EC_308_Solutions_to_Problem Set 2_Fall_2023

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EC 308 Intermediate Microeconomics Problem Set 2 Solutions Problem 1: Bob’s utility function for soda and pizza is 𝑢(? 1 , ? 2 ) = √ ? 1 √ ? 2 where ? 1 is the number of cans of soda and ? 2 is the number of slices of pizza that Bob consumes. a) Graph Bob’s indifference curve, labeling at least three points. b) If given the choice between 2 slices of pizza and 4 cans of soda or 4 slices of pizza and 2 cans of soda would Bob prefer the bundle with more pizza, the bundle with more soda, or be indifferent between the two bundles? Bob’s utility from bundle (4 sodas, 2 pizza slices) is: 𝒖(?, ?) = √? √? = ?√? . Bob’s utility from bundle (2 sodas, 4 pizza slices) is 𝒖(?, ?) = √? √? = ?√? . Since Bob’s utility from both bundles is the same, Bob is indiff erent between them. c) Bob was consuming 2 slices of pizza and 4 cans of soda. Alice offered to give Bob 1 slice of pizza in exchange for 1 can of soda. Would Bob want to make this trade? Bob’s utility before the trade is 𝒖(?, ?) = √? √? = ?√? . If Bob makes the trade, he will consume 3 slices of pizza and 3 cans of soda. Bob’s utility after the trade would be 𝒖(?, ?) = √? √? = ? . Since Bob’s utility after the trade is greater than his util ity before the trade, Bob would want to make the trade.

d) Derive the formula for Bob’s marginal rate of substitution bet ween pizza and soda. Bob’s marginal rate of substitution (MRS) is given by: 𝑴?? = − 𝑴𝑼 ? 𝑴𝑼 ? 𝑴𝑼 ? is Bob’s marginal utility from soda. It is the partial derivative of Bob’s utility functi on with respect to changes in the consumption of soda (holding Bob’s consumption of pizza constant). That is, using the ‘power rule’ for derivatives (i.e, the derivative of ? 𝒌 is 𝒌? 𝒌−? ), Bob’s marginal utility from Soda is:: 𝑴𝑼 ? = ?. ?? ? ?.? ? ? ?.? = ?. ? √ ? ? √ ? ? Similarly, 𝑴𝑼 ? = ?.? √ ? ? √ ? ? . Hence, Bob’s MRS is: 𝑴?? = − 𝑴𝑼 ? 𝑴𝑼 ? = − ?. ? √ ? ? √ ? ? ?. ? √ ? ? √ ? ? = − ( ?. ? √ ? ? √ ? ? ) ( √ ? ? ?. ? √ ? ? ) = −? ? /? ? Problem 2: Alice’s preferences over apples and oranges are represented by utility function: 𝑢(?, ?) = 10? 0.5 ? 0.5 where ? is the number of apples and ? is the number of oranges she consumes. a) What is the relationship between a consumer’s marginal rate of substitution and the consumer’s indifference curve? The marginal rate of substitution is the slope of the consumer’s indifference curve. b) Write a general formula for Alice’s marginal rate of substitution (MRS) between apples and oranges. Alice’s marginal rate of substitution (MRS) is given by: 𝑴?? = − 𝑴𝑼 ? 𝑴𝑼 ? 𝑴𝑼 ? = ?√? √? , 𝑴𝑼 ? = ?√? √? . Hence, Alice ’s MRS is: 𝑴?? = − 𝑴𝑼 ? 𝑴𝑼 ? = − ( ?√? √? ) ( √? ?√? ) = − ? ? . c) What is Alice’s marginal rate of substitution (MRS) when she is consuming bundle (25,5) ?

