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**Production and costs: Cost-Minimizing input combination**

AP Economics Mr. Bordelon

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**Alternative Input Combinations**

A firm can choose among a number of alternative combinations of inputs that will produce a given level of output. Joslyn Farms To produce the optimal quantity of wheat, they could choose to have a relatively capital-intensive operation by investing in tractors and hiring little labor. To produce the optimal quantity of wheat, they could choose to have a relatively labor-intensive operation by hiring more workers to do the planting and harvesting by hand.

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**Alternative Input Combinations**

Joslyn Farms To produce the optimal quantity of wheat, they could choose to have a relatively capital-intensive operation by investing in tractors and hiring little labor. To produce the optimal quantity of wheat, they could choose to have a relatively labor-intensive operation by hiring more workers to do the planting and harvesting by hand. The same amount of wheat can be produced using many different combinations of capital and labor. Kyle and Ryan must determine which combination of inputs will maximize their profits.

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**Substitutes and Complements**

Capital and labor can serve as both. Substitutes. If the price of one gets to be too expensive, the Joslyns will make a choice for the substitute to maintain a certain level of wheat production. AP: Tractors for farm workers. ATMs for bank tellers. Complements. If more of one increases the marginal product of the other. The Joslyns’ farm workers are more productive with tractors, and each tractor requires a worker. Office workers are more productive when they use faster computers. Doctors are more productive with modern X-ray machines. The quantity and quality of capital available affect the marginal product of labor, and thus the demand for labor.

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Cost Minimization Firms determine the combination of inputs that maximize profits by finding the input combination that costs the least—the cost-minimizing input combination. Publix can alternate self-checkout stations (capital) and cashiers (labor) to check out customers shopping at the store. If Publix puts in 20 self-checkout stations, Publix will need to hire 1 cashier to monitor every 5 stations. Trained cashiers are faster, so Publix could check out the same number of customers using 10 cashiers and 10 self-checkout stations.

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Cost Minimization Publix can alternate self-checkout stations (capital) and cashiers (labor) to check out customers shopping at the store. If Publix puts in 20 self-checkout stations, Publix will need to hire 1 cashier to monitor every 5 stations. Trained cashiers are faster, so Publix could check out the same number of customers using 10 cashiers and 10 self-checkout stations. Capital Labor Rental rate = $1,000/month Wage rate = $1,600/month a. 20 4 b. 10

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**Cost Minimization a. cost of capital 20 x $1,000 = $20,000**

cost of labor 4 x $1,600 = $6,400 TOTAL $26,400 b. 10 x $1,000 $10,000 10 x $1,600 $16,000 $26,000 Publix should choose option B, the lower cost combination.

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Cost Minimization When firms must choose between alternative combinations of inputs, they evaluate the cost of each combination and select the one that minimizes the cost of production. This can be done by calculating the total cost of each alternative combination of inputs. However, because the number of possible combinations can be very large, it is more practical to use marginal analysis to find the cost-minimizing level of output.

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**Cost-Minimization Rule**

Additional output results from hiring an additional unit of an input is the marginal product (MP) of that input. Firms want to receive the highest possible marginal product from each dollar spent on inputs. Cost-minimization rule. Firms adjust their hiring of inputs until the MP per dollar is equal for all inputs. For labor and capital:

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**Cost-Minimization Rule**

Assume MPL is 20 and MPK is 100. If wage is $10 and rental is $100, then MPL per dollar is 20/$10 = 2 units of output per dollar for labor and 100/$100 = 1 unit of output per dollar for capital. Publix is getting more output for its money by hiring labor, so it should hire more labor and less capital. Why?

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**Cost-Minimization Rule**

Assume MPL is 20 and MPK is 100. If wage is $10 and rental is $100, then MPL per dollar is 20/$10 = 2 units of output per dollar for labor and 100/$100 = 1 unit of output per dollar for capital. Publix is getting more output for its money by hiring labor, so it should hire more labor and less capital. Diminishing returns! As Publix hires more labor, MPL decreases and as it hires less capital, MPK increases. Publix will continue to substitute labor for capital until both are equal.

