Managerial Economics · Quiz 1 · May 2024

**1. Differentiate:** For \(q=4\sqrt L+4K\), \(MP_L=2/\sqrt L\) and \(MP_K=4\). **2. Compare:** With other inputs fixed, \(MP_L\) falls as labour rises, while \(MP_K\) remains constant. **Answer: C — MPL decreases; MPK is constant.** A: **Incorrect:** Labour’s marginal product varies with L. B: **Incorrect:** Labour’s marginal product decreases rather than increases. C: **Correct:** Labour has diminishing marginal product, while capital’s derivative is constant. D: **Incorrect:** Capital’s marginal product remains 4.

**1. Recognize a constant substitution rate:** Three units of labour always replace one unit of capital without affecting output. A suitable technology is \(q=aL+3aK\). **2. Identify the form:** Its isoquants are straight lines, representing perfect substitutes and a linear production function. **Answer: C — Linear production function.** A: **Incorrect:** Cobb–Douglas generally has a changing substitution rate. B: **Incorrect:** Leontief inputs are complements rather than continuously substitutable. C: **Correct:** A linear technology has a constant technical substitution rate. D: **Incorrect:** A linear function matches the stated technology.

**1. Identify fixed proportions:** Every machine requires exactly two workers. Extra workers without machines, or machines without enough workers, do not raise output. **2. Represent the bottleneck:** A suitable function is \(q=a\min\{L/2,K\}\), the Leontief form. **Answer: B — Leontief production function.** A: **Incorrect:** Cobb–Douglas permits input substitution. B: **Correct:** The minimum form requires the stated worker–machine ratio. C: **Incorrect:** Linear technology allows substitution rather than a fixed complement ratio. D: **Incorrect:** Leontief technology matches the condition.

**1. Hold the other inputs constant:** Labour’s marginal product is the change in output from one additional unit of labour: \(MP_L=\partial F/\partial L\). **Answer: C — the extra output produced by employing one more unit of labour while .** A: **Incorrect:** This is average product of labour. B: **Incorrect:** Changing other inputs does not isolate labour’s marginal effect. C: **Correct:** This is the marginal-product definition. D: **Incorrect:** This defines marginal product of capital instead.

**1. Differentiate average total cost:** \[\frac{dATC}{dq}=\frac{MC-ATC}{q}.\] **2. Interpret the crossing:** When MC crosses the usual U-shaped ATC curve from below, MC=ATC at its minimum. Below that point MC pulls ATC down; above it MC pulls ATC up. **Answer: D — average total costs are minimized.** **Convention note:** Equality alone establishes a stationary point. Calling it a minimum uses the standard U-shaped ATC and upward-crossing assumption. A: **Incorrect:** ATC falls when MC<ATC, not at the equality point. B: **Incorrect:** The standard U-shaped cost curve has a minimum at the crossing. C: **Incorrect:** ATC rises when MC>ATC. D: **Correct:** MC crosses the usual ATC curve at its minimum.

**1. Apply the tangency condition:** For \(q=10\sqrt{LK}\), \(MP_L/MP_K=K/L=w/r=20/80=1/4\), hence \(K=L/4\). **2. Meet the target:** \(LK=(100/10)^2=100\). Therefore \(L^2/4=100\), so \(L^*=20\), \(K^*=5\). **Answer: B — L=20, K=5.** A: **Incorrect:** This reverses labour and capital. B: **Correct:** These inputs satisfy both tangency and the output target. C: **Incorrect:** This combination produces more than 100 and has excessive cost. D: **Incorrect:** This does not have the cost-minimizing capital–labour ratio.

**1. Evaluate cost at zero output:** In \(TC=200+55q\), the constant term remains when q=0. Therefore fixed cost is 200. **Answer: 200.**

**1. Separate variable cost:** \(VC=55q\). **2. Divide by output:** At q=100, \(AVC=5500/100=55\). The 200 fixed cost is excluded. **Answer: 55.**

**1. Identify the best alternative:** Returning the cartons yields \(0.25(100)=25\) dollars. The original 100-dollar purchase is already sunk. **2. Compare the new offer:** Selling to the fraternity yields 55 and forgoes the 25-dollar return. The incremental gain is \(55-25=30\) dollars. **Answer: 25.**

**1. Identify the best alternative:** Returning the cartons yields \(0.25(100)=25\) dollars. The original 100-dollar purchase is already sunk. **2. Compare the new offer:** Selling to the fraternity yields 55 and forgoes the 25-dollar return. The incremental gain is \(55-25=30\) dollars. **Answer: A — Yes.** A: **Correct:** The offer exceeds the 25-dollar opportunity cost by 30 dollars. B: **Incorrect:** Rejecting would leave only the smaller 25-dollar return.

**1. Respect the axes:** Coconut is on X and fish on Y. The budget \(20C+10F=1000\) becomes \(F=100-2C\). **2. Read the slope:** \(-P_C/P_F=-20/10=-2\). **Answer: A — −2.** A: **Correct:** This is the negative ratio of horizontal to vertical prices. B: **Incorrect:** A positive sign is inconsistent with the budget trade-off. C: **Incorrect:** This reverses the relevant price ratio. D: **Incorrect:** Both its sign and ratio are incorrect.

**1. Set the other good to zero:** Coconut’s horizontal intercept is \(1000/20=50\); fish’s vertical intercept is \(1000/10=100\). **Answer: A — 50 on X-axis and 100 on Y-axis.** A: **Correct:** This assigns both correct intercepts to their stated axes. B: **Incorrect:** This swaps the axes. C: **Incorrect:** The fish intercept is 100, not 20. D: **Incorrect:** Neither intercept matches income divided by its price.

**1. Halve the horizontal good’s price:** Coconut now costs 10, while fish still costs 10. **2. Calculate the slope:** The new budget is \(F=100-C\), so its slope is −1. **Answer: C — −1.** A: **Incorrect:** The old slope is −2; it changes when coconut’s price falls. B: **Incorrect:** The slope must be negative. C: **Correct:** Equal prices give a one-for-one negative trade-off. D: **Incorrect:** This is positive and has the old magnitude.

**1. Use the new prices:** Both goods now cost 10, so \(C_{max}=F_{max}=1000/10=100\). **2. Assign axes:** The new intercepts are 100 on both X and Y. **Answer: C — 100 on X-axis and 100 on Y-axis.** A: **Incorrect:** Both intercepts are 100, not 50. B: **Incorrect:** Fish’s vertical intercept remains 100, not 50. C: **Correct:** The coconut intercept doubles and the fish intercept is unchanged. D: **Incorrect:** The coconut intercept is 100, not 10.

**1. Keep relative prices fixed:** With original prices, the slope remains \(-20/10=-2\). **2. Halve income:** Intercepts become \(500/20=25\) on X and \(500/10=50\) on Y. Both are half their original values. **Answer: B — Slope remains same and both intercepts are halved.** A: **Incorrect:** Lower income halves rather than doubles the intercepts. B: **Correct:** Income changes the intercepts but not the slope. C: **Incorrect:** Relative prices are unchanged and intercepts do not double. D: **Incorrect:** Income changes both intercepts.

**1. Use the substitute technology:** One chair requires \(L+K=4\). Each hour of machinery substitutes for one hour of labour. **2. Choose the cheaper input:** Labour costs 30 per hour and machinery 15. Thus \(C=30L+15(4-L)=60+15L\), minimized at \(L=0\), \(K=4\). **Answer: B — No.** A: **Incorrect:** Using positive labour wastes money because machinery substitutes at lower cost. B: **Correct:** The 3:1 labour-to-machine mix is more expensive than using only machinery.

**1. Use the substitute technology:** One chair requires \(L+K=4\). Each hour of machinery substitutes for one hour of labour. **2. Choose the cheaper input:** Labour costs 30 per hour and machinery 15. Thus \(C=30L+15(4-L)=60+15L\), minimized at \(L=0\), \(K=4\). **Answer: 0.**

**1. Use the substitute technology:** One chair requires \(L+K=4\). Each hour of machinery substitutes for one hour of labour. **2. Choose the cheaper input:** Labour costs 30 per hour and machinery 15. Thus \(C=30L+15(4-L)=60+15L\), minimized at \(L=0\), \(K=4\). **Answer: 4.**

**1. Write the technical substitution ratio:** \(MRTS_{L,K}=MP_L/MP_K=1/4\). **2. Solve for capital’s marginal product:** \[\frac{50}{MP_K}=\frac14\quad\Rightarrow\quad MP_K=200.\] **Answer: 200 kg per machine-hour.**