Problem Set 4

Problem Set 4

    1. Consider the following total cost (TC) function for a firm: TC = 6 + 2·Q + Q²
    a. Which part of the TC equation represents the firm's total fixed costs (TFC)? Its total variable costs (TVC)? Explain both answers briefly. How large will the firm's total cost (TC1) be if Q1=2?
    b. Complete the following table of marginal cost (MC) information for various intervals of output around Q1=2. The first two rows are completed as an example:

DQ	Q2=Q1+DQ	TC2=TC(Q1+DQ)	DTC= TC2-TC1	Estimated MC
+2	2+(+2)= 4	TC(4) = 30	(30-14)= 16	(16/2)= 8
+1	2+(+1)= 3	TC(3) = 21	(21-14)= 7	(7/1)= 7
+1/2
+1/4
-1/4
-1/2
-1
-2

    c. Sketch this total cost curve and illustrate how these various slope calculations show up in your diagram.
    d. Use the limits approach to slope estimation (the "delta" method) to derive the formula for this firm's marginal cost as DQ --> 0.
    e. Calculate the derivative of the TC equation and compare your result to your answer for part d.
    f. Use your formula for MC to calculate MC for Q=1,2,3,4,5.
    g. How would your answers to part f change if TC = 26 + 2·Q + Q²? Provide a brief economic interpretation of your answer.

    2. This problem makes use of Econ.102 computer exercise number 1, "The geometry of univariate functions." Start the program, click "Examples," then "View." In the "Sample equations" panel, click the arrow in the top white box, and then click each of the various examples listed below. Compute the derivative for each example, evaluate the function and the derivative at the x-values listed for the example (you can click the diagram at each x-value to check your results), and then provide the following information for each case:
        (1) variable names; (2) equation formula; (3) derivative;
        (4) economic term for the derivative; and
        (5) value of Y and the derivative at the two x-values listed.
The first example is completed as an illustration:
    a. PPF: Constant opportunity costs
               (1) X = Consumption goods (C), 1000s; Y = Investment goods (I), 1000s
               (2) I = 10 - 2C
               (3) dI/dC = d(10 - 2C)/dC = -2
               (4) (Marginal) opportunity cost of C in terms of I
               (5) C   I   dI/dC
                     2   6     -2
                     3   4     -2

		X1	X2
a.	PPF: Constant opportunity costs	2	3
b.	PPF: Increasing opportunity costs	1	2
c.	Total revenue: price taker	3	6
d.	Total revenue: monopoly w/ linear demand	3	7
e.	SR production with variable returns	2	3
f.	SR production with diminishing returns	4	9
g.	SR total costs	3	4
h.	Consumption function	4000	6000
i.	Saving function	4000	6000
j.	Total utility of a consumer	1	4

3. Use the product rule to prove that MR = P for a perfectly competitive firm and MR < P for a monopolist.

4. Consider the firm's total expenditure on labor (TEL): TEL = PL·L. In general, the wage rate (PL) is a function of the quantity of labor hired (L): PL = PL(L). A perfectly competitive firm in the labor market faces a constant PL, but a firm with market power in the labor market (a monopsonist) faces an upward sloping supply curve of labor. Use the product rule to prove that the marginal expenditure on labor (MEL) = PL for a perfectly competitive firm in the labor market but MEL > PL for a labor market monopsonist.

    5. Consider a firm facing a linear (inverse) demand curve: P = B₀ - B₁·Q.
    a. Compute the formula for the firm's total revenue.
    b. Calculate the derivative to prove the following: For a monopolist facing a linear demand curve, marginal revenue (MR) has the same y-intercept as the demand curve and is twice as steep as the demand curve.
    c. Diagram the firm's demand and marginal revenue curves.

    6. Consider the short-run production function, Q=Q(L), and related unit values, average product and marginal product of labor (APL and MPL, where APL=Q/L and MPL=dQ/dL). Use the quotient rule to prove the following:
        (1) If MPL > APL then APL rises;
        (2) If MPL < APL then APL falls;
        (3) If MPL = APL then APL remains constant.

    7. A firm's total revenue (TR) is a function of the output (Q) that the firm produces: TR = TR(Q). The firm's output is in turn a function of the labor the firm hires: Q = Q(L). Use the chain rule to prove that the marginal revenue labor of labor (MRPL) = MR·MPL, where MR=marginal revenue and MPL=marginal product of labor.

    8. Consider the following general formula for a linear demand curve:
                Qd = a - bP.
    a. Draw this demand curve. Label both intercepts in terms of the constants, a and b.
    b. Again in terms of a and b, label the value for Qd and P at the midpoint of the demand curve.
    c. What is the formula for the point elasticity of demand for this demand curve?
    d. Use your formula to show the value of the point elasticity of demand at each of the following points on the demand curve: (1) Qd = 0; (2) P = 0; (3) the midpoint of the demand curve

    9. Consider the following demand equation: Qd = B₀·P^-B1), where B₀ and B₁ are both constants greater than zero.
    a. Take the derivative of Qd and rearrange terms to prove that the price elasticity of demand for this equation is a constant equal to the parameter B₁.
    b. Indicate whether marginal revenue would be positive, negative or zero for each of the following possible values for B₁: 0.5, 1.0, 1.5. Explain briefly (hint: recall the rules relating elasticity and total revenue/expenditure).