Costs Lecture Notes

Costs August 2015 1 / 63 Introduction Last lecture, we discussed how …rms turn inputs into outputs. But exactly how much will a …rm wish to produce? 2 / 63 Introduction How much a …rm wishes to produce depends on the cost function. The …rm’s …rst step is to …nd the production process that is technically e¢ cient. Technical e¢ ciency is a necessary condition for pro…t maximization, but it is not su¢ cient. The …rm’s second step is to …nd the technologically e¢ cient production process that is also economically e¢ cient. A …rm is economically e¢ cient if it minimizes the cost of producing a speci…ed amount of output. 3 / 63 Outline Measuring Costs - How exactly do economists measure costs? Short-Run Costs - What does a …rm’s cost function look like when some inputs are …xed? Long-Run Costs - What does a …rm’s cost function look like when all inputs are variable? What is the …rm’s optimal input combination? Lower Costs in the Long Run - Firm has more ‡exibility in the long run, which implies lower costs. Cost of Producing Multiple Goods - Does producing multiple goods in the same factory make sense? 4 / 63 Measuring Costs To …nd the economically e¢ cient level of output, we need to know how to measure costs. It is easy to measure explicit costs. Paying a worker £ 7 an hour is an explicit cost. But we must look at all costs, including the implicit ones. A cost is implicit if it re‡ects forgone opportunity rather than current expenditure. 5 / 63 Measuring Costs Opportunity cost is an implicit cost. The opportunity cost is the value of the next best alternative. The opportunity cost of me being in school is $ 25, 000 salary. Opportunity cost is important when a …rm purchases capital, because it durable. 6 / 63 Measuring Costs Opportunity costs should in‡uence the …rms current decisions, but sunk costs should not. A sunk cost is an expenditure that cannot be recovered. A non-refundable movie ticket is an example of a sunk cost. 7 / 63 Measuring Costs EXAMPLE You can play tennis inside or outside. You can book the inside court for a non-refundable £ 20 fee in advance. Playing outside is free. You prefer to play inside if it is rainy but outside if it is sunny. You booked the inside tennis court in advance and it turns out it is sunny. Do you play inside or outside? 8 / 63 Short-Run Costs To maximize pro…t, the …rm needs to know how costs vary with output. A cost function C (q ) tells us how much it will cost to produce various levels of output. All points on the cost function are economically e¢ cient. 9 / 63 Short-Run Costs EXAMPLE Lets graph the cost function C (q ) = 10q + 10. If the …rm wants to produce 10 units of output, could it do so at a cost of £ 100? At a cost of £ 200? 10 / 63 Short-Run Costs Remember in the short run, at least one input is …xed. The cost of producing 10 units in the short-run is not always the same as the cost of producing 10 units in the long-run. We will …rst look at the …rm’s cost function in the short-run, then the cost function in the long-run. 11 / 63 Short-Run Costs It is useful to break up our costs into di¤erent types. One type of cost is a …xed cost (F ). Fixed costs do not vary with the level of output. For example, it costs £ 10,000 to heat a factory no matter how much you produce. 12 / 63 Short-Run Costs Fixed costs might be sunk or non-sunk. It is sunk if it cannot be recovered by shutting down. If you own a factory that has no alternative uses upon shutting down (you can’t sell it), that is a sunk …xed cost. 13 / 63 Short-Run Costs A Variable cost (VC ) is the production expense that does change with quantity produced. The cost of dough is a variable cost for a bakery. Total cost (C ) is the sum of …xed and variable cost. C = VC + F 14 / 63 Short-Run Costs EXAMPLE If our cost-function looks like C (q ) = 100q + 10, what are the variable costs and what are the …xed costs? 15 / 63 Short-Run Costs Marginal cost is the amount by which the total cost changes when we add more output. MC = dC (q ) dq 16 / 63 Short-Run Costs Average …xed cost is the …xed cost divided by the amount produced q. AFC = F q It declines with output because the …xed cost is spread over more units. Average variable cost is the variable cost per each unit produced. AVC = V q 17 / 63 Short-Run Costs Average cost is the sum of these. AC = C VC F = + q q q 18 / 63 Short-Run Costs EXAMPLE Suppose our cost function looks like. C = q2 100q + 1000 What is the variable cost, …xed cost, marginal cost, average variable cost, average …xed cost and average cost? 19 / 63 Short-Run Costs What do all these cost curves look like graphically? Fixed cost does not vary with