Managerial Economics · Quiz 1 · May 2026

**1. Differentiate separately:** For \(q=5L^{1/2}K^{1/2}\), \[MP_L=\tfrac52\sqrt{K/L},\qquad MP_K=\tfrac52\sqrt{L/K}.\] **2. Hold the other input fixed:** \(MP_L\) falls as L rises, and \(MP_K\) falls as K rises; both own-input second derivatives are negative. **Answer: D — Both marginal products are decreasing.** A: **Incorrect:** Neither marginal product is constant in its own input. B: **Incorrect:** Labour’s marginal product falls rather than rises, and capital’s is not constant. C: **Incorrect:** Capital’s marginal product also falls as capital rises. D: **Correct:** Each marginal product diminishes when its own input increases.

**1. Scale inputs:** \[F(tK,tL)=t^{p+q}F(K,L).\] **2. Classify:** When \(p+q=1\), output scales exactly with both inputs, giving constant returns to scale. **Answer: B — Shows constant returns to scale if p+q=1.** A: **Incorrect:** Decreasing returns occur when p+q<1. B: **Correct:** The homogeneity degree is one. C: **Incorrect:** p+q<1 implies decreasing, not increasing, returns. D: **Incorrect:** The capital derivative generally varies with the inputs.

**1. Isolate the capital effect:** Marginal product measures the extra output from an additional unit of capital with labour and other inputs fixed: \(MP_K=\partial F/\partial K\). **Answer: C — the extra output produced by employing one more unit of capital while .** A: **Incorrect:** This is average product of capital. B: **Incorrect:** Varying other inputs prevents isolating the effect of capital. C: **Correct:** This is the marginal-product definition. D: **Incorrect:** This concerns labour and is internally inconsistent.

**1. Divide output by labour:** Average labour productivity is \(AP_L=Q/L\), rather than the extra output from one additional input unit. **Answer: A — a firm’s total output divided by total labour input.** A: **Correct:** Total output per unit of labour is average product. B: **Incorrect:** This describes an input change rather than an average. C: **Incorrect:** This is marginal product of capital. D: **Incorrect:** This is not output divided by labour and uses contradictory wording.

**1. Add cash expenses:** Wages and supplies require actual payments. \[C_{explicit}=100000+80000=180000.\] **Answer: B — £180,000.** A: **Incorrect:** This is the foregone salary, an implicit cost. B: **Correct:** This sums wages and supplies. C: **Incorrect:** This includes implicit as well as explicit costs. D: **Incorrect:** This omits the supplies expense.

**1. Value the owner’s time:** By managing the business, the owner gives up a corporate salary of £75,000. This foregone alternative income is the implicit cost. **Answer: A — £75,000.** A: **Correct:** This is the best alternative use of the owner’s time. B: **Incorrect:** This is the cash operating cost. C: **Incorrect:** This combines implicit and explicit costs. D: **Incorrect:** This is an explicit wage expense.

**1. Include both resource costs:** Explicit expenses are \(100000+80000=180000\), and the owner gives up income of 75000. **2. Sum economic opportunity costs:** \[C_{economic}=180000+75000=255000.\] **Answer: C — £255,000.** A: **Incorrect:** This counts only the owner’s foregone income. B: **Incorrect:** This counts only explicit expenses. C: **Correct:** This includes both cash expenses and foregone earnings. D: **Incorrect:** This is not the sum of the stated resource costs.

**1. Hold K fixed and apply the chain rule:** For \(Q=(\sqrt L+\sqrt K)^2\), \[MP_L=2(\sqrt L+\sqrt K)\frac{1}{2\sqrt L}=(\sqrt L+\sqrt K)L^{-1/2}.\] **Answer: B — \((\sqrt L+\sqrt K)L^{-1/2}\).** A: **Incorrect:** The derivative includes L to the power −1/2, not L. B: **Correct:** This follows from the chain rule. C: **Incorrect:** This is marginal product of capital. D: **Incorrect:** Multiplying by K does not differentiate with respect to labour.

**1. Hold L fixed and apply the chain rule:** For \(Q=(\sqrt L+\sqrt K)^2\), \[MP_K=2(\sqrt L+\sqrt K)\frac{1}{2\sqrt K}=(\sqrt L+\sqrt K)K^{-1/2}.\] **Answer: C — \((\sqrt L+\sqrt K)K^{-1/2}\).** A: **Incorrect:** This is marginal product of labour. B: **Incorrect:** This is not the derivative with respect to capital. C: **Correct:** This follows from differentiating the square-root capital term. D: **Incorrect:** The derivative uses K to the power −1/2, not K.

