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ECON 211 FINAL EXAM FORMAT and REVIEW QUESTIONS

MATERIAL COVERED:   

• MODULES 1 through 16;  

• Assignments 1 through 7

SECTION A: Consumer Choice

PREFERENCES and UTILITY

1 What do we mean when we say preferences are Complete? Transitive?   

• What do we mean by Convex Preferences? Show an example of a Strictly Convex preference.  Show an example of a Weakly Convex preference.  Explain.   

• What do we mean by Monotonic Preferences? Show an example of a Strictly Monotonic preference.  Show an example of a Weakly Monotonic preference.  Explain.

• What is the relationship between Preferences and Utility Functions?

• What is the Marginal Rate of Substitution?  Explain what the MRS tells us about a consumer’s willingness to exchange goods.

2 Suppose we had the following Utility function:  U(X,Y) = min [3X,Y]

• Draw the indifference curve for U = 12

• Explain why the MRS is either 0 or - ∞  or undefined

• If we doubled utility to 24, what happens to the indifference curve?

• Confirm that these preferences are (weakly) Convex and (weakly) Monotonic

3 Suppose we had the following Utility function:  U(X,Y) = X^2/5  Y ^3/5  

• Draw the indifference curve for U = 20

• What is the MRS_xy? Interpret it for X the bundle (10, 10)  (hint MUx = ⅖ X^-3/5 Y^3/5   and  MUy = ⅗

X2/5 Y3=2/5  ).

• Explain why the MRS is always strictly within the range 0 to - ∞  · If we doubled utility to 40, what happens to the indifference curve?

• Confirm that these preferences are (strictly) Convex and (strictly) Monotonic

4 Suppose we had the following Utility function:  U(X,Y) = 3X + 2Y · Draw the indifference curve for U = 20

• Explain why the MRS is -3/2.  Interpret it.

• If we doubled utility to 40, what happens to the indifference curve?

• Confirm that these preferences are (weakly) Convex and (strictly) Monotonic

BUDGET SETS

1 Define a budget constraint, conceptually, mathematically, and graphically.

2 Let income M = $200, Px=$4 and Py = $5.

• Draw the budget line.

• What is the slope of the budget line?  What does this slope tell us about the market rate of exchange?

3 Let income M = $200, Px=$4 and Py = $5.

• Show the changes to the budget line if M doubles.

• Show the changes to the budget line if Px doubles  

• Show the changes to the budget line if M, Px, and Py doubles.  Explain.

4 Let income M = $200, Px=$4 and Py = $5.

• Discounts:  suppose the first 5 items purchased of each good are discounted by one dollar (eg Px=3 for X = 1 to 5; Py = 4 id Y = 1 to 5).  Draw the budget set.

• Suppose the government imposes a 10% tax on all goods.  What does the budget set look like now?  Explain.   

Consumer Optimization

1 Suppose we have a consumer with utility  U(X,Y) = X^2/5  Y ^3/5 .  She has income M = $200 and faces prices Px=$4 and Py = $5.

• Clearly state the Consumer’s Optimization (choice) Problem in words. · Sketch the problem (budget set, indifference curves, and optimal solution) · What two equations describe the solution to the problem.  Explain. · Solve for optimal consumption X* and Y*.

2 Suppose we have a consumer with utility  U(X,Y) = 3X + 2Y.  She has income M = $200 and faces prices Px=$4 and Py = $5.

• Clearly state the Consumer’s Optimization (choice) Problem in words. · Sketch the problem (budget set, indifference curves, and optimal solution) · Solve for optimal consumption X* and Y*.

3 Suppose we have a consumer with utility  U(X,Y) = min [3X,Y].  She has income M = $200 and faces prices Px=$4 and Py = $5.

• Clearly state the Consumer’s Optimization (choice) Problem in words. · Sketch the problem (budget set, indifference curves, and optimal solution) · What do we mean by a composite good and what is its price?

• Solve for optimal consumption X* and Y*

4 How does the solution to the Consumer’s Optimization (choice) Problem give us Demand Curves?  

Engel Curves?

5 Suppose we have a consumer with utility  U(X,Y) = X^2/5  Y ^3/5 .   

• What is X* and Y* if she has income M = $200 and faces prices Px=$4 and Py = $5.

• Suppose Px rises to $5.  What is the Hicks Substitution Effect? Hicks Income Effect?

• Sketch the substitution and income effects of the rise in Px.   

6 Suppose we have a consumer with utility  U(X,Y) = X^2/5  Y ^3/5 .   

• Find the Indirect Utility Function (U as a function of M and Prices).

• What is utility if she has income M = $200 and faces prices Px = $4 and Py = $5.

• Suppose Px rises to $5.  What is her utility now?  Compare to the original utility and explain why it changes.

• What is the Compensating Variation of the change in Px? · What is the Equivalent Variation of the change in Px?

7 Suppose we have a consumer with utility  U(X,Y) = min [3X,Y].  

• What is X* and Y* if she has income M = $200 and faces prices Px=$4 and Py = $5.

