Intermediate Microeconomics

Question 1: In the standard consumer theory model, we assume that individuals have a fixed income and then choose the allocation of goods that maximizes their utility. However, in many real-world scenarios, an individual’s income may itself be a function of their optimal choices. This is particularly true in rural areas where agricultural households are both producers and consumers of food. The model below requires you to re-examine a household’s optimal choice when its income is no longer fixed and is instead a function of its optimal choices. Consider an agricultural household that has preferences that can be represented by the following utility function: u = v L + p CF (1) where u refers to utility, L refers to the hours of leisure per day that this household enjoys, and CF is its total consumption of food per day. This household is endowed with T hours of time per day, which it can allocate between leisure and working on the family farm. Let the number of hours that this household spends working on the family farm be N. It follows that this household’s combined time spent on leisure and farm work cannot exceed T. This implies that we can write this household’s time constraint as L + N = T (2) 1 Unlike the standard consumer theory model, we will assume here that F is a food item that the household can buy from the market and/or produce on its own. Let QF represent this household’s production of food. This means that if QF > CF , this household is a net seller of food while if QF < CF , it is a net purchaser. Let this household’s net income (?) from producing food be: ? = P 2 F 4w (3) where PF is the market price of food and w is the hourly market wage. The latter represents the “cost” of using an hour of labour, including labour provided by family members. That is, even when a household uses a family member to work on the farm, it must compensate her by paying an hourly wage of w. 1 You can assume that this household takes both PF and w as given. To solve for this household’s optimal choice of L and CF , we need to first construct its budget constraint. This constraint will have two distinct components. First, there is this household’s cash expenditure and income. The former is simply PF × CF while the latter is its net income from farming, ?. In addition to being constrained by its cash income, ?, this household is also constrained by its time endowment, T. This means that, even if it wanted to, it cannot spend an unlimited amount of time on either leisure or farm work. It follows that this household’s budget constraint must also account for the “cost” of its leisure time and the “value” of its time endowment, T. This budget constraint can be written as: PF CF + wL = ? + wT (4) where the wage w is our measure of the “price” of an hour of time. (a) [2 marks] Use equation (4) to write down an expression for this household’s budget line. Fully illustrate this budget line with CF on the horizontal axis. (b) [1 mark] Add an indifference curve to your diagram in (a) and depict an initial equilibrium where this household optimally chooses positive values of CF and L. Label these optimal values as C * F and L * respectively. (c) [3 marks] Write down the expression for the marginal rate of substitution (MRS) of CF for L. What does this MRS tell us about the consumption and farm work trade off that this household is willing to engage in? 1 In reality this “payment” is often not a monetary one, but is instead an in-kind payment. The latter includes scenarios where a household member who works on the farm is compensated by not having to do other household chores. As long as the hourly value of such in-kind payments is equal to w, we can assume that the compensation for an hour’s work is w regardless of whether a monetary payment was made. 2 (d) [6 marks] Now suppose that w = 1. Let the price of food continue to be PF . Use your answers above to show that this household’s optimal consumption, C * F , is: C * F = (T + ?) × 1 PF + P 2 F (5) (e) [2 marks] Use your answer in (d) to derive an expression for the optimal leisure, L * . (f) [6 marks] Re-draw the diagram you created in parts (a) and (b) and fully illustrate the effect of an increase in PF on the budget constraint in this diagram. Does an increase in PF lead to an unambiguous change in CF ? Fully explain your answer. In explaining your answer, you should emphasize the difference between the effect you find here compared to the standard consumer theory model. You may find it useful to use (5) to help you with this explanation. (g) [5 marks] Now suppose that this household’s utility function is u = v L + p CF + p CO (6) where CO is the consumption of non-food items. Importantly, this household does not produce any of these non-food items. If it wants to consume them, it must purchase them in the market. Let the market price of the non-food items be PO. You can continue to assume that the market price of food is PF and the wage is equal to 1. Write down the expression for the optimal value of CO. How does an increase in PF affect this optimal value? Fully explain your answer. 3      

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