Problem Set 5 Answers. A grocery shop is owned by Mr. Moore and has the following statement of revenues and costs:

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1 Problem Set 5 Ansers 1. Ch 7, Problem 7. A grocery shop is oned by Mr. Moore and has the folloing statement of revenues and costs: Revenues $5, Supplies $5, Electricity $6, Employee salaries $75, Mr. Moore s salary $8, Mr. Moore alays has the option of closing don his shop and renting out the land for $1,. Also, Mr. Moore himself has job offers at a local supermarket at a salary of $95, and at a nearby restaurant at $65,. He can only ork one job, though. What are the shop s accounting costs? What are Mr. Moore s economic costs? Should Mr. Moore shut don his shop? The accounting costs are simply the sum: $5, + $6, + $75, + $8, = $186,. The economic costs also include the opportunity cost of the land rental ($1,) and the extra salary Mr. Moore ould earn if he selected his next best alternative ($95, - $8, = $15,). So, the economic costs are $186, + $1, + $15, = $31,. Should Mr. Moore shut don his shop? His economic costs exceed his revenues by $31, - $5, = $51,. Since his economic costs are greater than his revenues, he should shut don his shop. 1

2 . Ch 7, Problem 7.5 A firm uses to inputs, capital and labor, to produce output. Its production function exhibits a diminishing marginal rate of technical substitution. a. If the price of capital and labor services both increase by the same percentage amount (e.g., percent), hat ill happen to the cost-minimizing input quantities for a given output level? If the price of both inputs change by the same percentage amount, the slope of the isocost line ( / r ) ill not change. Since e are holding the level of output fixed, the point at hich the isocost line is tangent to the (fixed) isoquant does not change. Therefore, the cost-minimizing quantities of the inputs ill not change. b. If the price of capital increases by percent hile the price of labor increases by 1 percent, hat ill happen to the cost-minimizing input quantities for a given output level? If the price of capital increases by a larger percentage than the price of labor, the isocost lines become flatter (since / r decreases). This means that the point of tangency ill move to the southeast as shon in the diagram belo: K Original Cost-Minimizing (, K ) Ne Cost-Minimizing (, K ) Another ay to think about this is to realize that hen the price of capital increases by a larger percentage than the price of labor, labor has become cheaper relative to capital. The firm responds by substituting aay from capital in favor of labor. In sum, the cost-minimizing quantity of labor should increase, and the costminimizing quantity of capital should decrease.

3 3. Suppose the production of digital cameras is characterized by the production function Q = K, here Q represents the number of digital cameras produced. Suppose that the price of labor is $1 per unit and the price of capital is $1 per unit. a. Graph the isoquant for Q = 11,. K 11, 11 Q = 11, 11 11, b. On the graph you dre for part a, dra several isocost lines including one that is tangent to the isoquant you dre. What is the slope of the isocost lines? K Q = 11, The slope of the isocost lines is / r = 1/1 = 1. c. Find the cost-minimizing combination of labor and capital for a manufacturer that ants to produce 11, digital cameras. Mark this point on the graph you dre for part a. We recognize that the production function Q = K is Cobb-Douglas; therefore, e ill have an interior solution to this problem. We can begin by setting up the minimization problem: 3

4 Min TC = + rk K, st.. Q = K Using the quantity constraint e sub. in or K into the objective function. Q Min TC = + r TC rq = = solving rq = This is the labor demand function We can sub in for r, and Q to find *=11 Subbing the labor demand into quantity constraint e can obtain the capital demand function. K * = 11,/11= 11 Therefore, the cost-minimizing combination of labor and capital is, K = 11, 11. This point is marked in the graph belo: ( ) ( ) K 11 Q = 11, 11 d. See part c for labor demand function 4

5 = rq This is the labor demand function Subbing into constraint: K= Q = Q o rq = Q r e. Suppose, instead, that the technology for producing digital cameras is 1/ 1/ described by ( ) Q = + K. No hat is the cost-minimizing combination of labor and capital for a manufacturer that ants to produce 11, digital camera? This production function is unfamiliar, so e should first determine hether or not it meets the criteria for an interior solution: 1/ 1/ 1/ 1. Are the isoquants donard sloping? Since = ( + K ) > MP 1/ 1/ 1/ and MP K = ( + K ) K >, more is better of both inputs. Therefore, the isoquants are donard sloping.. Do the isoquants exhibit diminishing MRTS, K? Note that ( K ) 1/ MRTS, K = MP / MPK = /. We already kno that the isoquants are donard sloping, so moving don along an isoquant means increasing and reducing K. If e do that, then the denominator of MRTS, K increases and the numerator of MRTS, K decreases; so overall, MRTS, K decreases as e move don along an isoquant. Therefore, the isoquants do exhibit diminishing MRTS,. K 3. Are the isoquants smooth? Yes, since there is no eird break in the equation for the production function. 4. Do the isoquants touch the axes? Yes. The equation for the isoquant in 1/ 1/ question is ( ) 11, = + K, and the input combinations,, 11 are both points on that isoquant. ( 11 ) and ( ) In sum, e can go ahead ith trying to find an interior solution because criteria 1 through 3 have been met; hoever, since the isoquants touch the axes, e could get a corner solution. 5

