Ari owns an auto shop where mechanics apply their labor (L) to repair esrs using

Question

Ari owns an auto shop where mechanics apply their labor (L) to repair esrs using repsir kits (K). The number of cars repaired can be expressed as The shop currently owTs 8 kits and currently employs 8 mechanics Esch kit cost B when it was initially purchssed. They can now be used without paying any extra charge. There is also a rental market for kits, which can be rented on the market for r per day. owever, kits are difficult to transport, and it takes one month to move them between shops. Mechanics can also be hired for a daily wage of w, and enn be hired by the day (or portion of the day) (For this problem, do not worry about limiting your answers to whole numbers) A) Describe Ari's nccounting conts, opportunity oonts, and sunk oosts. Which are relevnnt for his produc- tion decision Explain why the cost of kitsB doesn't factor into profit maximizntion (b) What time frame describes the long run? Write Ari's short-run production function (e) In the short-run, what is the fixed oot of production? (Hint: what is K in the short run? (d) In the shoet run, what is the variable cost of produetion? (Hnt How much L is needed to produce a given amount of g? At what price in L purchnsed? (e) Compute the short-run average cost and average variable cost curves () Show that short-run aveage variable cost is equal to the inverse of the avernge product of labor times the wage rate. (nt: Finst, compate the average product of labor. Then, replnce g with a fanction of L. in the sbort-run average vnriable cout expression nnd rearrange terms) )Compute the short-run manginal cost (h) How does the short-run marginal cost relate to the short-run manginal product of lsbor? (You do not need to show it mathematically, bat it in asefal practlce for the exam) () On a single graph, plot the the short-run marginal cont, average total cout, nnd average variable cost when to3 and 5. Draw the curves to scale you may need to use a caleulator or website to do so. ) Write Arl's long-run production fanction. (k) Write n expression for n long-run isocost curve gmornly in terms of u, r·and some cost level C (1) Set up Ari's cost minimizntion problem as n Lngrangian. That in, write the Lagrnnginn that corresponds to Ari minimizing the cost of production subject to producing 8 fixed amount (m) Solve the cost-minimization problem to compute K 8nd L 8s·function of t, r, and g. (n) Maltiply the optimal levels of K nd by their respoctive prices to compute the longrun cost curve. (o) On s new graph, plot the long-run aver8ge cost when w = 3 and r = 5. On the same graph, plot the sbort-run average cost at the same prices when R is 5, 8, nd1. Draw the eurves to scale; you may need to use a caleulator or website to do so. (P) Explain the relntionship between the short-run and long-run avernge cost fanctions you plotted above.

Explanation / Answer

(a). Accounting cost = B*K+w*L

Opportunity cost = r*K per day

Sunk cost = B*K

Sunk cost are not relevant to the production deiosion. Accounting cost help determine the break even level of equilibrium or the short run levelof equilibrium while the opportunity cost is relevant in determining the long run level of output.

(b) One month time frame represents the long run when the kits are alos mobile between the shops. The short run production is iven by K = 8

That is q(8, L ) = 2(8)^1/3*(L)^1/2

q=4L^1/3

(c) fixed cost = B*8

(d) variable cost = 8*w

(e) short run TC= 8B+8w

Short run output at K=8 and L= 8

Q = 4*(8)^1/2

Q=11.31

ATC = 8B+8w/Q

=(8B+8w)/11.31

AVC = 8w/Q

=8w /11.31

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