Answers for Exercises

The answers below are sketchy.  Their purpose is to guide your thinking about these specific cases. You might want to change the values of some parameters and see ways in which the answers differ from those below, and ways in which they remain the same. The following general cases are considered:

Linear Demand ParametersLinear Demand Curve.  The table at the right contains a set of parameters that results in a linear demand curve. Change any parameters except the one set equal to zero. Other values may be substituted, as long as β = 0 and μ = 1.

In this familiar case, R" = 2p' (the MR curve has twice the slope of the demand curve).  From equation (6),
dp/dt = p'/(R" - C"),
we can conclude that
dp/dt > 1
only if -p' > -C". That is, the price rises by more than the tax only if the marginal cost curve is negatively sloped and it is more steeply sloped that the demand curve.
1 
 a
 Enter the values above into the "EquilibriumWithNoTax" worksheet and determine the effect of a tax. This exercise will involve a series of steps. First, enter the values and run Solver.(For now, leave the "Firm's Cost" parameters as they were in the initial case.) If Solver is not enabled on your computer's Excel, use the "Tools/Add-Ins" menu option to install it. Run Solver again in the "OptimalQuantity" and "EquilibriumWithTax" worksheets.
  • Discuss the size of the deadweight loss due to the monopolist's not producing a quantity such that MC = p.
  • Also, discuss the size of the added deadweight loss due to a tax.


Answer
  • In the absence of any tax, the deadweight loss due to the monopolist's production of a quantity less than the efficient one (the one at which MC = p) is $2484.20.
  • The tax of $200 per unit produced adds $462.19 to the deadweight loss because the equilibrium quantity moves still farther from the efficient quantity. This change brings total DWL to $2946.39.  The tax revenue generated by this tax is $1352.95, so it costs the private sector ($462.19 + $1352.95) or $1815.14 to deliver $1352.95 to the government.

b
 Now consider the case of a horizontal MC curve, coupled with the linear demand curve. Set λ = 2000,  δ = 0, and κ = 0 in the "EquilibriumWithNoTax" sheet. Run Solver. Run Solver again in the "EquilibriumWithTax" sheet. 
  • By what fraction of the tax does price increase? 
  • Will this fraction always pertain when the demand curve is linear and the MC curve is horizontal? Explain. 


Answer
  • The change in price per $1 change in the tax, Δp/Δt, is ($3600 - $3500)/$200 = 0.5.
  • This ratio will be the same for any tax rate because the MR curve's slope is twice the slope of the demand curve.  So if marginal revenue changes by the amount of the tax (movement along the MR), then the equilibrium price will rise by exactly one-half the amount of the tax (movement along the demand curve).


Constant-elasticity demand curve.  When α = γ = 0 the demand curve exhibits the same price elasticity of demand at every quantity. For the illustration to make sense, the demand curve must be elastic. Since the marginal revenue is MR = p(1 -  1/e), where e is the price elasticity of demand, the MR curve must lie below the demand curve. More precisely MR = p*(1 - 1/e)  at each q. 

2
 a
 Return to the "EquilibriumWithNoTax" sheet. .  Leave the cost coefficients as they were set for 1(b). Now set α = γ = 0, β = 5000, and ρ = 0.2. This corresponds to setting the price elasticity of demand equal to 5.  Run Solver.
  • Confirm that the MR curve is 0.8 as high as the demand curve, i. e., that MR = 0.8P.
  • Why 0.8?
Answer
  • In this case the height of the demand curve is p = $3177.93, and the MR is R' = $2542.35. Thus, R' = 0.8p.
  • The question can be solved analytically by using the formula above: MR = p*(1 - 1/5) = 0.8p.

b
 Run Solver again in the "EquilibriumWithTax" sheet. 
  • By what fraction of the tax does price increase? 
  • Will this fraction always pertain when the demand curve exhibits constant elasticity and the MC curve is horizontal? Explain.


Answer
  • The rate of change of the tax related to the price is from $2500 to $2750, so Δp/Δt is $250/$200, or 1.25.    
  • Since the excise tax adds a fixed number of dollars per unit sold, the new MC is a horizontal line parallel to the initial one. In this case the firm uses a "mark-up" rule and it applies the same mark-up to the tax that it does to MC. To see the nature of the markup rule, set MR = MC: p(1 - 1/e) = MC, so p = MC/(1 - 1/e). In this case (1 - 1/e) is 4/5, so the mark-up is p = (5/4)MC. When the tax is added, p = (5/4)(MC + t), so the price increase is always 5/4 the tax increase.


An odd case? Upward-sloping MR curve. Economists offer both theoretical and empirical reasons to expect that virtually all demand curves are downward sloping.  Textbook representations of demand curves and marginal revenue curves typically show the latter below the former and downward sloping as well. A downward-sloping demand curve need not, however, imply a downward-sloping marginal revenue curve.

Smith, Formby, and Layson (1982) show that many of the demand curves that authors of classic textbooks have drawn to illustrate the demand/marginal revenue relationship, in fact, imply marginal revenue curves that have some upward-sloping region (though the authors mistakenly drew their corresponding marginal revenue curves as downward sloping).

Equation 1 can generate a marginal revenue curve that is initially upward-sloping and then downward sloping: ρ > 0 and μ > 1 will generate such a function. For the current illustration, ρ = 1.5 and μ = 0.5, which generates a marginal revenue curve that slopes upward throughout.