6 Min TC = + r K, 1 1 st.. Q = ( + K ) Using the quantity constraint e sub. in or K into the objective function. Reriting the quantity constraint: 1 1 rq 1 1 K = ( Q ) 1 1 Min TC = + r ( Q ) TC 1 = + r ( Q )( ) = = ( + r) rq 11, * = = = 1, ( + r) 11 Subing * back into the quantity constraint: 1 1 K * = (11, 1 ) =1,. Therefore, the cost-minimizing combination of labor and capital is, K = 1, 1. ( ) ( ) 4. Ch 7, Problem 7.14 Suppose a production function is given by Q min{, K } =. Dra a graph of the demand curve for labor hen the firm ants to produce 1 units of output ( Q = 1). The production function Q min{, K } = indicates that the inputs are perfect complements. The cost-minimizing combination of labor and capital for a given level of output Q is (, K ) = ( Q/ a, Q/ b). In this case, the coefficients, a and b, are both 1; so, (, K ) = ( Q, Q). This indicates that the input demand curve for labor is = Q and that the input demand curve for capital is K = Q. Note that the demand for labor does not depend on the price of labor. In particular, if the firm ants to produce 1 units of output, its demand for labor is simply = 1. We sketch this demand curve belo: 6

7 1 To reiterate, the demand curve is vertical because the demand for labor does not vary ith the price of labor. 5. Ch 7 problem 7.19 A plant s production function is Q=K+K. The price of labor services is $4 and of capital services r is $5 per unit. a) In the short-run, the plant s capital is fixed at K =9. Find the amount of labor it must employ to produce Q=45 units of output. Since e kno K and Q e can use the quantity constraint to find the short-run amount of labor needed to produce Q= = (9) + 9, so *=. b) Ho much money is the firm sacrificing by not having the ability to choose its level of capital optimally? The optimal solution is the long-run cost minimizing amounts of and K. In the long-run capital is not fixed. Min TC = + rk K, st..q = K+ K Reriting the budget constraint and subbing it into the objective function e get: Q From the quantity constraint: k = ( + 1/ ) 7

8 Q Min TC = + r 1 ( + ) TC rq = = 1 ( + ) rq 1 5* K* = = (4.8 + ) * = = = 4.8 The firm s costs in the long-run are: TC = 4* *4.4 = $4.4 The firm s costs in the short-run are: TC = 4* + 5*9 = $53 Therefore, the firm loses about $1.6 because of the constraint on capital. TC Q = 4Q 1Q + Q. Over hat range of output does the production function exhibit economies of scale? Over hat range does it exhibit diseconomies of scale? At hat quantity is minimum efficient scale? 3 6. A firm s long-run total cost curve is ( ) Economies of scale occur hen MC < AC ; diseconomies of scale occur hen MC > AC ; and minimum efficient scale occurs here MC = AC (if MC is increasing at that point). Clearly, the first step is to calculate MC and AC : Economies of scale occur hen MC AC ( Q) = 4 Q + 3Q ( Q) = 4 1Q + Q MC < AC : 4 Q + 3Q Q Q 1Q < ( Q 1) < Hence, e have economies of scale hen Q < 5. < 4 1Q + Q Similarly, diseconomies of scale occur hen of scale hen Q > 5. MC > AC. Hence, e have diseconomies 8

9 And finally, since AC is decreasing (economies of scale) for Q < 5 and increasing (diseconomies of scale) for Q > 5, it must be the case that minimum AC occurs at Q = 5. Hence, the minimum efficient scale is Q = Ch 8 problem 8.1: For each of the total cost functions, rite the expression for the total fixed cost, average variable cost, and marginal cost: a. TC(Q)=1Q TFC =, AVC = 1, MC = 1. b. TC(Q)=16+1Q TFC = 16, AVC = 1, MC = 1. c. TC(Q)=1Q TFC =, AVC = 1Q, MC=Q d. TC(Q)=16-+1Q 1/ TFC = 16, AVC = 1Q, MC=5/Q 8. Ch 8 problem 8.11: A firm produces a product ith labor and capital as inputs. The production function is described by Q=K. et =1 and r=1. Find the equation for the firm s long-run total cost curve as a function of quantity Q. From Question 3 e kno the input demand functions: = rq K= Q r We sub these back into the total cost function: TC = rq + r Q r TC = rq + rq = rq hen r=1 and =1 TC = Q o 9

10 9. Ch 8 problem 8.13: A firm produces a product ith labor and capital. Its production function is described by Q=+K. et =1 and r=1 be the prices of labor and capital respectively. a. Find the equation for the firm s long-run total cost curve as a function of quantity Q hen the prices of labor and capital are =1 and r=1. With a linear production function, the firm operates at a corner point depending on hether < r or > r. If < r, the firm uses only labor and thus sets = Q. In this case, the total cost (including the fixed cost) is Q. If > r, the firm uses only capital and thus sets K = Q. in this case, the total cost is rq. When = r = 1, the firm is indifferent among combination of and K that make + K = 1. Thus, e have TC(Q) = Q. b. Find the solution to the firm s short-run cost-minimization problem hen capital is fixed at a quantity of 5 units (i.e. K=5), and =1 and r=1. Derive the equation for the firm s short-run total cost curve, average cost curve and marginal cost curve as a function of Q. When capital is fixed at 5 units, the firm s output ould be given by Q = 5 +. If the firm ants to produce Q < 5 units of output, it must produce 5 units and thro aay 5 Q of them. The total cost of producing feer than 5 units is constant and equal to $5, the cost of the fixed capital. For Q > 5 units, the firm increases its output by increasing its use of labor. In particular, to produce Q units of output, the firm uses Q 5 units of labor, for a cost of Q 5, and 5 units of capital, for a cost of 5. Thus, STC(Q) = Q = Q 1