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**Cost-Minimization Rule**

Assume MPL is 20 and MPK is 100. If wage is $10 and rental is $25, then MPL per dollar is 20/$10 = 2 units of output per dollar for labor and 100/$25 = 4 unit of output per dollar for capital. Publix is getting more output for its money by hiring capital, so it should hire more capital and less labor. Diminishing returns! As Publix hires more capital, MPK decreases and as it hires less capital, MPL increases. Publix will continue to substitute capital for labor until both are equal.

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**Cost-Minimization Rule**

Similar to optimal consumption rule, which has consumers maximize their utility by choosing the combination of goods so that MU$ is equal for all goods.

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Review Firms combine inputs, like labor, capital and land, to produce output and minimize costs. Construction. Carpenters use tools to build houses, but there are different combinations of labor and capital that will get the same house built. One man with a nail gun could be more productive than several men with hammers and nails, and the firm must decide if that more expensive, but more productive, nail gun is a better choice than several men with inexpensive hammers.

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Review Substitutes. Two factors of production that can do essentially the same work. An ATM machine dispenses cash, accepts deposits and allows you to transfer money. The ability to perform these banking tasks makes the machine a substitute for a bank teller. A Caterpillar backhoe with one driver can dig holes and ditches. A team of men with shovels (or a team of students with spoons) can also dig holes and ditches. These are also substitutes in production. Two types of labor can also be substitutable. An American computer programmer and a Korean programmer could do the same work. A group of union autoworkers in Indiana could be considered substitutable with a group of non-union workers in Tennessee.

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Review Complements. Two factors of production that must be combined to produce output. The presence of one factor increases the marginal product of the other. The Caterpillar backhoe and the driver are complements. Not much digging gets done if they aren’t combined in production. A team of pilots and a 747 passenger jet are complements. An 18-wheeler and a truck driver are complements.

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Review Firms will choose the combination of inputs that can produce the output at the lowest cost—least-cost combination of inputs. Orange City needs to dig a 100-foot drainage ditch and the city hires Dana’s Ditch Diggers for the job. DDD has been experimenting with two combinations of labor and capital that can each get the ditch dug in the same amount of time. Combo 1: Rented backhoe and skilled driver. Combo 2: 10 unskilled workers each with a shovel.

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**Review Ditch diggers are preferred and DDD will choose Combo 2.**

What if DDD underestimated the productivity of the backhoe and driver and discovered Combo 1 could actually produce a drainage ditch 300 ft long in the same amount of time as 10 men and 100 ft? Cost of Labor Cost of Capital Total Cost of Producing 100 ft of ditches Combo 1 1 driver = $500 1 backhoe = $2,500 $3,000 Combo 2 10 workers = $1,000 10 shovels = $250 $1,250

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**Review In this case, Dana’s getting a backhoe.**

Firms need to consider the prices of labor and capital, as well as productivity before choosing the combination of inputs that produces output at lowest possible cost. Cost of Labor Cost of Capital Total Cost of Producing 300 ft of ditches Combo 1 1 driver = $500 1 backhoe = $2,500 $3,000 Combo 2 10 workers = $1,000 10 shovels = $250 $3,750

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Review Suppose DDD has hired labor to the point where MPL=50 and capital to the point where MPK=40. MPL/w = 50 units per dollar MPK/r = 20 units per dollar, so we know that MPL/w > MPK/r If the firm takes $2 away from hiring a unit of capital, it could hire 2 more units of labor. Total costs would remain the same. Lost production from one less unit of capital = approximately 20 units. Gained production from two more units of labor = a little less than 50 units. So DDD would see more production, at the same cost. DDD would continue to hire more labor (which causes MPL to decline), and less capital (which causes MPK to rise), until output can no longer rise.

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Review Suppose DDD has hired labor to the point where MPL=10 and capital to the point where MPK=60. MPL/w = 10 units per dollar MPK/r = 30 units per dollar, so we know that MPL/w < MPK/r If DDD takes $2 away from hiring two units of labor, it could hire one more units of labor. Total costs would remain the same. Lost production from two fewer units of labor = approximately 10 units. Gained production from one more units of capital = about 30 units. So DDD would see more production, at the same cost. Great deal! DDD would continue to hire more capital (which causes MPK to decline), and less labor (which causes MPL to rise), until output can no longer rise.

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Review Anytime the marginal product per dollar is not equal, the firm can reshuffle employment of labor and capital to increase output while keeping costs unchanged. Once the firm has found the least-cost combination of labor and capital, the firm has found the combination that produces that output at the lowest possible cost.

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