output, so it is a straight line. Average …xed cost falls as output decreases. Average cost is the vertical sum of average …xed cost and average variable cost. 20 / 63 Short-Run Costs Average cost slopes downward at …rst because average …xed cost declines. Average cost begins to slope upward because of diminishing marginal returns. Marginal cost intersects average cost at the minimum of average variable cost. Why? 21 / 63 Short-Run Costs 22 / 63 Short-Run Costs The production function we saw Ch. 6 and the cost function are basically mirror images of each other. We can …nd the cost function from the production function and vice versa. For example, the production function tells us we need 10 units of labour to produce 6 units of output. The cost of one unit of labour is £ 5. The cost of producing 6 units of output is then £ 5 *10 = £ 50. 23 / 63 Short-Run Costs Suppose we have the following short-run production function. q = f (L, K ) = g (L) We are in the short-run, so capital is …xed. Labour is the variable cost and capital is the …xed cost. VC = wL 24 / 63 Short-Run Costs If we invert the production function we can …nd the amount of labor needed to produce any amount of output L=g 1 (q ) Plugging this in we can see our cost function is now C (q ) = V (q ) + F = wg 1 (q ) + F 25 / 63 Short-Run Costs Suppose our production function is as follows: q = L.5 K .5 Capital is stuck at 16 units in the short-run so the short-run production function can be written as: q = 4L.5 Suppose the price of capital is $ 1 per unit of capital. 26 / 63 Short-Run Costs Solve this for L q2 16 This tells us how much labour we will use to produce each amount of output. L= If we want to produce 4 units of output, we must use 1 unit of labour. 27 / 63 Short-Run Costs We can express the short-run cost function as C (q ) = w q2 + 16 16 28 / 63 Short-Run Costs EXAMPLE Suppose capital is …xed at 19 units and the price of capital is $ 2 per unit. The production function is q = LK + L What is the short-run cost function? 29 / 63 Short-Run Costs If we know marginal product of labour, we can easily …nd the marginal cost. Recall that variable cost in the short run is V (q ) = wL MC = dV (q ) d (wL) dL = =w dq dq dq We know the marginal product of labour is relationship as 1 MC = w MPL dq dL so we can write the 30 / 63 Short-Run Costs If we know the average product of labour, we can easily …nd average variable cost. Remember AVC = VC q = wL q The average product of labour is qL , so we can write the relationship as AVC = w APL 31 / 63 Short-Run Costs EXAMPLE Suppose your short-run cost function is C (q ) = q + q 2 + 10 The wage rate is £ 1. What is the average product of labour and the marginal product of labour when q = 4? 32 / 63 Short-Run Costs The government can a¤ect a …rm’s cost curves through various forms of taxation. Di¤erent types of taxes a¤ect the cost curves in di¤erent ways. A speci…c tax will shift the …rm’s variable and marginal costs up, but won’t a¤ect the …xed costs. A franchise tax will a¤ect the …rm’s …xed costs. 33 / 63 Long-Run Costs Now let’s turn to the long-run. Firms can vary everything in the long run. There are no …xed costs (technically they can have avoidable …xed costs in the long run but we assume they don’t). Long run cost is just C (q ) = VC 34 / 63 Long-Run Costs Now that both inputs are free to vary, what input combination should the …rm select? Remember that isoquants show us all the technologically e¢ cient input combinations. The …rm must pick the technically e¢ cient input combination that is the cheapest (economically e¢ cient). 35 / 63 Long-Run Costs The Isocost line shows all the input combinations that cost exactly the same. You hire L units of labour at a price of w and K units of capital at a price of r . We can write the isocost line as C = wL + rK The isocost is a lot like the budget line in consumer theory, the di¤erence being that the …rm has many isocosts and the consumer has only one budget line. 36 / 63 Long-Run Costs EXAMPLE What is the equation for an isocost if w = 4, r = 5 and we want to spend $ 1, 000? What is the slope of the isocost line? What happens if we want to spend $ 2, 000? What happens if the wage increases? 37 / 63 Long-Run Costs Suppose the …rm wants to produce Q units. How does the …rm …nd the cheapest input combination? There are three equivalent ways the …rm can …nd this out. Lowest Isocost Rule pick the isocost closest to the origin that touches the isoquant. 