Review Objective: Derive firm’s Supply curve Price Supply Review: 1. What is a production function? 700 Eu • 2. What is the marginal product of some input? 3. What is the MRTS? 400 Eu • 4. What are decreasing returns to scale? 0 600 1500 Quantity Review How much does it cost the firm to produce q units? The firm’s cost function specifies the cost to produce q units Total costs are determined by the production function and the costs of inputs But there are many different combinations of inputs that can produce q units. The firm’s problem is to choose the combination of inputs that minimizes costs. The Cost Function Example 1. A firm requires a single input to produce output, with production function f(L)=√L. If the wage is w=$10, how much does it cost to produce 3 units? How much does it cost to produce Q units? Example 2: A firm needs 2 workers and 1 machine to produce a single unit. The wage is w = 5 and rent is r = 10. What is the cost of producing 10 units? What is the cost function? Example 3: A firm can produce 1 unit of output with either 2 workers or with 1 machine. If the wage is w=5 and rent is r = 5. What is the cost of producing 10 units? Derive the cost function TC(Q)? The Cost Function When the firm hires L workers and K units of capital the total cost is: w = wage rate L = Quantity of Labor r = price per unit of capital services K = Quantity of Capital The total costs are: TC wL rK The Cost Function If are many different combinations of inputs that can produce Q units. The firm’s problem is to choose the best combination of inputs (that minimizes costs). The firm’s problem is to choose L and K that solve, First Order Conditions: w MPL w MPL MPK OR MRTSL,K OR r MPK r w r The Cost Function Definition: The iso-cost line is the set of combinations of labor and capital that yield the same total cost for the firm. Example: The wage is w = $10 per hour and the rent is r = $20 per hour. The iso-cost line for TC = $1,000 is all the combination L and K such that: 10L + 20K=1,000 or K = 50-0.5L The Cost Function The iso-cost lines is K=(TC/r)+(w/r)L K TC2/r TC2 TC1/r TC0/r TC1 TC0 L TC0/w TC1/w TC2/w The Cost Function The Firm’s problem: K min wL rK The isocost line TC2/r s.t.F(K, L) Q TC1/r The isoquant for Q. TC0/r The firm must choose a point on the isoquant. Q L TC0/w TC1/w TC2/w Deriving The Cost Function Key explanation: How to derive the cost function The Cost Function Example: Suppose a firm needs 10 workers and 1 machine to produce a single unit. What is the production function? The wage is w = 5 and rent is r = 100. What is the cost of producing 50 units? What is the cost function? Solution: The production function F(L,K)= Min[0.1L,K]. To produce 50 units we need 50 machines and 500 workers, and the total cost will be 500*5+50*100=7,500 The Cost Function Example: Suppose F(K,L)=L + 2K, the wage is w = 1 and rent is r = 1. What is the cost of producing 1000 units? Derive the cost function TC(Q)? Do the same if the cost of capital increases to r=4 Solution: If MPL/w > MPK /r the productivity of a dollar spent on labor is larger than the productivity of a dollar spent on capital. If MPL/w < MPK /r the productivity of a dollar spent on labor is smaller than the productivity of a dollar spent on capital. Deriving The Cost Function Example: Suppose F(K,L)= 50L1/2K1/2, the wage is w = $5 and rent is r = $20. What is the cost of producing 1,000 units? Derive the cost function TC(Q)? To get the cost of 1,000 units we need to ficure out how many workers and machines to use: 1. Optimality condition: MPL/MPK = w/r or K/L = 5/20 2. Use the production constriant: 50L1/2K1/2=1,000 K = 10; L = 40; And the total cost is 5*40+20*10=400 The Cost Function MP MP At point E L K w r The firm can increase labor and decrease capital, and produce for less. MP MP At point F L K w r The firm can decrease labor and increase capital, and produce for less. The Cost Function Example: As Quantity increases from 1 million to 2 million the cost minimizing input combination moves from point A to point B The Cost Function What happens when the price of input changes? Suppose price of capital r = 1 and quantity of output Q0 is constant. When price of labor w = 1 the isocost line is C1, optimal point A. But, when price of labor rises w = 2 isocost line is C2, optimal point B Key point: When the price of an input rises, we use less of that input and more of the other - An increase in the wage decreases the optimal quantity of labor and increases the quantity of capital. - An increase in rent decreases the optimal quantity of capital and increases the quantitty labor The Cost Function Question: How does the total cost curve shift if the price of all inputs rises by the same amount? The price of inputs increases proportionately by 10%. 1. Cost minimization input stays same (Why?). 2. TC curve shifts up by the same 10 percent Fixed and Variable Inputs Some inputs are fixed and some vary Total Variable Costs – the costs of variable inputs Total Fixed Costs – the cost of fixed inputs; does not vary with output •Fixed costs but not sunk: become zero if no production takes place Q=0 • Fixed and sunk: positive even if Q=0 Fixed and Variable Inputs Example: Q 50 LK capital fixed at K Q2 L* (Q) 2500 K Q2 TC(Q) w* L* (Q) rK w* rK 2500 K Long Run Cost Curve Definition For given input prices: The average cost function is the long run total cost function divided by output, Q. ATC(Q) = TC(Q)/Q The marginal cost function is the rate of change of total cost as output varies MC(Q) = TC(Q)/Q Long Run Total Cost Key points: When marginal cost is lower than average cost, average cost is decreasing in quantity. When marginal cost is greater than average cost, average cost is increasing in quantity. When marginal cost equals average cost, average cost is minimized. Economies Of Scale Definitions We say that a cost function exhibits economies of scale, if the average cost decreases as output rises, all else equal. We say that a cost function exhibits diseconomies of scale, if the average cost increases as output rises, all else equal. The smallest quantity at which the long run average cost curve attains its minimum point is called the minimum efficient scale. Economies Of Scale Economies of scale, diseconomies of scale, minimum efficient scale. AC ($/yr) AC(Q) Q (units/yr) 0 Q* = MES Economies Of Scale Why are there economies of Scale? 1. Increase returns to scale- Fixed costs - More production permits more specialized inputs. 2. Decreasing return to scale- Organizational structure and managerial costs - Adding software engineers increases communication costs: If there are n engineers, there are ½n*(n – 1) pairs, so that communication costs rise at the square of the project size - “The Peter Principle”: Workers move up until they become incompetent - System slack: it is easier to hide inefficiencies in a large organization than in a small one Economies Of Scale 24 Economies of scale and Returns to scale - When the production function exhibits decreasing returns to scale, the long run average cost function exhibits diseconomies of scale. - When the production function exhibits increasing returns to scale, the long run average cost function exhibits economies of scale. - When the production function exhibits constant returns to scale, the long run average cost function is flat: it neither increases nor decreases with output..