Her MRS is − ? ?? = − ? ? . d) What is Alice’s marginal rate of substitution when she is consuming bundle (25,25) ? Her MRS is − ?? ?? = −? . Problem 3: Suppose the price of good 1 is 𝑝 1 = $5 , the price of good 2 is 𝑝 2 = $2 , and four consumers, Alice, Bob, Charlotte, and David, each have 𝑚 = $100 available to spend on these two goods. a) Alice has utility function 𝑢(? 1 , ? 2 ) = √ ? 1 √ ? 2 . Find Alice’s demand functions for her optimal quantities to consume of goods 1 and 2 (i.e., find formulas for her optimal values of ? 1 and ? 2 ). Given the prices and the money she has available, what is Alice’s optim al consumption of goods 1 and 2? We solve the utility maximization problem for utility function U(x 1, x 2 ) = √ ? ? √ ? ? = (x 1 1/2 )(x 2 1/2 ). Step 1: Write down the budget constraint: p 1 x 1 + p 2 x 2 = m. Step 2: Write down the optimality condition: MRS = - MU 1 /MU 2 = - p 1 /p 2 . Step 3: Find MU 1 and MU 2 which are the partial derivatives of U(x 1, x 2 ) with respect to x 1 and x 2 : MU 1 = 0.5x 1 -1/2 x 2 1/2 MU 2 = 0.5x 1 1/2 x 2 -1/2 Recall from the first day of class that the rules of exponents imply: ? ? ? ? = ? ?−? . So MU 1 /MU 2 = 𝑴𝑼 ? 𝑴𝑼 ? = ?.?? ? −?.? ? ? ?.? ?.?? ? ?.? ? ? −?.? = ( ?.? ?.? ) (? ? −?.?−?.? )(? ? ?.?−(−?.?) ) = ? ? −? ? ? ?.?+?.? = ? ? ? ? . So the optimality condition is: x 2 /x 1 = p 1 /p 2 . So x 2 = x 1 p 1 /p 2 = x 1 p 1 /p 2 Step 4: Substitute the above formula for x 2 into the budget constraint: p 1 x 1 + p 2 x 2 = p 1 x 1 + p 2 (x 1 p 1 /p 2 ) = p 1 x 1 + p 1 x 1 = 2p 1 x 1 = m. Step 5: Solve for x 1 : x 1 = m/2p 1. Step 6: Substitute the solution for x 1 into the formula for x 2 and solve for x 2 : ? ? = ? ? 𝒑 ? 𝒑 ? = ( 𝒎 ?𝒑 ? ) ( 𝒑 ? 𝒑 ? ) = 𝒎 ?𝒑 ? Step 7: Substitute the given values into the formulas for x 1 and x 2 : p 1 = $5, p 2 = $2, m = $100. So x 1 = m/2p 1 = 100/10 = 10 and x 2 = m/2p 2 = 100/4 = 25. The optimal bundle is (x 1 ,x 2 ) = (10, 25).

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b) Bob has utility function 𝑢(? 1 , ? 2 ) = min{? 1 , 2? 2 } . Find Bob’s demand functions for his optimal quantities to consume of goods 1 and 2 (i.e., find formulas for his optimal values of ? 1 and ? 2 ). Given the prices and the money he has available, what is Bob ’s optim al consumption of goods 1 and 2? We solve the utility maximization problem for utility function U(x 1, x 2 ) = min{x 1 ,2x 2 }. Step 1: Write down the budget constraint: p 1 x 1 + p 2 x 2 = m. Step 2: Write down the optimality condition: x 1 = 2x 2 . Step 3: Substitute the above formula for x 1 into the budget constraint: p 1 x 1 + p 2 x 2 = p 1 2x 2 + p 2 x 2 = x 2 (2p 1 + p 2 ) = m. Step 4: Solve for x 2 : x 2 = m/(2p 1 + p 2 ). Step 5: Solve for x 1 : x 1 = 2x 2 = 2m/(2p 1 + p 2 ) Step 6: Substitute the given values into the formulas for x 1 and x 2 : p 1 = $5, p 2 = $2, m = $100. So x 1 = 2m/(2p 1 + p 2 ) = 200/(10+2) = 16.67 and x 2 = m/(2p 1 + p 2 ) = 100/12 = 8.33. The optimal bundle is (x 1 ,x 2 ) = (16.67, 8.33). c) Charlotte has utility function 𝑢(? 1 , ? 2 ) = ? 1 + ? 2 . Find Charlotte’s demand functions for her optimal quantities to consume of goods 1 and 2 (i.e., find formulas for her optimal values of ? 1 and ? 2 ). Given the prices and the money she has available, what is Charlotte ’s optim al consumption of goods 1 and 2? The utility function U(x 1 ,x 2 ) = x 1 + x 2 represents perfect substitutes preferences. Since this consumer cares only about the total quantity of goods 1 and 2 and views them as interchangeable, the consumer will buy only the less expensive good. Since p 1 = $5 > p 2 = $2, the consumer will only buy good 2. How much of good 2 will she buy? We use the budget constraint. Since x 1 = 0, the budget constraint is: p 1 x 1 + p 2 x 2 = p 2 x 2 = m. The optimal quantity of good 2 to purchase is: x 2 = m/p 2 . For this problem, m = $100. So x 2 = 100/2 = 50. The optimal bundle is (x 1 ,x 2 ) = (0, 50). d) David has utility function 𝑢(? 1 , ? 2 ) = 10 ln(? 1 ) + 10ln(? 2 ) , where ln(? 1 ) is the natural logarithm of ? 1 . Find Dave’s demand functions for his optimal quantities to consume of goods 1