3
 a Upward sloping MRPlace the values at the right in the appropriate cells in the "EquilibriumWithNoTax" sheet, and execute Solver. (Solver can be sensitive to the initial value. Setting a high initial value can cause Solver to look at negative values of q. In such cases, it can fail to converge. Set the initial value in this case below 12. See the note on selecting an initial value in the "EquilibriumWithNoTax" sheet.)  Numbers for this case are the same as in problem four, do not match case scenario.
  • What does R" = 4.48 mean?
  • Why should the firm not produce more than q = 12.65 units even though the marginal revenue of subsequent units is higher than at q = 12.65 units


Answer
  • R"= 4.48 means that each additional unit sold adds $4.48 to Marginal Revenue.  This is consistent with the fact that the marginal revenue curve slopes upward.
  • At 12.65 units the firm will have maximized its profits in the absence of the tax.  The fact that marginal revenue increases as quantity increases is irrlevant. The relevant comparison to make is with marginal cost. As quantity increases, marginal cost rises above marginal revenue so producing additional units causes profits to decrease.
b Impose a tax of $200 on this commodity.
  • What is the predicted effect on price, using the value of dp/dt in the "EquilibriumWithNoTax" sheet?
  • What is the actual effect?

Answer
  • In the first worksheet, dp/dt has a value of  0.0567. Thus, the price is projected to rise by 0.0567*$200 = $11.34.
  • The actual effect, $3088 -$3075.51 = $12.49. 


Another odd case? Downward-sloping MC curve. Bishop (101) points out that a tax can raise price by more than the tax rate if the marginal cost curve is sufficiently downward sloping. Such might be the case with a "natural monopolist."

4
 a.








 Downward-sloping MCPlace the values at the right in the appropriate cells in the "EquilibriumWithNoTax" sheet, and execute Solver. This firm experiences dramatic economies of scale.
  • As a result, dp/dt = ___________.
  • Why is dp/dt greater than 1.0 in this case? That is, how does the existence of a downward-sloping MC lead to this result?

Answer
  • As a result of a downward marginal cost curve under "natural monopolist" conditions, dp/dt = 1.072. so the price will rise by more than the tax.  
  • In this case, when the monopolist reduces its output in response to the tax, it moves upward along its downward-sloping marginal cost curve. This increased MC adds to the upward pressure that the tax itself has on the product price..
b. In the "OptimumQuantity" sheet run Solver. What quanttity/price combination must occur for economic efficiency to be realized? What difficulty would be required if that combination were imposed?

Answer
  • The efficient quantity is about 8243 units, more than twice what the profit-maximzing monopolist would produce. To sell this much, the price must be $302.62, but the average cost is $1413.49. Accordingly, the firm would lose money with this outcome. This, of course, is the dilemma of regulating a natural monpoly. One solution would be to allow a two-part pricing scheme in which each buyer pays a per-period fee plus $302.62. per unit produced.
c. Return to the "EquilibriumWithNoTax" sheet. Change the value of α to shift the demand curve. Select values in the range of 3500 to 3700 or so. How can it be that increasing the demand causes the price to fall, while decreasing demand causes it to rise?
Answer
  • This relates to the answer in (b). Increasing demand causes the MR curve to shift rightward, which means that its intersection with the MC curve is to the right of, and below, the initial value. Because of the paramater values selected, the decrease in MC (downward movement along the curve) overwhelms upward pressure from the veritical upward shift in the MR curve.



A final case: Concave demand curve. This case is much like the linear case in terms of its predictions. In particular, dp/dt < 1. As with other cases, the deadweight loss of a tax on the monopolist's output is quite high.

5.         a.         Concave Demand CurvePlace the values at the right in the appropriate cells in the "EquilibriumWithNoTax" sheet, and execute Solver. The demand for this firm's product is concave.
What does p' = -27.69 mean?
What does p" = -10.00 mean?
What was p" in the case of the linear demand curve?
What was p" in the case of the connstant-elasticity demand curve?

Answer
  • If p' = -27.69, that means that, in the neighborhood of q*,  at for each unit beyond the profit maximizing q*, the price will drop by $27.69.  
  • p" = -10.00 not only confirms the downward concavity of the demand curve through the second derivative test, but also indicates that if q* increases, the rate at which prices will drop at each q increases by $10.00 at each additional unit sold.
  • For a linear demand curve, p" = zero, since the rate of decrease of the price is the same for all values of q.
  • For the constant elasticity case p" > 0; the demand curve becomes flatter as q increases.
b. Impose a tax of $200 on this commodity.
  • What is the predicted effect on price, using the value of dp/dt in the "EquilibriumWithNoTax" sheet?
  • What is the actual effect?
  • What is the size of the deadweight loss relative to the amount of revenue raised by the tax?
Answer
  • The predicted effect on price is that for each $1 of tax, the market price will increase by about $0.10 (dp/dt = 0.1047).  
  • The actual effect in the "Equilibriumwithtax" worksheet is that for each dollar of tax the market price increases by about $0.12 (∆p/∆t = ($385.82 - $361.65)/$200).  
  • The tax generates $336.79 in revenue and $178.31 in additional deadweight loss.

The following general cases are considered above: