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Homework answers / question archive / 1)Prudent Saving” Sharon's preferences over two pe- riod consumption streams can be represented by the additively separable utility function: In (co) + B In (C) Sharon's current income in period 0 is m
1)Prudent Saving” Sharon's preferences over two pe- riod consumption streams can be represented by the additively separable utility function: In (co) + B In (C) Sharon's current income in period 0 is m. In addition to purchasing consumption (at a normalized price of 1), she can buy bonds at a constant unit price p. The purchase of a bond in period 0 (which entails herl forgoing p units of consumption in period 0) allows her to finance one unit of consumption in period 1. (a) [15 points] Given her income in period 1 is also m, derive Sharon's demand for bonds as a function of its price p. Page 2 of 3 - Advanced Microeconomic Analysis (ECON8025) Now suppose p = B and instead of her income in period 1 being certain, it is now contingent on whether the economy is booming or in recession in period 1. In particular, suppose her income will be m+e in a boom but only m-e in a recession. Further suppose her preferences over intertemporal contingent consumption streams can be represented by the additively separable utility function 1 In (Co) +B In (CB) + In (CR) where CB denotes her consumption in period 1 in the event the economy is booming and CR denotes her consumption in period 1 in the event the economy is in recession. (b) [15 points] Derive the first order necessary condition characterizing her optimal purchase of bonds. Without explicitly solving for Sharon's optimal quantity of bonds, determine whether it is be greater than or less than your answer from part (a) for p= 8?
2)“Short-Answer Questions”. (a) (15 points] Suppose the following function e (P1, P2, u) = (1 + P2) u is a consumer's expenditure function obtained from that consumer's expenditure minimization problem. Derive this consumer's indirect utility function and her Mar- shallian (uncompensated) demand. (b) (15 points) Cathy's aim is to raise as much money as possible in the T units of time available to her. She is able to allocate her time between two activities 21 and 22. Suppose both activities are equally productive' in generating money in the sense that if she spends x e [0,T] units of time in activity i = 1, 2, she will raise f (x) dollars. Assume f (.) is a strictly concave and strictly increasing function with f(0) = 0. State whether you agree or disagree with the following statement and explain your reasoning "The solution to Cathy's problem is obvious, she should spend T/2 units of time in each activity since that equalizes the amount she raises per unit of time in each activity: that is, if x1 = C2 = T/2, then we have f (21)/21 = f (x2)/22" (c) (15 points) If h (2) is homogenous of degree 1 then f (x) = g(h (2)) is homothetic if g() is an increasing function of one variable. Homogeneous functions are a subset of the set of homothetic functions. (Note that for the purpose of production functions we need to make sure that g(x) is a positive monotonic function for x > 0.) An identifying characteristic of a homothetic function is that the marginal rate of technical substitution (MRTS) between any two inputs is the same at x and tu Vt > 0. (i) [9 points] Verify that for any input vector x > 0 the MRTS between any pair of inputs of f(x) and of f(tx) are the same for any t > 0. (ii) [6 points] Show that the production function f (11,82) = ln (1 + x? + x) is homothetic.
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In microeconomics, a consumer's Marshallian demand function (named after Alfred Marshall) specifies what the consumer would buy in each price and income or wealth situation, assuming it perfectly solves the utility maximization problem. Marshallian demand is sometimes called Walrasian demand (named after Léon Walras) or uncompensated demand function instead, because the original Marshallian analysis refused wealth effects.
According to the utility maximization problem, there are L commodities with price vector pand choosable quantity vector x. The consumer has income I, and hence a budget set of affordable packages
where 
The consumer's Marshallian demand correspondence is defined to be
(b) I totally agree on the statement as Cathy is able to allocate the time between the two activities and is able to get the production through which she is earning a good amount of money by doing more production in lesser time.
(c) (i) In microeconomic theory, the marginal rate of technical substitution —or technical rate of substitution —is the amount by which the quantity of one input has to be reduced when one extra unit of another input is used, so that output remains constant.
(ii) Homothetic production function have the property that f(x) = f(y) implies f(x)= f(y) homogeneous production dunctions have the property that for some k Homogeniety of degree one is constant returns to scale. Homogeneous implies homothetic but not conversly.
