Costs. Accounting Cost{stresses \out of pocket" expenses. Depreciation costs are based on tax laws.


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1 Costs Accounting Cost{stresses \out of pocket" expenses. Depreciation costs are based on tax laws. Economic Cost{based on opportunity cost (the next best use of resources). 1. A selfemployed entrepreneur's economic cost includes the opportunity cost of his time (the wage he could have received elsewhere). 2. A rm's economic cost includes its cost of capital, even if the rm owns the capital.
2 Example: Say you bought a building last year, and paid $1,,. Suppose the current market value is $8,, the interest rate is i, and the rental rate is r. (ignore depreciation) Then the economic cost of the building for this year is i 8; = r 8;. One alternative to owning the building is selling it and buying a bond, receiving i 8;. Another alternative is renting the building to someone else, and receiving r 8;. The price of capital is r = i. If r > i, everyone should borrow the money to buy their capital instead of renting. If i>r, everyone should sell their capital andrentitback.
3 Cost Minimization: the Importance of Marginal Analysis Plant 1: 1 workers, produce 1, widgets Plant 2: 5 workers, produce 4 widgets A consultant is hired, who reasons as follows. AP L (1) = 1 and AP L (2) = 8. Therefore, the rm should send workers from plant 2 to plant 1, where they will be more productive. What should the rm do? Fire the consultant. To know whether workers are allocated e±ciently, we need to know about marginal products. For example, if MP L (1) = 1 and MP L (2) = 15, send a worker from plant 1 to plant 2, for a net gain of 5 widgets.
4 Cost Minimization (Long Run) Here we solve the problem of how to produce a given amount of output at the minimum cost. Letting w be the price of labor (dollars per labor hour) and r be the price of capital (dollars per machine hour), the equation for the isocost line corresponding to a total cost of TC is wl + rk = TC:
5 1.8.6 K L An Isocost Line (w=2, r=1)
6 Graphically, we can see that the input bundle that minimizes the cost of producing a given amount of output is where the isoquant is tangent to an isocost line..8.6 K L The Cost Minimizing Input Bundle (w=2, r=1) In the above gure, the cost of getting to this isoquant is minimized by choosing L=1/4 and K=1/2, at a total cost of 1.
7 \Duality": Cost minimization subject to a minimum output constraint ( nding the lowest isocost intersecting an isoquant, shown above) is really the same as output maximization subject to a budget constraint ( nding the highest isoquant intersecting an isocost, shown below)..8.6 K L The Cost Minimizing Input Bundle (w=2, r=1)
8 The slope of the isoquant is MRTS,andtheslope of the isocost is w r. Therefore, the optimal input bundle satis es MRTS = MP L MP K = w r : (1) To derive condition (1) formally, we x output at x and solve min wl + rk subject to f(k; L) = x Set up the Lagrangean, Lagr: = wl + rk + [x f(k; L)] The rst order conditions = = @K = = (3) = = x f(k; L)
9 Solving (2) and (3) for, wehave = w = r : (5) MP L MP K Equation (5) is the same as equation (1). Equation (1) says that the ratio of marginal products should equal the ratio of input prices. The interpretation is that the internal willingness to trade K for L (the ratio of marginal products) should equal the external rate atwhichkcanbetradedforl. The interpretation of (5) is that the marginal cost of producing output using labor should equal the marginal cost of producing output using capital. For example, if w = 1 and MP L = 2, then one more labor hour yields 2 more units of output and costs $1, so the marginal cost of one more unit of output is $5. The multiplier,, has an economic signi cance, corresponding to the marginal cost of output. In any Lagrangean problem, the multiplier has the interpretation of the marginal (objective) of (constraint). Here, theobjectiveiscostandtheconstraintisoutput.
10 Equations (4) and (5) can be used to solve for the generalized (conditional) input demand functions, L (w; r; x) and K (w; r; x). CobbDouglas Example: x = AK 1=3 L 2=3 (where A is a positive constant) From (5), we have = w 2 3 AL 1=3 K 1=3 = r 1 (6) 3 AL2=3 K 2=3: Equation (6) can be simpli ed to L = 2rK w : (7) Now plug (7) into (4), x = AK 1=3 L 2=3, which yields x = AK 1=3 ( 2rK w )2=3 : (8)
11 We are interested in the demand for K and L, so solve (8) for K: K =( 2r w ) 2=3 A 1 x: (9) Plugging (9) into (7), we have the generalized demand function for L: L =( 2r w )1=3 A 1 x: (1)
12 Deriving the Total Cost Function Since (9) and (1) tell us the amounts of K and L to choose in order to produce x units of output, we can derive the total cost of producing x, assuming the rm chooses its inputs optimally to minimize costs. TC = wl + rk (11) For our example, (11) becomes TC = w( 2r w )1=3 A 1 x + r( 2r w ) 2=3 A 1 x = [2 1=3 +2 2=3 ]w 2=3 r 1=3 A 1 x Notice that total cost is proportional to x. This is a property of any constantreturnstoscale production function. Since all inputs are variable for this calculation, TC is sometimes called the long run total cost function, LRTC.
13 We can also de ne long run average cost and long run marginal cost: LRAC = LRT C d(lrt C) and LRMC = x dx For constant returns to scale, these cost functions are constant, independent of x
14 1 8 $ x LRTC (constant returns) $/unit x LRAC and LRMC (constant returns)
15 $ x LRTC (increasing returns) $/unit x LRAC and LRMC (increasing returns)
16 1 8 $ x LRTC (decreasing returns) $/unit x LRAC and LRMC (decreasing returns)
17 1 8 $ x LRTC (Sshaped) $/unit x LRAC and LRMC (Ushaped LRAC) There are IRS for x<2anddrsforx>2.
18 If the LRAC curve is falling, then at the margin, we are bringing down the average as we increase x. Thus, themarginalcostmustbebelowaveragecost(inorder to be bringing down the average). If the LRAC curve is increasing, then marginal cost must be above average cost. For a Ushaped LRAC curve, then at the minimum point, the curve is at. Since one more unit is not changing the average, it must be that marginal cost is equal to average cost.
19 Returns to Scope and Joint Production (We will not treat this topic formally.) Sometimes there can be cost savings by expanding the set of products produced, rather than expanding the amount of a given good produced. Examples include: (1) airlines, spoke and hub networks. By adding the Milwaukee to Chicago route, more passengers will want to y the Chicago to New York route. Federal Express. (2) Gas station and convenience store. (3) Timeshare condos, hotels. (4) Produce intermediate as well as nal products.
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