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Unformatted text preview: fied do we have a consumer equilibrium corresponding to constrained utility maximization. Let's do an example, Suppose you have a square root utility function and the following data on prices and income: U(X,Y) = X0.5Y0.5 PX = 1 PY = 1 M = 10 32 We calculate the MRS as: MRS = Y X = 0.5Y 0.5X =Y X Now we can say, MRS = PX so, PY Y=1 X 1 or, X=Y and, X + Y = 10 thus, X + X = 10 2X = 10 X=5 Y=5 Thus, the optimal solution for this problem is to choose X*=Y*=5 for a maximum utility of (X0.5Y0.5) = (50.550.5) = 51 = 5. We also have an exactly satisfied budget constraint of: X + Y = 10 5 + 5 = 10 Consumer Demand Functions Let's take this a step further and generate a consumer's demand function. To do this we need to calculate the consumer equilibrium for every value of prices PX and PY. This can be done by solving for the quantities of goods demanded in terms of the prices alone. 33 So now we change our example above as follows: U(X,Y) = X0.5Y0.5 PX = ? PY = ? M = 10 We calculate the MRS as: MRS = Y X = 0.5Y 0.5X =Y X Now we can say, MRS = PX so, PY Y = PX X PY PXX + PYY = 10 Now, given the two relationships above, we can solve the optimal quantities of X and Y as before (this time, however, will yield a demand function not just a single consumer equilibrium). Y = PX X PY PXX = PYY PXX + PXX = 10 2PXX = 10 X = _5__ PX This gives us the demand for good X as a function of price PX. Similarly, we can get the demand function for good Y as a function of price PY. Using these functions, and varying the prices for each, one can trace out the demand curve for each good. and, leads to... so, 34 Rational Behaviour Producer Equilibrium Since the theory of producer equilibrium does not differ significantly from that of the theory of consumer equilibrium, we will examine producer theory in the context of a comparison with consumer theory. CONSUMER EQUILIBRIUM goods utility given prices given income indifference curve budget constraint X,Y U = U(X,Y) PX , PY MRS = -slope PX X+ PY Y = PRODUCER EQUILIBRIUM inputs K,L output Q = f(K,L) given prices PX r...
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