38 / 63 Long-Run Costs Tangency Rule Assuming we have an interior optimum, the lowest isocost is where the isocost is tangent to the isoquant. The slope of the isocost is w r . The slope of the isoquant is the MRTS = MP L MP K . The optimal input combination occurs where w MPL = r MPK 39 / 63 Long-Run Costs Last Dollar Rule We can rearrange the tangency condition MPL MPK = w r cost is minimized when the last dollar spent on labour adds as much extra output on the last dollar spent on capital. 40 / 63 Long-Run Costs EXAMPLE If w = 2, MPL = 10, r = 10, MPK = 10, what should we do to lower costs? 41 / 63 Long-Run Costs 42 / 63 Long-Run Costs EXAMPLE Suppose w = 5 and r = 20. 1 1 If our production function is q = K 2 L 2 , what is the optimal input combination? If we want to produce 10 units, how much labour and capital will we use? 43 / 63 Long-Run Costs We can use math to get the same tangency condition. We want to minimize costs wL + rK such that we produce q = f (L, K ) min L = wL + rK + λ[q Lets prove together that this results in f (L, K )] MP L MP K = w r . 44 / 63 Long-Run Costs Rather than minimize costs for a desired level of output, what if we want to maximize output for some given cost? Our Lagrangian becomes max L = f (L, K ) λ(wL + rK C) If you do this, you get exactly the same result that MP L MP K = w r This is a "dual" problem... two sides of the same coin. 45 / 63 Long-Run Costs EXAMPLE Suppose your production function is: q = KL The wage rate is w = 1 and the rental rate is r = 1. Use the Lagrangian method to …nd the cheapest way of producing 25 units of output. 46 / 63 Long-Run Costs What happens when one factor becomes relatively cheaper? The slope of the isocost is wr , so the slope changes and we have a new cost minimizing combination. 47 / 63 Long-Run Costs How does the …rm’s cost change when we increase output? The expansion path shows us all the tangency points for each level of output. 48 / 63 Long-Run Costs The expansion path tells us the same thing as the long-run cost function essentially. When you produce qo units, use Ko and Lo units of capital and labour. When you produce q1 units, use K1 and L1 units of capital and labour. 49 / 63 Long-Run Costs Suppose we found from our expansion path that K = q and L = the optimal input combination. q 2 is We can plot the long run cost of producing di¤erent levels of q.. Suppose the wage is 24 and capital costs 8. C (q ) = wL + rK = w q + rq = 20q 2 50 / 63 Long Run Cost Functions 51 / 63 Long-Run Costs Remember the short-run cost curve is U shaped. It slopes downward at …rst because average …xed cost declines. It slopes upward because of diminishing marginal returns. There are no …xed costs and diminishing marginal returns in the long run. 52 / 63 Long-Run Costs If the LRAC curve is downward sloping, the cost function exhibits economies of scale. The average cost falls as you produce more. If the LRAC curve is upward sloping, the cost function exhibits diseconomies of scale. The average cost increases as you produce more. If it is ‡at there are no economies of scale. 53 / 63 Long-Run Costs Recall that returns to scale refers to how much your output will change when you scale up your inputs. Do economies of scale imply returns to scale? What do you think? 54 / 63 Lower Costs in the Long-Run Firms have more degrees of freedom in the long run. Suppose the optimal combination for some level of output is K = 10 and L = 5, if we are in the short run K might be stuck at the "wrong" level that is not cost minimizing. Costs in the short-run are always at least as high as in the long-run. In the long run, we can change K to be where we want. 55 / 63 Lower Costs in the Long-Run 56 / 63 Lower Costs in the Long-Run We can further illustrate this by comparing the short-run and long-run expansion paths. 57 / 63 Lower Costs in the Long-Run Another reason why costs are lower in the long is learning by doing. As workers gain experience and managers learn to organize, the average cost tends to fall over time. 58 / 63 Lower Costs in the Long-Run 59 / 63 Cost of Producing Multiple Goods If …rms produce multiple goods, the cost of one good might depend on the output of another. It is less expensive to produce poultry and eggs together than separately. If it is less expensive to produce goods jointly, the …rm enjoys economies of scope. If it is more expensive to produce goods jointly, the …rm experiences diseconomies of scope. 60 / 63 Cost of Producing Multiple Goods We can illustrate this by a production possibilities frontier 61 / 63 Summary What is an opportunity cost? What are …xed and variable costs? What are sunk costs? What are avoidable costs? 62 / 63 Summary How does a …rm …nd the cost minimizing input combination in the long run? What are economies of scale? What are economies of scope? What is learning by doing? 63 / 63