and 2 (i.e., find formulas for his optimal values of ? 1 and ? 2 ). Given the prices and the money he has available, what is Dave ’s optim al consumption of goods 1 and 2? We solve the utility maximization problem for utility function U(x 1, x 2 ) = ?? 𝐥𝐧(? ? ) + ?? 𝐥𝐧(? ? ) . Step 1: Write down the budget constraint: p 1 x 1 + p 2 x 2 = m. Step 2: Write down the optimality condition: MRS = - MU 1 /MU 2 = - p 1 /p 2 . Step 3: Find MU 1 and MU 2 which are the partial derivatives of U(x 1, x 2 ) with respect to x 1 and x 2 : Recall that the derivative of 𝐥𝐧(?) is ?/? . So MU 1 = 10/x 1 . MU 2 = 10/x 2 So MU 1 /MU 2 = 𝑴𝑼 ? 𝑴𝑼 ? = ( ?? ? ? ) ( ? ? ?? ) = ? ? ? ? . So the optimality condition is: x 2 /x 1 = p 1 /p 2 . So x 1 = x 2 *p 2 /p 1 . Step 4: Substitute the above formula for x 1 into the budget constraint: p 1 x 1 + p 2 x 2 = p 1 (x 2 *p 2 /p 1 ) + p 2 *x 2 = p 2 *x 2 + p 2 *x 2 = 2p 2 x 2 = m. Step 5: Solve for x 2 : x 2 = m/2p 2 . Step 6: Substitute the given values into the formulas for x 1 and x 2 : p 1 = $5, p 2 = $2, m = $100. So x 2 = 100/(2*2) = 25 and x 1 = x 2 *p 2 /p 1 = 25*(2/5) = 10. The optimal bundle is (x 1 ,x 2 ) = (10, 25).

Problem 4: Alice has the utility function 𝑼(𝑭, 𝑪) = 𝑭 ?/? 𝑪 ?/? . She has $35 of income and faces a price of food, P F = $6 and a price of clothing, P C = $10. Compute the utility maximizing choice of food and clothing consumption (denoted F and C respectively). Use calculus and show all steps. Hint: Things will work out more easily if you keep writing the coefficients from the utility function as fractions. Show and explain all work. It may be helpful to review the Handout on Utility Maximization posted in the “Lecture Slides” folder on Blackboard. Solution: (1) The budget constraint is: 𝒑 𝑭 𝑭 + 𝒑 𝑪 𝑪 = 𝒎 . (2) The optimality condition is: MRS = − 𝑴𝑼 𝑭 𝑴𝑼 𝑪 = − 𝒑 𝑭 𝒑 𝑪 Given U(F,C), we have: 𝑴𝑼 𝑭 = ? ? 𝑭 −?/? 𝑪 ?/? and 𝑴𝑼 𝑪 = ? ? 𝑭 ?/? 𝑪 −?/? . Then, MRS = − ( ? ? 𝑭 − ? ? 𝑪 ? ? ) ? ? 𝑭 ? ? 𝑪 − ? ? = − ? ? ? ? (𝑭 − ? ? − ? ? ) (𝑪 ? ? −(− ? ? ) ) = −?𝑭 −? 𝑪 = − ?𝑪 𝑭 . By the optimality condition: MRS = − ?𝑪 𝑭 = − 𝒑 𝑭 𝒑 𝑪 . So 𝑪 = ( 𝒑 𝑭 ?𝒑 𝑪 ) 𝑭 Substituting for C in the budget constraint gives us: 𝒑 𝑭 𝑭 + 𝒑 𝑪 𝑪 = 𝒎 = 𝒑 𝑭 𝑭 + 𝒑 𝑪 ( 𝒑 𝑭 ?𝒑 𝑪 ) 𝑭 = 𝒑 𝑭 𝑭 + 𝒑 𝑭 ? 𝑭 = 𝑭 (𝒑 𝑭 + 𝒑 𝑭 ? ) = 𝑭 ( ?𝒑 𝑭 ? ) = 𝒎 . Solving for the optimal amount of food, F, to purchase gives us: 𝑭 ∗ = ( ? ? ) ( 𝒎 𝒑 𝑭 ) Substituting for F in the optimality condition gives us: 𝑪 ∗ = ( 𝒑 𝑭 ?𝒑 𝑪 ) 𝑭 ∗ = 𝑪 = ( 𝒑 𝑭 ?𝒑 𝑪 ) ( ? ? ) ( 𝒎 𝒑 𝑭 ) = 𝒎 ?𝒑 𝑪 . Since we are given 𝒎 = ??, 𝒑 𝑭 = ?, 𝒑 𝑪 = ?? , the optimal consumption bundle is: (𝑭 ∗ , 𝑪 ∗ ) = ( ?𝒎 ?𝒑 𝑭 , 𝒎 ?𝒑 𝑪 ) = ( ?(??) ?(?) , ?? ?(??) ) = ( ??? ?? , ?? ?? ) = ( ?? ? , ? ? ).

