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Note_04M.5-6 - where the first term on the rhs represents...

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5 How do we find the equilibrium? Assume that the utility function is a Cobb-Douglas utility function where A , and are preference parameters that are given. Method 1. Trial and error (i) Choose a range of . In the example, this will be an interval . (ii) Pick a value of in the interval and compute . (iii) Compute the utility level at these consumption level. (iv) Repeat (ii) and (iii) for many different values of . (v) Compare the utility levels and pick the value of and that yield the highest utility level. Method 2. Use the equilibrium condition : A bundle on the budget line at which the slope of the budget line is equal to the slope of one of the indifference curves. The solutions are . To show this, note that the slope of the budget line is -(1+r). Slope of an indifference curve is defined by the total differential equation
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Unformatted text preview: where the first term on the rhs represents the change in utility as changes by , and similarly for the second term. As increases by , utility increases by . To stay on the same indifference curve, must decrease by , the effect of which is sufficient enough to offset the effect of on the utility, so that there is no change in utility. The partial derivatives are Similarly, Solving the total derivative equation, we derive 6 Equating the two slopes and using the expression for , we derive and the solution for is . For example, if , , A =1, and , then , , and the maximum utility level is . Method 3. Use computer optimization program We can use Excel’s SOLVER to find directly the optimum solution for and subject to the budget constraint. See Excel file, “Consumption” sheet...
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