MP_C = 4MP_D. Since C and D are perfect substitutes, we know the production function has the form B = aC + bD, where a and b are positive constants.  Thus we can write the production function as B = C + (1/4)D.  Note that this is consistent with generating one blueprint (B = 1) from the following combinations of inputs: (C, D) = (1, 0), (C, D) = (0, 4), and (C, D) = (0.5, 2).

15. b) Notice that . That is, the marginal product per dollar spent on computer time is always higher than the marginal product per dollar spent on draftsman time.  So the optimal input combination involves D = 0 and C =   The graph below illustrates the (dotted) isocost lines with slope = –p_C / p_D = –2, along with the (solid) B = 15 isoquant with slope = –MP_C / MP_D = –4.