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b. Illustrate your answer to part (a) using a graph with a budget line and indifference curve. Label your diagram fully and explain your diagram fully. This includes interpreting the slope of the budget line, indicating the intercepts of the budget line, indicating consumption of both goods and labeling the indifference curve with the level of utility achieved in part a. Let’s break this problem into specific parts: 1) Illustrate your answer to part (a) using a graph with a budget line and indifference curve. To graph the budget line (6F + 10C = 35), identify the two endpoints and draw the line that connects them. The endpoints are: F = 35/6 (which is approximately 6) if C = 0. C = 35/10 = 3.5 if F = 0. So the budget line connects the points (F,C) = (35/6, 0) and (0, 3.5). 2) Interpret the slope of the budget line The budget line can be written in slope-intercept form, which gives us: C = -(6/10)F + 35/10. The slope of the budget line is − 𝒑 𝑭 𝒑 𝑪 = −?. ?. The interpretation of the slope is that a one unit increase in food (i.e., if F increases by 1), results in a 0.6 decrease in units of clothing (i.e., C decreases by 0.6). 3) Indicating the intercepts of the budget line. (F,C) = (35/6, 0) (F,C) = (0, 3.5) F C (F*,C*) = (35/8, 7/8) U = ( 35 8 ) 3 4 ( 7 8 ) 1 4 = 2.93

These are shown in the figure and are labeled (F,C) = (0, 3.5) and (F,C) = (35/6, 0). 4) Indicating the consumption of both goods. This is labeling the point (F*,C*) = (35/8, 7/8) in the figure. It corresponds to the consumer’s optimal level of consumption. 5) Labeling the indifference curve with the level of utility achieved in part a). This is shown by labeling the indifference curve with U = ( 35 8 ) 3 4 ( 7 8 ) 1 4 = 2.93. This equation simply inserts the optimal consumption bundle (F*,C*) = (35/8, 7/8) into the utility function that we are given in the problem: 𝑼(𝑭, 𝑪) = 𝑭 ?/? 𝑪 ?/? . 6) Also keep in mind the big picture: The graph illustrates the point on the indifference curve that is tangent to (i.e., has the same slope as) the budget constraint. That is why we use the equation MRS = − 𝒑 𝑭 𝒑 𝑪 since MRS is the slope of the indifference curve and − 𝒑 𝑭 𝒑 𝑪 is the slope of the budget constraint. Hence, by using the optimality condition along with the budget constraint equation, we can directly find the optimal choice for the consumer by setting the slope of the indifference curve equal to the slope of the budget constraint and finding which point that equation corresponds to. In essence, the consumer is choosing the best bundle (the one on the highest indifference curve) that the consumer can afford (i.e., that does not exceed her budget). 7) As one other note, when you draw the indifference curve for an example with convex preferences (like this one), the indifference curve should not curve back up, it should not intersect the horizontal or vertical axis and it should intersect the budget line at only one point.

EC_308_Solutions_to_Problem Set 2_Fall_2023

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