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6,5 C : lucomé EFFGCI Answer Key for Part III 1. 1. (a) This is the problem of a short run decision since the amount of bauxite is fixed. The
firm chooses the amount of electricity that equates the marginal revenue product of
electricity to the price of electricity. The optimal use of electricity can be represented in
either way of the two below. (In the following PA denotes the price of one unit of
aluminum.) MPE MRPE PE/PA ............................................................ pE :E :E (Grading Policy : A partial credit was given when it was not indicated what the sloped
line meant but the graph was right. When one misunderstood the sloped line and mistakenly write PB, P or PA in place of PE , no credit was given.) (b). The firm chooses the optimal level of the two inputs, Bauxite and electricity, such
that minimizes the cost incurred by producing the given output level Q. Therefore, the
firm will choose an isocost line that is tangent to the isoquant curve of the given output
level Q where the slope of the isocost line is determined by the ratio of the price of inputs. Optimal Choice of Inputs
13* Isocost Line : C = E* PE+ B* PB Isoquant at Q E F (c). The tangency implies that two equations hold at the optimal choice of inputs, E* and
B :C=E*PE+B PB (orQ=F(E ,B ))andMPﬂPE=MPB/PB. (Grading policy: A full credit was given to any answer that contains the tangency
condition MPE/PE = MPH/PB although one needs one more equation : cost line equation or
production function. When only the cost line equation or production function equation
was given, I gave a partial credit.) ((1) Since the firm needs to remain at the same level Q, the optimal choice of inputs
should be on the same isoquant but at a different tangency point because the relative price
of inputs has changed. In order to maintain the same level of output, the firm would
increase use of bauxite and decrease use of electricity because the latter has become
relatively Cheaper than the former. Graphical representation of this logic is as follows: New Isocost Line : C = E P’E+ B PB Optimal Choice of Inputs After Change in PE
Optimal Choice of Inputs Before Change in PE Old Isocost Line : C = E PE + B PB Isoquant at Q =. s E
E’ E* (Grading policy: I gave a partial credit that simply says E goes down and B goes up. I
gave a full credit to any answer that contains a correct explanation of why B needs to go
up, either graphically or verbally.) Part III
9% Given optimal inputs (bauxite and electricity, or labor and capital, etc.) as a function of
input prices and output, i.e. B*( p1,,pe, Q) and E *(pb, pg, Q), we know that the total cost as a
function of output is: C(Q) =pb' B *( pupa Q) We E *(pwe, Q)
Marginal cost is then just the slope of the total cost function, i.e. the cost of producing
one more unit: MC= AC(Q) = _AB*(pb,pe,Q)+pe_AE*(pb,pe,Q)
AQ AQ AQ Note that ﬁnding marginal cost does not depend on the P=MC proﬁt—maximizing rule for
a perfectively competitive ﬁrm. This rule is used to determine the optimum output level. b om:
Cost function a+bQ+dQ2 (a) Fixed cost is all costs that must be incurred regardless of how much output is
produced. Fixed cost is a. (b) Variable cost is total costs minus ﬁxed costs. So variable costs are bQ+dQZ. (c) Average cost is total costs divided by output: 2
Ac=E=M=g+b+dQ Q Q Q cm This question asks you to express proﬁts in terms of price, average cost, and output. We
know that proﬁts are deﬁned to be total revenues (P Q) minus total costs (TC). Since
total cost is simply average cost (AC) times output, we have the following relationship
between proﬁts, average cost, and output: PROFITS = PQ— AC  Q = Q(P—AC). \WVE Qw/s’h‘m :1 \ W [47?) q at)" Méx‘Q—WQ Law/4's) (Ls QWM badges ‘.
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This note was uploaded on 04/08/2008 for the course ECON 115 taught by Professor Stevenberry during the Fall '07 term at Yale.
 Fall '07
 StevenBerry
 Economics

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