• Suppose Px rises to $5.  What is the Hicks Substitution Effect? Hicks Income Effect? · Sketch the substitution and income effects of the rise in Px.   

8 Suppose we have a consumer with utility  U(X,Y) = min [3X,Y].  

• Find the Indirect Utility Function (U as a function of M and Prices).

• What is utility if she has income M = $200 and faces prices Px=$4 and Py = $5.

• Suppose Px rises to $5.  What is her utility now?  Compare to the original utility and explain why it changes.

• What is the Compensating Variation of the change in Px?

• What is the Equivalent Variation of the change in Px?

SECTION B: FIRM’S CHOICES

1. Suppose we have a production function Q(L) = 10 L² – L³/10.

• What is maximum output and what L achieves it?

• Show that the Marginal Product of Labour (MP_L) rises then falls when Labour rises. Relate this to the shape of the production function ( eg shape of Q versus L)

• Show that the Average Product of Labour (MP_L) rises then falls when Labour rises. Relate this to the shape of the production function ( eg shape of Q versus L)

• Does the production function have increasing returns to Labour?  Decreasing? Explain.   

2. Let w = $10 and a fixed cost of $100.   

• Show/plot costs, average costs (AC), average variable costs (AVC), and average fixed costs (AFC).

• What is the relationship between Marginal Costs (MC) and AC, AVC, and AFC.  Show and discuss.

3. Let  w = $1000 and a fixed cost of $1000.  Let P = $5 · Identify the profit maximizing level of L (and hence Q).

• Identify the break-even level of Labour and Output.   

• Show that, if P falls below $4, the firm will shut-down.  Explain.

COBB-DOUGLAS

4. Suppose we have a production function Q(L,K) = L^1/2 K^1/2

• What do we mean by short-run and long-run production functions?

§ Let K = 16 be fixed in the short-run.  Show that more Labour raises Output but at a diminishing rate.  What does this imply for the MP_L?  Try to explain why.

• Does this production function have constant, increasing, or diminishing returns to scale (RTS)?  Show.

• Now suppose K is variable, show the isoquants for Q = 100 and Q = 200.  Explain how we can use the isoquants to establish the degree of RTS.

• Can you show that the Marginal Rate of Technical Substitution is MRTS_LK = L/K.   (hint  MP_L = ½ L^-1/2 K 1/2 and MPK = ½ L1/2 K -1/2 )

• Interpret the MRTS in words.

5. Now let w= $10 and r = $20

• What do we mean by short-run and long-run costs · Let K = 16 be fixed in the short-run.   

• What is the firm’s cost minimizing problem for producing Q = 100?  Eg what is L*?

• Find the cost of Q = 100.  Identify the variable costs and fixed costs.  

• Now find the short-run cost function.

• Show that AC, MC, and AVC are all increasing if output increases.  Express this in terms of Economies of Scale

6. Let w= $10 and r = $20 Let P = $100

• Let K = 16 be fixed in the short-run.  State the firm’s profit maximization problem.

• What is the firm’s optimal output, revenues, costs, and profits? (hint Q*= 400) · Show that optimal Q rises if P rises.  Eg set P = $200 and solve · Show that higher prices raise profits.  Explain.

• What happens to optimal output, revenues, costs, and profits if w rises to $20?

Long-run:   

7. Let w= $10 and r = $20 Let P = $100

• What is an isocost line?  What is its slope?

• Let K be variable.  What is the firm’s cost minimization problem?  

• What two conditions must be met to ensure that the firm is minimizing costs?  Show in a diagram.  Hint, use the isocost-isoquant figure.

• We know that the MRTS = L/K.  What does this imply for the optimal choice of inputs given w and r above?

• Find input demand for L and K as a function of Q, w, and r.  

• Show that, if we want to double Q, we also have to double K and L too.  What does this imply about average costs and economies of scale?

• Suppose, w rises to $15.  What happens to input demands?  Show in a diagram.  (hint show that K rises and L falls for any choice of Q).  Show also using the MRTS = r/w condition. · What happens to average costs in the long-run when w rises?  Explain.  

• Let K be variable in the long run.  What is the firm’s profit optimization problem?

• Let Q = 1.  What is optimal L and K ?  What is the cost of Q =1?

• You can show that the cost of Q = 100 is 100 times more than the cost of Q = 1.  Why is this the case? Explain.   

• What does this imply for the equilibrium market price in the long run?  Explain.

• Now suppose w rises to $15.  What happens to the costs of Q = 1 and Q = 100?  Explain the process.

PERFECT COMPLEMENTS:   

8. Suppose we have a production function Q(L,K) = min [L, 2K] · Let K = 100 be fixed in the short-run.   

§ Show that more Labour cannot raise Output beyond a fixed level.  Why?

• Now suppose K is variable, show the isoquant for Q =100.  Q = 200.  Explain how we can use the isoquants to establish the degree of RTS

• Does this production function have constant, increasing, or diminishing returns to scale (RTS)?  Show.

• Why is the concept of MRTS not useful in this production function?

9. Now let w= $10 and r = $20

• What are the input demands for K and L?   