**1. Set equal marginal product per dollar:** With wage 10 and rental rate 1, \(MP_L/MP_K=10\). For this CES function the ratio is \(\sqrt{K/L}\), so \(K=100L\). **2. Meet the output target:** \[(\sqrt L+\sqrt{100L})^2=121L=121000.\] Thus \(L^*=1000\), \(K^*=100000\). **3. Calculate cost:** \(C=10(1000)+1(100000)=110000\). The interior allocation is cheaper than either single-input corner. **Answer: 1000.**

**1. Set equal marginal product per dollar:** With wage 10 and rental rate 1, \(MP_L/MP_K=10\). For this CES function the ratio is \(\sqrt{K/L}\), so \(K=100L\). **2. Meet the output target:** \[(\sqrt L+\sqrt{100L})^2=121L=121000.\] Thus \(L^*=1000\), \(K^*=100000\). **3. Calculate cost:** \(C=10(1000)+1(100000)=110000\). The interior allocation is cheaper than either single-input corner. **Answer: 100000.**

**1. Set equal marginal product per dollar:** With wage 10 and rental rate 1, \(MP_L/MP_K=10\). For this CES function the ratio is \(\sqrt{K/L}\), so \(K=100L\). **2. Meet the output target:** \[(\sqrt L+\sqrt{100L})^2=121L=121000.\] Thus \(L^*=1000\), \(K^*=100000\). **3. Calculate cost:** \(C=10(1000)+1(100000)=110000\). The interior allocation is cheaper than either single-input corner. **Answer: 110000.**

**1. Use the stated axes:** Fish is on X and coconut on Y. The budget is \(40F+10C=1000\). **2. Solve for the vertical good:** \(C=100-4F\), so the slope is −4. **Answer: A — −4.** A: **Correct:** This is −Pf/Pc with the stated axes. B: **Incorrect:** A positive slope would not describe this budget trade-off. C: **Incorrect:** This inverts the price ratio. D: **Incorrect:** This has both the wrong sign and ratio.

**1. Put each good at zero:** At \(C=0\), \(F=1000/40=25\); at \(F=0\), \(C=1000/10=100\). **2. Match axes:** These are 25 on X and 100 on Y. **Answer: A — 25 on X-axis and 100 on Y-axis.** A: **Correct:** Both intercepts use the appropriate price and axis. B: **Incorrect:** This reverses the axes. C: **Incorrect:** These intercepts would require twice the income. D: **Incorrect:** This reverses the axes and doubles the attainable quantities.

**1. Halve the coconut price:** Now \(P_C=5\), while \(P_F=40\) and income remain unchanged. **2. Compute the slope:** \(C=200-8F\), so \(-P_F/P_C=-40/5=-8\). **Answer: C — −8.** A: **Incorrect:** Halving the vertical good’s price doubles the slope magnitude, not halves it. B: **Incorrect:** The slope must remain negative. C: **Correct:** This is the new negative price ratio. D: **Incorrect:** The magnitude is correct but the sign is not.

**1. Use the new prices:** With \(P_C=5\), coconut’s intercept is \(1000/5=200\); fish’s remains \(1000/40=25\). **2. Assign axes:** X=fish, Y=coconut, so the intercepts are (25,200). **Answer: C — 25 on X-axis and 200 on Y-axis.** A: **Incorrect:** Fish’s intercept remains 25, and coconut’s rises to 200. B: **Incorrect:** Neither listed intercept matches the new budget. C: **Correct:** The fish intercept is unchanged and the coconut intercept doubles. D: **Incorrect:** These swap the axes.

**1. Keep relative prices fixed:** The slope remains \(-40/10=-4\). **2. Reduce purchasing power:** Income 500 gives intercepts \(500/40=12.5\) and \(500/10=50\), half the original values. **Answer: B — Slope remains same and both intercepts are halved.** A: **Incorrect:** Income falls, so the intercepts do not double. B: **Correct:** Income changes intercepts proportionately but leaves the price ratio unchanged. C: **Incorrect:** The slope does not change and intercepts do not double. D: **Incorrect:** Both intercepts change when income changes.

**1. Identify substitutes:** One chair requires \(L+K=5\). Labour and machinery replace each other hour for hour. **2. Minimize cost:** \(C=20L+30K=100+10K\). Cost is smallest at \(K=0\), \(L=5\). **Answer: 5.**

**1. Identify substitutes:** One chair requires \(L+K=5\). Labour and machinery replace each other hour for hour. **2. Minimize cost:** \(C=20L+30K=100+10K\). Cost is smallest at \(K=0\), \(L=5\). **Answer: 0.**

**1. Separate fixed cost:** In \(TC=300+25Q+2Q^2\), the fixed component is 300. **2. Divide by output:** \(AFC=FC/Q=300/100=3\). **Answer: 3.**

**1. Differentiate total cost:** \(MC=dTC/dQ=25+4Q\). **2. Evaluate:** At \(Q=25\), \(MC=25+4(25)=125\). **Answer: 125.**

**1. Divide variable cost by output:** \(VC=25Q+2Q^2\), so \(AVC=25+2Q\). **2. Evaluate:** At \(Q=25\), \(AVC=25+50=75\). **Answer: 75.**