• What is the firm’s cost minimizing problem for producing Q = 100?  Find the cost of Q = 100.   

• What is the relationship between output and costs?  That is, find the long-run cost function.

• Show that, if we want to double Q, we have to double K and L too.  What does this imply about average costs and economies of scale?

• Suppose, w rises to $15.  What happens to input demands?  Show in a diagram.   

• What happens to average costs in the long-run when w rises?  Explain.   

• What does this imply for the equilibrium market price in the long run?  Explain.

• Now suppose w rises to $15.  What happens to the costs of Q = 1 and Q = 100?  Explain the process.

PERFECT SUBSTITUTES

10. Suppose we have a production function Q(L,K) = L + 1.5 K with w = $10 and r = $20

• Let Q = 1.  What is optimal L and K ?  What is the cost of Q =1?

• Now let Q = 100.  What is optimal L and K ?  What is the cost of Q =100?

• You can show that the cost of Q = 100 is 100 times more than the cost of Q=1.  Why is this the case?

• What does this imply for the equilibrium market price in the long run?  Explain.

• Now suppose w rises to $15.  What happens to the costs of Q =1  and Q=100?  Explain the process.

SECTION C:  Market Equilibrium

Perfect Competition

1 Let market demand be P = 150 – ¼ Q^D and market supply P = ½  Q^S.

• Sketch the (inverse) demand and supply curves.  

• What do we mean by Equilibrium in this context?

• Find equilibrium price and quantity in the competitive market.

• Calculate the consumers’ Willingness To Pay (WTP), expenditures, and Consumer Surplus.

• Calculate the firms’ Costs, Revenues, and Producer Surplus.

• What is the Social Surplus at the equilibrium?  Recall SS = CS + PS or SS = WTP-COSTS.   · Show that any deviation from the market equilibrium reduces social surplus.   

2 Let market demand be P = 150 – ¼ Q^D and market supply P = ½  Q^S.

• Suppose the government imposes a $20 tax on consumers.  Show how demand shifts due to the tax.  

• What is the effect on equilibrium (P, Q) if the government imposes a $20 tax on consumers?   

• What is the DWL of the Tax?

• Show that the DWL rises exponentially with rising taxes.  Eg. with a small tax the DWL is small.  But if I double the tax, the DWL is more than doubling.  Why?

• Suppose instead that the government imposes a $20 tax on production.  Show how supply shifts due to the tax.  

• What is the effect on equilibrium (P, Q) if the government imposes a $20 tax on production?   · Show that the tax on consumption has identical effects as the tax on production.

3 Suppose we have n firms each with an individual supply curve of Q^S = 2P. Assume that firms have a quasi-fixed cost of $10,000 (that is COST = 0 if they shut down but costs = 10,000 + variable costs if they are open). There are 500 consumers with individual demand Q^D = 600 – 4P.   

• Sketch the individual inverse demand and supply curves.  

• Let n = 200 firms.  What is the equilibrium market price, output per firm, and consumption per consumer?  Calculate CS per consumer and Profits per firm.   

• Now suppose the number of firms rises to 300.  What is the new equilibrium market price, output per firm, and consumption per consumer?  Compare and discuss.

• How does total Consumer Surplus, Total Producer Surplus, and Total Social Surplus change?  Discuss.   

4 Suppose firms have individual supply curves of Q^S = 2P. Assume that firms have a quasi-fixed cost of $10,000 (that is COST = 0 if they shut down but costs =10,000 + variable costs if they are open).

There are 500 consumers with individual demand Q^D = 600 – 4P (as in above).   

• What do we mean by Long-run Equilibrium in this context?

• Find the number of firms that ensures that the incentive to enter or exit is gone.  Eg, what number of n insures that profits per firm = 0

• Show that SS is maximized at his equilibrium.

Monopoly and Oligopoly

1 Suppose a monopolist has the following cost function C(Q) =  100 + ⅓ Q² (with marginal cost MC(Q) =  ⅔ Q).  Suppose they face demand is P = 150 – ¼ Q.   

• Sketch the market demand, marginal revenues, and marginal costs.

• What is the monopolist’s optimal level of output, price, and profits?

• Calculate the firm’s markup.   

• What is the DWL associated with the monopoly output?

• Why would a subsidy improve Social Surplus?

• Explain how two-part pricing achieves the Social Optimum.

2 Suppose there are two firms each with the following cost function C(Q) =  30Q (with marginal cost MC(Q) =  30).  Suppose they face demand P = 150 – ¼ (Q₁ + Q₂).   

• Set Q₂ =  0.  What is optimal Q₁, prices, and profits?  (eg firm 1 is a monopolist).

• Now set Q₂ = 480.  What is optimal Q₁, prices, and profits? (show  Q₁= 0) · From the above, show that firm 1’s reaction function is Q₁ = 240 - ½ Q₂.   

• Solve for the Cournot equilibrium outputs (Q₁ and Q₂), price, and profits per firm.  (hint Q₁=Q₂=160) · Explain why Cournot completion leads to lower profits but higher Social Surplus.